Procedia Computer Science 57 ( 2015 ) 960 – 969

1877-0509 © 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)doi: 10.1016/j.procs.2015.07.493

ScienceDirectAvailable online at www.sciencedirect.com

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

Performance Study of LDA and KFA for Gabor Based Face Recognition System

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) has consistently been one of the most formidable problems in Pattern Recognition and many diverse techniques have been proposed over the past few years to enhance its effectiveness. Feature extractors play an instrumental role in establishing recognition accuracy of FR systems and hence in this paper, we ardently focus our attention on the popular Gabor Filter feature extraction technique and comprehensively compare its linear variant, Gabor-LDA with its nonlinear counterpart, Gabor-KLDA to ascertain which technique is better suited for FR tasks. The findings proffered by this work, along with our other studies, form a series intended to aid developers in making informed decisions while building proficient FR systems. © 2015 The Authors. Published by Elsevier B.V. Peer-review under responsibility of organizing committee of the 3rd International Conference on Recent Trends in Computing 2015 (ICRTC-2015). Keywords:Gabor; Gabor, Kernel Fisher, Face Recognition.

1. Introduction and Related Work

Face Recognition (FR) has witnessed monumental developments over the past few years and has proffered a number of cross-domain applications that range from commercial applications such as Access Control and Identity Authentication to law enforcement specific applications such as Surveillance and Criminal Apprehension [3]. In spite of its ubiquitous presence and prevalent success, it is only effective in certain constrained scenarios and the general task of face recognition still poses a number of challenges [2]. The performance of FR systems dwindles in the presence of extreme variations in terms of expression, illumination, occlusion and so on [1]. Thus, FR remains one of the most formidable problems in Pattern Recognition proffering several daunting challenges, and hence is an effervescent area for researchers to immerse themselves into. Recent developments [3] emphasize the emerging need for augmenting the proficiency of FR Systems using more robust techniques to effectively address these lingering shortcomings.

The performance of FR systems extensively relies on the selection of the Facial Feature Extractor, which is

of crucial importance while building any FR system. Several seminal techniques exist for carrying out the extraction of facial features and an in-depth survey of the prevalent extraction mechanisms can be found in [4]. Gabor Filters [1] has distinguished itself as being one of the most robust techniques to carry out seamless

© 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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extraction of facial features in an efficient manner. It is particularly effective due to its biological relevance, standout computational properties [5] and embodies of a gamut of notable characteristics such as the ability to capture salient visual properties such as orientation selectivity, spatial localization and spatial frequency [6][7]. In spite of Gabor’s potency and widespread adoption, a fundamental concern in its application is that, its face images are convolved with a bank of Gabor filters, yielding high dimensionality to the Gabor feature space [1][5]. Although, the application of several down-sampling techniques were attempted to selectively choose feature points [8], they were not feasible, as they still contained a substantial number of high dimensions and led to considerable amount of feature discriminative information being lost, in addition to causing accuracy reduction in the classification stage. Subsequently, PCA [12] and LDA [10][11] techniques were explored to reduce the dimensionality of the Gabor feature space and have been shown to be remarkably effective [1] [9].

LDA is a highly robust technique that is widely used for dimensionality reduction and de-noising. Recent

studies have shown that linear techniques such as PCA and LDA may not be adequately equipped to effectively handle FR in certain scenarios, as face is fundamentally non-linear [13]. Since the images contains higher order dependencies among the pixel intensity values i.e. the pixel relationships around edges and curves that have important information needed for effective recognition and these higher order statistics are capable of distinctly representing complex patterns [13]. The requisite for such higher order statistics has led to the advent of non-linear techniques such as Kernel PCA [14] and Kernel Fisher Analysis [18][19][23], which consists of non-linear, mapping and operate by extracting non-linear features. These have been demonstrated to be capable of achieving higher recognition rate and also yield reduced error rate. Furthermore, as kernel methods have the added advantage of not needing non-linear optimization, they were considered as a viable option for our comparison study [14][15]. Moreover, for our deliberations, we assume that that at least one large face is present in the given complex background (image) and do not perform face detection. This assumption has been shown to produce reasonable results in a few seminal works [16]. The images used for the experimentations do not need to be of high resolution, unless the images being compared have high matching likeness [17].

In this paper, we employ LDA (linear) and KFA (non-linear) to perform dimensionality reduction of the

Gabor filtered face images, in order to ascertain which technique is more competent in an effort to aid developers in making informed decisions and thus design proficient FR systems.

The rest of the paper is organized in the following manner: Section 2 illustrates the adopted methodology

along with the pertinent background information; Section 3 details the experimentation and results; Section 4 proffers a discussion of the proposed approach and outlines future work.

2. Methodology

The various sequence of steps followed in our approach is illustrated in Fig .1.

Fig.1.Framework for the Applied Methodology

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2.1 Feature Extraction using GABOR Filters We initially load the images from the ORL Database [21] and compute the Gabor magnitude features. We construct a Gabor filter bank of 40 filters (8 orientation x 5 scales) in order to extract the Gabor magnitude features (facial features from the images). The employment of 5 frequencies and 8 orientations (as depicted in Fig.2) has been shown to be an effective choice [1] [22]. The Gabor filters [6] have kernels similar to the 2D receptive field profiles of the mammalian cortical simple cells and are often employed in computer vision and image analysis owing to its optimal localization properties in both spatial analysis and frequency domain [1].

Fig.2 Gabor Filter with 5 frequencies and 8 Orientations

A Gabor filter can essentially be viewed as a complex exponential modulated by a Gaussian function [1][20] i.e. a two-dimensional Gabor wavelet is a Gaussian kernel function modulated by a sinusoidal plane wave [1] and can be represented as follows:

Wherex = x cosΘ + y sinΘand y = -xsinΘ + ycosΘ where (x, y) is the position of the pixel in the spatial domain, the orientation of Gabor filter is denoted by , the radial center frequency of the sinusoidal plane wave is denoted by ω and σ represents the standard deviation of the round Gaussian function(sharpness) along the co-ordinate axes [1]. An ideal value for the standard deviation is suggested to be: σ≈π/ω [1][20] and is capable of distinctly defining the relationship between σ and ω[1].

Choosing 5 frequencies (m = 1,2,..5) and 8 orientations (n = 1,2,…8) for the Gabor filter bank results in the

following equations:

Θn =

The Gabor feature representation of an image I (x,y) is essentially the convolution of the image with the Gabor filter bank ψ (x,y, n) [1]and can be represented as follows: Omn(x,y)=I(x,y) ⊗ ψ(x,y, Θn) (4)

2.2 Dimensionality Reduction

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 as the verification set. The subsequent step consists of the computation of training, evaluation and verification feature vectors using the LDA and KFA methods for dimensionality reduction. For both LDA and KFA, we

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compute the subspace using the training data from the ORL database. The training data comprised of 120 samples (images) with 10240 variables (pixels). After the subspace construction (LDA/KFA) was completed, we performed evaluation and test image projection. 2.2.1 Linear Discriminant Analysis (LDA)

LDA [10][11] is a widely adopted supervised learning method used to reduce high dimensionality by combining linear features, that operates by projecting high dimensional data onto the lower dimensional space. It utilizes the category information associated with each sample. LDA seeks a projection that best separates the data in a least-squares sense [1].

LDA aims to minimize the within-class scatter (Sw) and maximize the between-class scatter (Sb) which are

defined as follows [1]:

Sw = (5)

Sb = (6)

Where is the ith sample of class j and j denotes the mean of class j. μ represents the mean image of all the classes and c is the number of classes with Nj representing the number of samples in class j [1]. Let us consider that a set of N sample images {x1,x2,x3……xN}is represented by a t-dimensional feature vector (In our case, Gabor). Then LDA 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 [1][10][11]. The resultant feature vector yi€ Rf is: yi= xi(i=1,2,……N) (7) [1] signifies that the best way to select Wlda is to maximize the ratio: det|Sb|/det|Sw|. If Sw is a nonsingular matrix then this aforementioned ratio is maximized, when the transformation matrix Wlda consists of g generalized eigenvectors corresponding to the g largest eigenvectors of . Furthermore, it also states that there are at the most c-1 non-zero generalised eigenvalues and hence an upper bound on g is c-1. 2.2.2 Kernel Fisher Analysis (KFA)

Kernel Fisher Analysis [18][19][23] aims to yield a non-linear discriminant analysis in the higher space. It operates by projecting the input data into an implicit feature space using the nonlinear mapping, : x € RN→ f € F and then seeks to find a nonlinear transformation matrix that can maximize the between-class scatter (SB) and minimize the within-class scatter (Sw) in F [23]. The dot product in F can be defined as follows [23]:

k(x,y) = Φ(x).Φ(y) (8) SB and Sw can be defined in the feature space F as follows:

Sw= (9) SB= (10) Where ui represents the sample mean of class i, u denotes the mean of all the samples in F and p(wi) is the prior probability [23]. In order to perform LDA in F, we need to maximize the following: J(w) = wSBwT/wSwwT (11)

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Furthermore, since any solution thus obtained (W) must lie in the span of all the samples in F, the following must exist [23]: W= (12)

Subsequently, by maximizing Eqn.12, we obtain: J(w) = wKBwT/wKwwTY (13)

Further details on KB and Kw can be found in [18, 19]. This problem, similar to LDA can be solved by finding the leading eigenvectors of (Kw)-1KB according to techniques detailed in [18][19] to yield the generalized Kernel Fisher Discriminant. In [23] using the technique of pseudo inverse of the within-class covariance matrix and the projection of a point x onto w in F is given by:

w.Φ(x) = (14)

2.3 Similarity Measures

The feature matching has been conducted using distance classifiers such as Euclidean [26], Cosine [27], City-Block [26] and Mahalanobis Cosine (MAHCOS) [24] [25], in order to exhibit the validity of our results. 2.3.1 Euclidean Distance

Euclidean Distance is one the most commonly used distance measures and has become synonymous with distance [26]. It calculates the root of the square difference between the coordinates of a pair of objects and is employed to calculate the distance between the probe image and the image from the database to be identified. It can be announced that the images are similar, if the distance is small i.e. the distance between the image projection and the known projections is calculated and subsequently, the face image can then be classified as one of these faces with the minimum Euclidean distance [26]. Euclidean distance, d2 can be calculated in the following manner:

(15) Where x, y are in the data set X and xi, yiare the ith coordinates of x and y. This measures the dissimilarity on X. 2.3.2 Cosine Distance

This method is similar to the Euclidean distance, but has been proven to be more effective in certain scenarios where former gives unsatisfactory results [27]. It focuses on similarity rather than dissimilarity [27] [28] as Euclidean does. It can be calculated in the following way:

Similarity = cos (θ) = = (16)

2.3.3 City-Block Distance

City-Block Distance Classifier signifies the distance between the points in a city road grid. It examines the absolute difference between the coordinates of a pair of objects and is calculated as follows [26]:

(17)

2.3.2 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

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MAHCOS has been demonstrated in [24] and has also shown promising results in [25]. Let us consider that Γ1,Γ2.….ΓNare 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 [25] of ATA as ATAvi = μivi..

We then multiply both sides by A to obtain AATAvi = 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 itheigen face, we have the projection coefficients of the face as u=[w1, 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 Eigenface. It is necessary to know the transformation between the image space and the Mahalanobis space before computing MAHCOS and 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 [25]: mi= and ni= (18)

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: DMahCosine(u,v) =cos (Θmn) = (19)

3. Experimentations and Results

The approaches have been compared exhaustively on each database using three distinct sets: Training set, Evaluation set and the Verification set. The first three images of the subject were used for the training set. Subsequently, the next three images were utilized for the Evaluation set, which were used to ascertain if the expected outcome been achieved for a given input. Finally, the remaining images were 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 [21] database to compare the performance of the Gabor-LDA and Gabor-KFA techniques. 3.1GABOR-LDA Results 3.1.1 Using Euclidean Distance In the identification experiments, the rank one recognition rate of the GABOR-LDA experiments equalled 93.33%. The verification experiments on the evaluation data yielded the following: the EER (equal error rate) on the evaluation set was 1.73%, the minimal HTER (half total error rate) was 1.60% and the verification rates at 1%, 0.1% and 0.01% FAR was found to be 96.67%, 93.33% 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 99.38% and 93.75% respectively.

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3.1.2 Using Cosine Distance In the identification experiments, the rank one recognition rate of the GABOR-LDA experiments equalled 94.17%. The verification experiments on the evaluation data yielded the following: the EER (equal error rate) on the evaluation set was 1.67%, the minimal HTER (half total error rate) was 1.44% and the verification rates at 1%, 0.1% and 0.01% FAR was found to be 97.50%, 93.33% 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 99.38% and 93.13% respectively. 3.1.3 Using City-Block Distance In the identification experiments, the rank one recognition rate of the GABOR-LDA experiments equalled 93.33%. 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 1.98% and the verification rates at 1%, 0.1% and 0.01% FAR was found to be 95.83%, 92.50% 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 96.88% and 90.00% respectively. 3.1.4 MAHCOS Distance In the identification experiments, the rank one recognition rate of the GABOR-LDA experiments equalled 94.17%. The verification experiments on the evaluation data yielded the following: the EER (equal error rate) on the evaluation set was 1.67%, the minimal HTER (half total error rate) was 1.54% and the verification rates at 1%, 0.1% and 0.01% FAR was found to be 97.50%, 93.33% 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 99.38% and 91.25% respectively. The results are corroborated by the corresponding sample ROC (Fig.3), CMC (Fig.4), EPC (Fig.5) and DET (Fig.6) curves on the ORL dataset.

Fig.3 ROC Curve for Gabor-LDA on ORL Fig.4 CMC Curve for Gabor-LDA on ORL

Fig.5 EPC Curve for Gabor-LDA on ORL Fig.6 DET Curve for Gabor-LDA on ORL.

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3.2 GABOR-KFA Results 3.2.1 Using Euclidean Distance In the identification experiments, the rank one recognition rate of the GABOR-KFA experiments equalled 92.50%. The verification experiments on the evaluation data yielded the following: the EER (equal error rate) on the evaluation set was 3.53%, the minimal HTER (half total error rate) was 2.91% and the verification rates at 1%, 0.1% and 0.01% FAR was found to be 95.00%, 86.67% 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 91.88% and 76.88% respectively.

3.2.2 Using Cosine Distance In the identification experiments, the rank one recognition rate of the GABOR-KFA experiments equalled 95.00%. 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 1.75% and the verification rates at 1%, 0.1% and 0.01% FAR was found to be 96.67%, 90.83% 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 96.25% and 87.50% respectively. 3.2.3 Using City-Block Distance In the identification experiments, the rank one recognition rate of the GABOR-KFA experiments equalled 92.50%. The verification experiments on the evaluation data yielded the following: the EER (equal error rate) on the evaluation set was 3.32%, the minimal HTER (half total error rate) was 2.60% and the verification rates at 1%, 0.1% and 0.01% FAR was found to be 93.33%, 86.67% 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 90.63% and 77.50% respectively. 3.2.4 Using MAHCOS Distance In the identification experiments, the rank one recognition rate of the GABOR-KFA experiments equalled 94.17%. The verification experiments on the evaluation data yielded the following: the EER (equal error rate) on the evaluation set was 2.51%, the minimal HTER (half total error rate) was 1.74% and the verification rates at 1%, 0.1% and 0.01% FAR was found to be 97.50%, 93.33% 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 99.38% and 93.13% respectively. The results are corroborated by the corresponding sample ROC (Fig.7), CMC (Fig.8), EPC (Fig.9) and DET (Fig.10) curves on the ORL dataset.

Fig.7 ROC Curve for Gabor-KFA on ORL. Fig.8 CMC Curve for Gabor-KFA on ORL.

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Fig.9 EPC Curve for Gabor-KFA on ORL. Fig.10 DET Curve for Gabor-KFA on ORL.

4. Discussion and Future Work

We have proffered a comprehensive comparison of the GABOR-LDA and GABOR-KFA variants in order to ascertain which technique is more effective. Our experimentations were carried out on the publicly available ORL database and were corroborated with the corresponding ROC, CMC, EPC and DET curves. Our results illustrated that the GABOR-LDA method outperformed GABOR-KFA using Euclidean (0.83%) and City-Block (0.83%) distance measures. GABOR-KFA outmatched GABOR-LDA using the cosine distance (0.83%) and both these techniques were equally matched using MAHCOS distance. The variation in performance among them was minor and these findings convey that both of the techniques are viable options and would fare more or less in a similar manner in FR tasks.

Future work is currently being directed towards carrying out several similar studies (to be conducted in the forthcoming academic year) on the performance comparison 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. References [1] Meshgini, Saeed, Ali Aghagolzadeh, and Hadi Seyedarabi. "Face Recognition Using Gabor Filter Bank, Kernel Principle Component

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