Extended interval type-II and kernel based sparse representation method for face recognition

Accepted Manuscript

Extended Interval Type-II and Kernel Based Sparse Representation Method for Face Recognition Sudesh Yadav , Virendra P. Vishwakarma PII: DOI: Reference:

S0957-4174(18)30609-2 https://doi.org/10.1016/j.eswa.2018.09.032 ESWA 12220

To appear in:

Expert Systems With Applications

Received date: Revised date: Accepted date:

26 September 2017 12 September 2018 13 September 2018

Please cite this article as: Sudesh Yadav , Virendra P. Vishwakarma , Extended Interval Type-II and Kernel Based Sparse Representation Method for Face Recognition, Expert Systems With Applications (2018), doi: https://doi.org/10.1016/j.eswa.2018.09.032

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Highlights

Unseen information is extracted using extended interval type 2 membership function It also works where other Gaussian based membership functions are not applicable. Computational efficient: using sparsity concept which makes FR systems fast. It deals with uncertainty originating from non-linear variations in the features. It gives better results as compared to other state-of-art methods.

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Extended Interval Type-II and Kernel Based Sparse Representation Method for Face Recognition Sudesh Yadav1

Virendra P. Vishwakarma2

University School of Information, Communication & Technology, Guru Gobind Singh Indraprastha University, Sector 16-C, Dwarka, New Delhi, India1,2 [email protected], [email protected]

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Abstract:

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In the world of ubiquitous computing, fuzzy logic has been emerged as an important research area in the field of face recognition (FR) applications. In this paper, a new efficient and advanced method; inspired from interval type-II fuzzy membership concept of fuzzy logic is proposed. The motivation behind our method is to exploit the benefit of an extended interval type-II membership function: a new concept to fuzzy logics; in collaboration with kernel based sparse representation for FR. To integrate all the pluton, we propose a method called: extended interval type-II and kernel based sparse representation method (ExIntTy2KBSRM). In our proposed method, first we figure out the measure of participation of individual pixels in identifying face images using a variant of Interval type-II fuzzy logic i.e. extended interval type-II membership function in place of type 1 fuzzy logic. Next, we calculate K nearest neighbor to training specimens using simple Euclidean distance metric for sparse representation of each test specimen as a combinatorial of calculated K nearest training specimens. After this, we do classification based on contribution made by calculated train specimens in representation results. The efficacy and effectiveness of our proposed method is shown based on experiments performed on various standard databases. The experimental results show that our method deal with challenges of face recognition more efficiently as compared to other state of art methods, as it integrates the pluton of two different membership function based extended interval type-II fuzzy logics in collaboration with kernel sparse representation. Also, our experimental analysis tells that our proposed method improves the classification accuracy to 2-10% greater than the other existing relevant methods.

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Impact and significance of the proposed method: The main impact of the proposed method in FR based expert and intelligent systems is that, it considers unseen information available in pixel values of a face image present due to non-linear variations and overlapping of pixels. It also contains the advantage of spatial similar structure information present in face images. This makes the system more effective and efficient in processing face images for FR. Our method also works where Gaussian membership function does not work and discretizes linear and non-linear functions appropriately. Thus, the proposed method is universally applicable to solve more challenges of FR viz. illumination, occlusion, expression etc. and also has application in other areas viz. medical image processing, decision making problems, hand written words recognition, speech processing, watermarking etc. Also, our method makes FR systems computationally more efficient and cost effective by using sparse concept to matrices, which makes system to consume less memory and process data faster. Keyword: Fuzzy Logics; Kernel Methods; Sparse Representation; Face Recognition.

Introduction:

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1.

With the passage of time, face recognition has become an important area of research in machine learning and intelligence. It has been used in various fields such as robotics, computer vision, biometric security etc., having numerous applications such as: identity authentication, bank and airport security, human computer vision interaction, laws enforcement etc. With such a huge advancement in this area, there is always a hope of improvement. In literature, a lot of attempts for improving classification accuracy has been done. FR systems mainly identifies a subject or an image from a list of intended users or a provided database of images (Cavalcanti, Ren, & Pereira, 2013). A basic framework for FR systems consists of following phases: detection, feature extraction, and recognition or verification of face images. Recognition system identifies and verifies a person or an image from a list of users or database of images (Cavalcanti et al., 2013). For accomplishing the above discussed tasks, researchers use many data classification methods from decades. The basic purpose of data classification methods is

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to automatically assign data specimens to a set of predefined class of specimens. However, in real world, data distribution is unknown in most cases. To overcome this problem, various methods has been purposed in literature like nearest neighbor classification (NNC), fuzzy nearest neighbor classification (FNNC), Fuzzy k-nearest neighbor classification (FKNNC) etc. (Cavalcanti et al., 2013; Cover & Hart, 1967; Keller & Gray, 1985). In this paper, we present a method based on kernel sparse and fuzzy logic concept to improve the classification accuracy of pattern recognition. In past few decades, face recognition has been widely used as an application of pattern classification (Vishwakarma, Pandey, & Gupta, 2010; Sudesh Yadav & Vishwakarma, 2016). We also use here face recognition as an application of pattern recognition. From last few decades, the concept of kernel has become an interesting method for FR systems. The name kernel came from kernel functions, which is capable of handling high dimensional feature space (Bergman, 1970). In the field of machine learning kernel methods has been considered in the class of pattern analysis algorithms (Elisseeff & Weston, 2002). The most important task handles through pattern analysis algorithms are: clusters, ranking, principle components, correlations and classifications. In this paper, we mainly focus on pattern classification problem of FR systems. For the advancement in kernel based FR systems, researchers from various countries came forward and proposed various kernel based algorithms like kernel principal component analysis (KPCA) and kernel fisher discriminate analysis etc. for the development of improved FR systems (Bishop, 2006). These algorithms map the database into a high dimensional feature space by a nonlinear mapping. For classification problem of face recognition, Yu et al. proposed a kernel nearest neighbor classifier (Kernel-NNC) (Yu, Ji, & Zhang, 2002) . By using this method, one could change the distribution of specimens by non-linear mapping. The classifier used in kernel nearest neighbor performed better than nearest neighbor classifier.

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Researchers (Adam Glowacz, 2015) proposed a method using wavelet transform and k- nearest neighbor classifier (k-NNC) for fault diagnosis and analysis of failures occurring in a DC motor due to acoustic signals. This makes motor more resistant to fault and remaining motor in acceptable stage for a long time. In Adam et al. (A Glowacz et al., 2017), they proposed a technique for diagnosis and reduction of frequent faults occurring in equipment used in metallurgy industry based on recognition of currents. The whole process is done using feature extraction method and classification techniques viz. bayes classifier, linear discriminant analysis and nearest neighbor classifier. Researchers (Adam Glowacz & Glowacz, 2017) proposed a technique to prevent early financial and downtimes occurring in companies due to failure of induction motors. They used frequency base feature extraction method and classification was done by KNNC, k-means clustering and linear perceptron. In (Hernandez-Matamoros, Bonarini, Escamilla-Hernandez, Nakano-Miyatake, & Perez-Meana, 2016) researchers proposed a low complexity complex classifier for FR using clustering and fuzzy logic, which is computationally more efficient than other classifiers. Researchers (Melin, Castillo, Gonzalez, Castro, & Mendoza, 2016) used fuzzy edge detectors based on general typeII for FR systems based on neural network. Experimental results were compared to other edge detectors viz. sobel operator, type-1 and type-II fuzzy edge detectors. Researchers (Polyakova & Lipinskiy, 2017) performed a deep analysis on ensemble of various data mining techniques to solve decision making problems using fuzzy logic and data mining algorithms viz. artificial neural network, decision tree and support vector machine. Researchers (Ezghari, Belghini, Zahi, & Zarghili, 2017) proposed a technique for gender recognition using OWA- operator and RIM quantifier to define weightage of features, which they called fuzzy-similarity based classification (FSBC). Researchers in (Ezghari et al., 2017) proposed a technique called fuzzy logic ternary pattern to identify faces and emotions from a video sequence and various experiments were performed to prove the efficacy of the proposed technique.

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In today’s world of ubiquitous computing, sparse representation using kernel methods has become a famous method. Sparsity concept is come from sparse matrix representation, in which most of the elements are zero (Hackbusch, 1999). If we consider it in the context of systems, then it shows that systems are loosely coupled. The main reason behind using sparsity is that, it consumes less storage space, hence computationally more efficient and fast (Bank & Douglas, 1993). Sparsity concept also helps in improving the classification accuracy of pattern recognition applications. First time the researchers (Wright, Yang, Ganesh, Sastry, & Ma, 2009) proposed a method, which was called sparse representation to deal with problems came across FR systems such as illumination problems, facial detail and expression variations. They consider the problem of classification of test specimens; for this, they represent each test specimen in the form of identified train specimens, where some of the coefficients corresponding to identified train specimens are zero. In this way, sparse concept of matrix is included in classification methods defined for pattern recognition applications. A new concept of representation called collaborative representation (CR) was proposed by researchers (Lei, Yang, & Feng, 2011), which used the concept of regularized least square solution based method for improving classification. Experiments performed on various state-of-art databases showed

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the improvement in recognition accuracy. For the first time in research, Jun Yin (Yin, Liu, Jin, & Yang, 2012) proposed a method using kernel method as well as sparsity concept for solving face recognition problems. In their method, each test specimen was represented in the form of identified nearest train specimens using an appropriate kernel. Experiments performed on standard databases showed that the proposed method gave promiscuous results as compare to simple sparse representation concept. After this, researchers (Y. Wang, Wang, Chen, & Zhu, 2014), proposed a method called virtual sample space based sparse representation that showed the limitation of less training specimens in representation of test specimens and its effect on classification accuracy. A new method for semi supervised classification was proposed by researchers (Gu, Wang, Fan, & Meng, 2014), where they consider cost function for preserving the sparse coefficients and experimental results calculated on various standard databases showed the effectiveness of the method. Researchers (Kang, Liao, Xiang, & Pan, 2014) proposed a method using benefits of two prime methods: local feature based and kernel decent coordinate (KDC) in a single framework for FR systems. The method proposed was robust to various distortions in face image representation and was tested on Extended Yale B, CMU-pie and AR databases. To nullify the effect of illumination in classification problems, researchers (Vishwakarma, 2015) proposed a method based on discrete cosine transform (DCT) integrating the pluton of fuzzy filters applied over low frequency DCT coefficients. They used simple k-nearest neighbor classification (k-NNC) for performance measurement. To show and prove the superiority of proposed method over existing methods, the method was tested on various standard databases. Researchers in ((J. J.-Y. Wang, Huang, Sun, & Gao, 2015) proposed a method named graph regularized nonnegative matrix factorization (GRNMF) to resolve the margins by constructing nearest neighbor graph using regularized NMF for input databases. They further proposed two iterative algorithms to solve optimization problem. The method was tested on standard databases to show the superiority of the method to existing methods. Researchers in (Zhang, Xu, Yang, Li, & Zhang, 2015) discussed all sparse representation algorithms viz. greedy strategy approximation, constrained optimization, proximity algorithm based optimization, and homotopy algorithm-based sparse representation and are analyzed with experimentally comparative study of these sparse representation algorithm. A method for sparse representation using both l1 and l2 regularization was proposed by researchers (Zeng, Gou, & Deng, 2017). The main consideration of the proposed method was that a well-established face recognition system had to tune with noise before applying for any purpose. Authors in (Liu, Li, Peng, Qiu, & Lei, 2017) proposed a method to improve sparse representation classification by using approximation technique in nearest neighbors considered in sparse representation and an objective function for more discrimination of information between classes. Experimental results proved the significance of method for classification problems. To deal with data imperfection in classification problems, researchers (Cadenas, Garrido, Mart\’\inez, Muñoz, & Bonissone, 2018) proposed a fuzzy k-nearest neighbor classifier and the method was tested on both synthetic and real-world databases to prove the efficacy of the method. Researchers in (M. Wang, Hu, Sun, & Zhao, 2018) proposed a method based on kernel and l2 regularization-based classifier for reducing the effect of sensitivity on classification algorithms generated due to unconstrained cases (i.e. occlusion and noise). Researchers in (Sudesh Yadav & Vishwakarma, 2018) proposed a method in which uncertainty present in face images due to nonlinear variation within an intended subject or similarities between different subjects and also produced due to absence of enough and overlapping features for representing a subject was dealt and investigation results proved the efficacy of the method. Whereas in present method, in addition to uncertainty produced due to various reasons in pixels of a face image, it also considers spatial similar information present in the database. Also, the method is computationally more efficient and cost effective by using sparse concept to matrices, which makes system to consume less memory and process data faster as compare to other methods available in literature. The results obtained using our method are also better than that of most of the existing sparse methods discussed in (Zhang et al., 2015) and fuzzy logic based FR methods available in literature.

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In this paper, we propose a new robust, efficient kernel based sparse method using fuzzy logic concept. Our method integrates the pluton of extended interval type-II fuzzy logic: a modified concept of interval type-II membership functions. In existing works, various types of research have been done on sparsity for face recognition, but there is no research available till now which uses the concept of extended interval type membership functions in collaboration with kernel sparse representation for FR. The effect of extended interval type-II is that, it makes method more simple, efficient and computationally reasonable and, measures the unseen information available in features. Also, various face recognition problems which occurs due to uncertainty are solved to major extent. For successful accomplishment of our method we do following tasks: first we divide the whole database in two parts: train and test part. Next, we find the unseen information available in train and test part using extended interval typeII membership function i.e. fuzzify the train and test part. After this, we do kernel sparse representation of obtained fuzzified data. All the process of implementation is done in MATLAB version 8.1 (Davis & Hu, 2011). For measuring the efficacy of our proposed method various experiments on standard databases viz. AT&T, Yale and

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Georgia Tech face databases are done. For performing experiments on AT&T face database, we take first 1-8 face images as train specimens and remaining images as test specimens. A significant improvement in recognition accuracy is obtained for 1, 2, 5, 6, 7 and 8 images as train specimens as compare to other standard methods viz. simple sparse (Xu & Zhu, 2013), kernel sparse (Zhu & Li, 2014), FPIE (Vishwakarma et al., 2010), IT2FPIE (S. Yadav & Vishwakarma, 2016), discriminative sparse representation (Yong, Zhong, Jian, You, & Zhang, 2016) and NIntTy2FPIE (Sudesh Yadav & Vishwakarma, 2018) available in literature, whereas corresponding to first 3 and 4 images as train specimen recognition accuracy does not improve. Similarly, for Yale and Georgia Tech face databases a significant improvement is obtained corresponding to first 2-5 images for Yale and 4-8 for Georgia tech face database as train specimens. After seeing this significant improvement in recognition accuracy on these standard databases, we can say that our method is more significantly efficient and accurate other than available methods. One more significant importance of our method is that it decreases the computational cost and also works universally irrespective of places where other fuzzy membership functions does not work. The whole paper is organized as follows: in next part, we discuss in detail about our proposed method, after this we do the experimental analysis of the proposed method, next we discuss about rationale of our method and in last we conclude our method with some future directions. 2.

Preliminaries:

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In this section, we discuss about the pre-concepts used in our proposed method. First of all we discuss about fuzzy logic concept, because our method mainly emphasizes on extended interval type-II fuzzy logics (S. Wang, 1994); next we discuss about sparsity: a concept to matrix, used in our proposed method. The detailed discussion about both is given as follows:

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2.1 Fuzzy Logics: Fuzzy logic is mainly concern about degree of genuineness, rather than the concept of exact true or false. It is an important branch of soft-computing. In today’s era of computing, fuzzy logic has been used in lower-level machine controls especially in consumer products. The main idea of fuzzy logic is drawn by Prof. Lotfi Zadeh (Zadeh, 1965), when he was working on a problem ‘computer understanding of natural language’. In today’s world of research fuzzy logic has been emerged as an efficient and improved area of ubiquitous computing.

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2.2 Sparsity: Sparsity word is come from sparse matrix, which is a concept of numerical analysis and computer engineering. It mainly corresponds to those systems which are loosely coupled. Sparsity finds its application in fields where low density of important data is available i.e. most of the data elements are not so much important in context of application. A sparse matrix is typically stored in a modern computer as a two dimensional array, and each value in array represent an element like ai,j, where i and j are indices for accessing the value of elements in sparse matrix (Gilbert, Moler, & Schreiber, 1992). Example:

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where, A is sparse matrix, in which most of the elements are zero.

3. Proposed Method:

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To deal with uncertainty problems come across in FR systems, researchers from all over the world came forward and gave various methods like kernel algorithms, sparsity concept, kernel sparse, and the fuzzy logic concept. In addition to all these, there is no method available till now, which uses the concept of fuzzy logics along with sparsity. The main benefit behind using fuzzy logic concept is that, it makes method computationally efficient and measures the unseen information available in features and it also deals with the problems occurring due to uncertainty. So, to integrate all these, we propose a method called extended interval type-II and kernel based sparse representation method (ExIntTy2KBSRM) for FR. For doing the work presented in this paper, we perform following tasks:  

First divide the whole sample databases in two parts. Where one part corresponds to training and other one for testing purpose. Next, we fuzzify the database train as well as test using extended interval type-II fuzzy logics concept.

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  

Thereafter, identify first K nearest training specimens (NTSs) corresponds to every test specimen using simple norm distance metric. After this, substitute every test specimen as a combination of identified K NTSs. Next, compute the deviation between each test specimen and sum of contribution made by the jth NTS in representing that test specimen. Here a minimum value of deviation of an NTS indicates that it has greater ability to represent the test specimen, and the test specimen is classified to the same class of NTS.

The complete process of our proposed method is depicted by flowchart as follows: Apply two different modified π- membership functions for generation of extended interval type-II membership functions

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Database Image (Train & Test Data)

Find t-norm of obtained pixel values

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Apply kernel sparse representation-based method (KBSRM)

Classify each test specimen on basis of deviation

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Figure 1 Flow chart of proposed method named (ExIntTy2KBSRM)

Select K nearest training specimens (NTSs) to every test specimen using Euclidean metric (l2 norm)

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Fuzzified pixel values (train & test images).

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𝑇𝑒𝑆𝑚𝑝𝑗

𝐾 𝑗=1 𝐶𝑀𝑎𝑡𝑗 𝑁𝑇𝑆𝑗 ,

where, 𝑇𝑒𝑆𝑚𝑝 is test specimen, and 𝐶𝑀𝑎𝑡𝑗 is coefficient matrix, obtained using Eq. (8)

Find contribution of K NTSs in representing each test specimen

Classify the test specimen to the same class of nearest training specimen, having minimum deviation

Figure 2 Flow chart of KBSRM for face recognition

3.1 Extended Interval type-II Membership Function: In this part, we discuss in detail about Extended interval type-II fuzzy logics. As we know extended interval type-II membership function is not as much popular as triangular, trapezoidal, and Gaussian. Mathematically all the three type1 membership functions are written as follows in fuzzy logics set theory:

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Tapezoidal mf is a function of a vector x, and depends on four scalar parameters a, b, c, and d, as given by: Trapmf(x)= max (min (

(1)

where a and d locate the feet of the function curve and b and c denotes the shoulders. Similarly, ̅ and ̅ membership function can be depicted in mathematics as follows:

) (

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(

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( ̅ {

̅ (

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and

(2)

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{

and points of ̅ and ̅ membership functions are called inflexion points. If parameters are same, like a = c & b = d, then ̅ and ̅ membership functions are complement to each other. If we take different values of parameters, then concatenation of these two membership functions is called type membership function.

̅

(

1 1

( )

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For the first time Liao et al. (Liao, Celmins, & Hammell, 2003) proposed an extended type membership function to cover a range of commonly known membership functions including triangular, trapezoidal, s-shaped and z-shaped as special cases depending upon the value of parameter t. As we know type membership functions are the concatenation of the ̅ and ̅ membership functions. So mathematically, we can write extended ̅ and ̅ membership as follows:

1 1

̅

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(

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( )

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On basis of extended ̅ and ̅ membership, extended

type membership function can be written as follows:

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1 1

( )

(

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( )

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̅

(6) 1 1

(

(

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( ) {

On basis of Eq. (6), we can infer that extended type membership i.e. ̅ is the concatenation of extended ̅ and ̅ membership function i.e. ̅ ̅ ̅ , when a< b<=c< d. As we know, till now two different Gaussian membership functions were used for the generation of interval type-II membership function, but in our method, we use two different extended type membership functions for the generation of extended interval type-II membership function, denoted as below: ̅

̅

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where, ̅ is the lower membership function and ̅ is the upper membership function. In this way, we generate lower membership function and upper membership function values corresponding to input face database. For the generation of fuzzified matrix, we do t-norm on the output produced by applying two different extended interval type-II membership functions. The same is done as follows: ̿

(7)

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3.2 Kernel Sparse Representation based Method:

In this part, we discuss in detail about kernel sparse representation of fuzzified input matrix. The whole process of kernel sparse is done in mainly two phases:

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Initially, we identify first K NTSs corresponding to each test specimen. Next, substitute each training specimen as a linear combination of identified K NTSs. The same is done as follows:

where, specimen.

(8)

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denotes the jth test specimen and CMatj is the corresponding coefficient matrix for the jth test

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For doing the first step of kernel sparse representation, we take fuzzified matrix as an input and corresponding to each test specimen, identify K NTSs from all train specimens by using non-linear kernel metric. Let n1, n2, …, nr (r is the total number of train specimens) are train specimens and m1, m2,…, ms (s is the total number of test specimens) are the test specimens from kth class( where CNo.= v1, v2,…..,vn ). We identify b is the class label, when train specimen is from bth class. Corresponding to every test specimen, first K NTSs are identified as follows: (

| (

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( )| (

(( ( ( (

( ) (( ( (

( (

( ( ) ( ( )

( ( )

( )

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Using Eq. (9), we calculate K NTSs on the basis of distance between train and test specimen. Train and test specimen having least value of distance is the 1st NTS and on increasing order of distance we identify first K NTSs. In next step of our proposed method, we represent each test specimen in terms of identified K train specimens. Let we take jth test specimen i.e. w for representing in terms of identified train specimens. This can be done as follows: 1

(10)

1 th

th

where wk (k= 1, 2, 3, …, j) is the k NTS and vk (k=1, 2, 3, …, j) is the k coefficient corresponding to intended NTS.

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Eq. (10) can be rewritten in vector- matrix form, a generalized way as follows: (11) where (11) is:

and

1

. If W is a non-singular matrix then least square solution of Eq.

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And if W in Eq. (11) is singular, then least square solution is calculated by using following Eq:

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1

( (13)

where is a small positive constant and I is the identity matrix, used as a regularization operator. With the help of v, we represent each test specimen in the form of train specimens. Next, we find the contribution of each class of training specimens in representing the test specimens. The same can be calculated by finding the sum of the contribution of each class in representing test specimen using following formula: ̅

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̅ (14)

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where , and C is the set of class labels of the different nearest neighbor training specimens obtained corresponding to each test specimen. Let if all K nearest neighbor training specimens from rth class are ̅ ̅; then contribution will be from rth class i.e. Contr. In this way, we find the contribution of each class in representing the test specimens. After this, we find the deviation between contribution of each class and test specimen using simple Euclidean distance metric. A class having smaller the value of deviation makes larger contribution in representing test specimen and it is classified to that class. The same can be calculated by using following formula: |

|

4.

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(15)

Experimental results:

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In this section, we discuss in detail about various experiments performed on different face databases such as: AT&T face database, Yale face database and Georgia Tech face database. We performed our experiments on MATLAB version 8.1. Experimental results obtained by using ExIntTy2KBSRM are shown in below tables. For the verification and justification of our proposed method, we do comparison of ExIntTy2KBSRM with other existing methods available in literature. 4.1.1 AT&T face database: AT&T face database is a collection of 400 specimen images of 40 different people. This database is a standard database available in face rec repository for the experimental verification and validation of various proposed methods. We also use the same database for the intended purpose. In this database, images were captured in different situations at different time such as in a dark room, various positions viz. frontal, upright etc. and with different facial expressions and details. Some of the face images of AT&T face database are given in Figure 3.

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Experimental results obtained on AT & T face database and comparison to other state-of-art methods present in literature viz. (Vishwakarma et al., 2010; Xu & Zhu, 2013; S. Yadav & Vishwakarma, 2016; Sudesh Yadav & Vishwakarma, 2018; Yong et al., 2016; Zhu & Li, 2014) are given in Table 1. Results in the table prove the efficacy and competitiveness of the method.

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Figure 3 Some of the captured face images of AT&T face database with different facial expressions and details.

4.1.2 Yale Face database:

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Yale face database is a collection of 165 grayscale images of 15 different people. Everyone has 11 specimen images taken in different configuration or facial expressions such as some images were taken with glasses or some are without glasses. Different configures are: sleepy, happy, open/ close mouth, normal, center light and surprised etc. Some of the face images of freely available Yale face databases are given in Figure 4. Experimental results obtained on Yale face database and comparison to other state-of-art methods present in literature viz. (Vishwakarma et al., 2010; Xu & Zhu, 2013; S. Yadav & Vishwakarma, 2016; Sudesh Yadav & Vishwakarma, 2018; Yong et al., 2016; Zhu & Li, 2014) are given in Table 2. Results in the table prove the efficacy and competitiveness of the method.

Figure 4 Some of the captured face images of Yale face database with different facial expressions and configuration of camera.

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4.1.3 Georgia Tech Face database: Georgia tech face database is a collection of 750 specimen images of 50 different people. Here each people is having 15 different specimen images taken between 06/01/99 and 11/15/99. Most of the specimen images of this database were captured in two different sessions for considering the variations in different illumination conditions due to lightening. Also, in this database, we consider different scales and orientations of faces while capturing the images. Some of the face images of Georgia Tech face database are given in Figure 5. Experimental results obtained on Georgia Tech face database and comparison to other state-of-art methods present in literature viz. (Vishwakarma et al., 2010; Xu & Zhu, 2013; S. Yadav & Vishwakarma, 2016; Sudesh Yadav & Vishwakarma, 2018; Yong et al.,

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2016; Zhu & Li, 2014) are given in Table 3. Results in the table prove the efficacy and competitiveness of the method.

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Figure 5 Some of the captured face images of Yale face database with two different sessions for considering various illumination problems.

Table 1 Comparison of percentage percentage error rate of our proposed method to existing methods available in literature on AT&T face database No. of train specimen subject

per

1

2

3

4

5

6

7

8

15.31

12.85

9.58

6.50

1.25

1.66

1.25

26.11

Method given in kernel sparse (Zhu & Li, 2014)

29.56

18.02

10.00

7.50

7.00

6.25

5.00

5.00

Method given in simple sparse(Xu & Zhu, 2013)

32.50

18.52

11.97

8.42

7.00

6.25

5.00

5.00

Method given in (Vishwakarma et al., 2010)

26.11

16.56

13.57

11.67

9.00

7.50

5.00

1.25

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Proposed method

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Method given in (Sudesh Yadav & Vishwakarma, 2016)

-

12.85

8.75

7.50

6.25

3.80

-

26.94

17.18

10.71

9.58

7.50

5.63

3.33

1.25

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Method given in (Sudesh Yadav & Vishwakarma, 2018)

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Table 2 Comparison of percentage percentage error rate of our proposed method to existing methods available in literature on Yale face database No. of train specimen images per subject

1

2

3

4

5

Proposed method

-

29.63

10.00

7.61

4.44

Method given in kernel sparse (Zhu & Li, 2014)

-

-

13.17

12.03

11.40

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Method given in simple sparse (Xu & Zhu, 2013)

-

14.69

13.01

12.11

-

-

15.00

12.38

10.00

Method given in (Vishwakarma et al., 2010)

-

31.11

20.00

15.23

14.44

Method given in (Sudesh Yadav & Vishwakarma, 2016)

-

34.07

20.00

18.09

14.44

Method given in (Sudesh Yadav & Vishwakarma, 2018)

-

29.63

19.17

14.28

14.44

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Method given in discriminative sparse representation (Yong et al., 2016)

Table 3 Comparison of percentage percentage error rate of our proposed method to existing methods available in literature on Georgia tech face database 4

5

6

7

8

40.00

37.00

28.88

27.50

22.57

Method given in kernel sparse (Zhu & Li, 2014)

42.37

39.40

29.33

26.75

23.14

Method given in discriminative representation (Yong et al., 2016)

42.73

38.40

31.33

28.75

26.29

Method given in (Vishwakarma et al., 2010)

45.27

42.60

29.55

25.25

24.57

Method given in (Sudesh Yadav & Vishwakarma, 2016)

48.36

44.80

33.56

27.50

24.57

Method given in (Sudesh Yadav & Vishwakarma, 2018)

45.09

41.00

29.11

26.75

23.71

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No of train specimen images per subject

sparse

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Proposed method

4.2 Performance comparison of extended interval type-II fuzzy logic over other fuzzy logic: In this section, we investigate the effect of extended interval type-II fuzzy logic and other fuzzy logics (type-1 and interval type-II) on three standard databases viz. AT&T, Georgia Tech and Yale face databases to show the superiority of extended interval type-II fuzzy logic. The experimental results depict that when we use type-1 fuzzy logic (viz. πmf, smf, zmf, trapmf and gaussmf) and interval type-II (πmf ) fuzzy logic instead of extended interval type-II fuzzy logic, the performance deteriorates for all the three AT&T and Georgia Tech and Yale face databases. The performance comparison is shown in Tables 4, 5 and 6. After analyzing the tables, we can say that the proposed extended interval type-II fuzzy logic gives better results as compare to other fuzzy logics because it deals with

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uncertainty occurring due to non-linearity present within interclass and intraclass overlapping features for representing a class, which type-1 fuzzy logic is not able to deal (Wu, 2012; Sudesh Yadav & Vishwakarma, 2018). Table 4 Performance comparison of percentage error rate using extended interval type-II fuzzy logic and other fuzzy logic in feature extraction phase of proposed method on AT&T face database Method

No of specimen images taker per subject for training 2

3

4

5

6

7

8

using extended interval type-II fuzzy logic

26.11

15.31

12.85

9.58

6.50

1.25

1.66

1.25

using (Πmf)

28.06

15.31

14.64

9.17

8.50

4.37

2.50

2.50

using (smf)

30.8333

22.50

16.43

10.83

11.00

5.63

2.50

2.50

using (zmf)

33.89

20.94

19.64

14.58

13.00

7.50

5.00

5.00

using (trapmf)

27.22

15.00

14.28

9.58

8.50

5.00

3.33

1.25

using (gaussmf)

27.78

17.81

18.93

12.50

6.50

2.50

2.50

2.50

using (interval type-II (πmf))

34.17

33.44

26.79

23.75

22.00

18.75

16.67

20.00

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Table 5 Performance comparison of percentage error rate using extended interval type-II fuzzy logic and other fuzzy logic in feature extraction phase of proposed method on Georgia tech face database No of specimen images taker per subject for training 5

6

7

8

using extended interval type-II fuzzy logic

40.00

37.00

28.88

27.50

22.57

using (Πmf)

42.00

37.60

29.44

23.00

21.14

using (smf)

39.27

35.60

30.22

27.75

27.71

using (zmf)

68.36

66.20

59.77

53.50

49.14

using (gaussmf)

43.45

42.40

33.11

28.50

23.43

81.09

78.40

72.67

69.00

65.71

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CE

4

PT

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Method

using (interval type-II(πmf))

Table 6 Performance comparison of percentage error rate using extended interval type-II fuzzy logic and other fuzzy logic in feature extraction phase of proposed method on Yale face database Method

No of specimen images taker per subject for training

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3

4

5

using extended interval type-II fuzzy logic

29.63

10.00

7.61

4.44

using (Πmf)

38.52

22.50

20.95

13.33

using(smf)

29.63

10.83

9.52

8.89

using (zmf)

40.74

23.33

20.95

13.33

using (trapmf)

28.89

20.00

12.38

12.22

using (interval type-II (πmf))

32.59

19.17

17.14

13.33

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4.3 Comparison of Average Computing Time:

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In this section, we show the superiority of the proposed method in terms of computational efficiency. To prove it, we measure the average computation time taken by all the testing specimens on methods (viz. proposed method, extreme learning machine (ELM)(Huang, Zhu, & Siew, 2006), regularized-ELM (Zong, Huang, & Chen, 2013) , principal component analysis (PCA) (Turk & Pentland, 1991)) on AT&T, Georgia Tech and Yale face databases. All the computations are performed on 1.6GHz Pentium CPU with 4-GB RAM and the software used is MATLAB version 8.1. The results presented in Tables 7, 8 and 9 clearly reveal that the proposed method has a fast computational speed.

Number of training specimens per subject 3

4

5

6

Proposed method

3.59

2.69

2.96

2.51

2.99

ELM

4.33

4.45

4.42

4.48

4.95

regularized-ELM

7.45

5.21

6.14

7.21

6.55

PCA

17.47

16.71

18.65

18.29

18.33

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ED

2

PT

Method

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Table 7Average time (seconds) of all the test specimens on the AT&T face database

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Table 8 Average time (seconds) of all the test specimens on the Georgia tech face database

Method

Number of training specimens per subject 4

5

6

7

8

Proposed method

0.81

0.75

0.64

0.71

0.46

ELM

2.29

1.39

1.55

1.76

1.91

regularized-ELM

1.03

1.69

1.37

1.53

1.44

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PCA

42.22

21.11

20.01

20.73

21.04

Table 9 Average time (seconds) of all the test specimens on the Yale face database Method

3

4

5

proposed

4.14

3.30

2.96

2.43

ELM

35.68

28.08

11.69

regularized-ELM

15.77

19.43

27.52

PCA

6.42

7.20

7.82

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13.61 10.52 9.09

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5.

Number of training specimens per subject

Analysis of the proposed method:

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After analyzing the results shown in tables 1, 2 and 3; following observations can be drawn. First and most important, the proposed method has been tested with various standard databases, which are prepared under varying constraints like: varying light, facial emotions and positions, with/without wearing accessories etc. It is observed that our method outperforms for small as well as large scale databases viz. AT&T, Yale and Georgia Tech face database respectively. For example, when first four images per class from the Yale face database were used as training specimens and the others were used as test specimens, we attained the classification error using our method (ExIntTy2KBSRM), simple sparse (Xu & Zhu, 2013),kernel sparse (Zhu & Li, 2014), FPIE (Vishwakarma et al., 2010), IT2FPIE (S. Yadav & Vishwakarma, 2016), discriminative sparse representation (Yong et al., 2016) and NIntTy2FPIE (Sudesh Yadav & Vishwakarma, 2018) are 7.61, 13.01, 12.03, 15.23, 18.09 and 14.28 respectively. Here we see that the difference between the classification errors for discriminative sparse representation (Yong et al., 2016) and our method is 10.46%, and with NIntTy2FPIE (Sudesh Yadav & Vishwakarma, 2018) is 6.67%, which is quite large. This shows that our method obtains a much lower classification error rate as compared to other state-of art methods available in literature. Last but not the least, our method is independent of linear and nonlinearity present in the databases and handles unseen information present in the databases very efficiently and process the database very fast.

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Next, we analyze the proposed method on basis of NTSs i.e. K. By varying the value , where img_sub= total number of images per subject and c is an integer multiplier; a different classification error rate is obtained. In Figure 6, we analyze the percentage error rate versus number of training images per subject; for AT&T face database corresponding to K= 10, 20 and 30. Whereas in figures 7 and 8, we do it for Yale (K= 11, 22, 33) and Georgia Tech face databases (K= 15, 30, 45). After analyzing the results obtained, we have come to conclusion that different value of K leads to different recognition accuracy.

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Percentage error rate

Analysis of percentage error rate corresponding to different value of K on AT&T face dataset 30 K=10 25 K=30 K=20 20 15 10

0

1

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5

2

3 4 5 6 No. of train specimens per subject

7

8

Figure 6 Analysis of percentage error rate corresponding to different value of K on AT&T face database

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20 15

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Percentage error rate

Analysis of percentage error rate corresponding to different value of K on Yale face dataset 35 K=11 30 K=22 K=33 25

10

PT

5 0

2

2.5

3 3.5 4 No. of train specimens per subject

4.5

5

Analysis of percentage error rate corresponding to different value of K on Georgia Tech face dataset 45 K=15 K=30 40 K=45

Percentage error rate

AC

CE

Figure 7 Analysis of percentage error rate corresponding to different value of K on Yale face database

35

30

25

20

4

4.5

5

5.5 6 6.5 No. of train specimens per subject

7

7.5

8

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Figure 8 Analysis of percentage error rate corresponding to different value of K on Georgia Tech face database After this, we also investigate the effect of using a different distance metric viz. Manhattan distance, cosine similarity, Chebychev distance, and correlation other than Euclidean to compute the deviation in Eq. (15). For example, Eq. (15) can be represented as follows for the case of cosine distance: (16)

(

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√(

The experimental evaluation for all the three AT&T and Georgia Tech and Yale face databases represents that classification error rate varies when other variant of distance metric is used in Eq. 15. The performance comparison is shown in Tables 10, 11 and 12 respectively for these databases. After analyzing the tables, it is found that the proposed method gives better results on Euclidean distance metric. Table 10 Performance comparison of percentage error rate after using different distance metrics other than Euclidean to compute the deviation on AT&T face database

Euclidean

26.11

Manhattan

26.94

Cosine similarity

35.83

Chebychev

48.61

Correlation

26.94

2

3

4

5

6

15.31

12.85

9.58

6.50

1.25

15.93

12.85

10.83

7.50

2.50

23.12

18.21

12.92

7.00

3.75

32.81

27.50

19.17

13.50

8.75

13.93

10.00

9.00

3.75

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Distance Metric

CE

PT

ED

16.56

AC

Table 11 Performance comparison of percentage error rate after using different distance metrics other than Euclidean to compute the deviation Georgia Tech face database Distance Metric

4

5

6

7

8

Euclidean

40.00

37.00

28.88

27.50

22.57

Manhattan

41.82

39.80

29.56

22.75

20.86

Cosine similarity

46.00

41.80

32.00

24.50

21.43

Chebychev

54.36

53.40

42.44

35.75

31.14

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Correlation

43.64

41.00

30.44

24.75

22.00

Table 12 Performance comparison of percentage error rate after using different distance metrics other than Euclidean to compute the deviation on Yale face database 3

4

Euclidean

29.63

10.00

7.61

Manhattan

31.85

16.67

13.33

Cosine similarity

35.5556

20.83

16.19

chebyshev

60.74

33.33

26.67

Correlation

31.1111

19.17

14.29

5

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2

4.44 7.78

11.11 25.56 13.33

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6.

Distance Metric

Rationale of the proposed method:

2. 3. 4.

It covers the unseen information available in pixel features using extended interval type-II membership functions. It also works where other Gaussian based membership functions are not applicable. It also discretizes the linear and non-linear functions. Computationally efficient: as our method uses the sparsity concept, so having less number of non-zero values that make FR systems fast.

ED

1.

M

In this paper, we mainly draw the effect of extended interval type-II membership function and kernel sparse representation-based FR. Experimental efficacy shown in the Tables 1, 2 and 3 proves the importance of the proposed method. Our proposed method integrates the following plutons:

7.

CE

PT

Irrespective of above mentioned benefits, our method has also some drawbacks. Our method uses two different membership functions for the generation of extended interval type-II membership function which is parameter dependent. Second, in kernel sparse representation, by varying the value of K, a different value of recognition accuracy is obtained. Third, we use identity matrix as regularization matrix, which does not account spectral decomposition properties of other diagonal matrices viz. scaled finite difference matrix. Conclusion and future scope:

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In this paper, we propose a new efficient and robust method for solving the recognition accuracy issue to FR systems. The main motivation behind our proposed method is to integrate the pluton of two different concepts viz. very first is a different concept to fuzzy membership functions called extended interval type-II fuzzy logics and second one is concept inspired from matrices called kernel sparse. Benefit of using these two is that they make method simple, efficient and computationally reasonable. Also, the extended interval type-II concept extracts the unseen information available in the features. In addition to these, we also observe the experimental efficacy of the proposed method on various databases viz. AT&T face database, Yale faces and Georgia Tech face database. In Yale faces and Georgia tech face databases, experimental results are significantly improved to other state-of-art methods. But when we take AT & T face database a significant improvement in recognition accuracy is obtained as compare to other standard methods, but when first 3 and 4 number of images are taken as a train purpose sequentially it does not show improvement from kernel sparse method, although in overall, our method gives better

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results. The comparison of our method to other state-of-art methods is also done and results show the competitiveness of our method to other methods.

8.

References:

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Number of possible explorations are there to extend our proposed method. As we have used in our procedure, two different membership functions to capture the uncertainty present in face images, our method can be further extended by varying the exponent value and by shifting the mean of the membership function. Second, instead of using identity matrix, we can use scaled finite approximation of an identity matrix as a regularization operator. Third, we can apply our method (ExIntTy2KBSRM) to other applications viz. damage detection, multi-attribute decision making, big data analysis, control engineering, hand written word recognition and many more. Last but not the least, as there is no method available to defuzzify extended interval type-II fuzzy membership functions, a method for defuzzification of extended interval type fuzzy membership functions can be explored.

Bank, R. E., & Douglas, C. C. (1993). Sparse matrix multiplication package (SMMP). Advances in Computational Mathematics, 1(1), 127–137. Bergman, S. (1970). The kernel function and conformal mapping. American Mathematical Soc.

AN US

Bishop, C. M. (2006). Pattern recognition and machine learning. springer.

Cadenas, J. M., Garrido, M. C., Mart\’\inez, R., Muñoz, E., & Bonissone, P. P. (2018). A fuzzy K-nearest neighbor classifier to deal with imperfect data. Soft Computing, 22(10), 3313–3330. Cavalcanti, G. D. C., Ren, T. I., & Pereira, J. F. (2013). Weighted Modular Image Principal Component Analysis for face recognition. Expert Systems with Applications, 40(12), 4971–4977. https://doi.org/10.1016/j.eswa.2013.03.003

M

Cover, T., & Hart, P. (1967). Nearest neighbor pattern classification. IEEE Transactions on Information Theory, 13(1), 21–27. https://doi.org/10.1109/TIT.1967.1053964

ED

Davis, T. A., & Hu, Y. (2011). The University of Florida sparse matrix collection. ACM Transactions on Mathematical Software (TOMS), 38(1), 1. Elisseeff, A., & Weston, J. (2002). A kernel method for multi-labelled classification. In Advances in neural information processing systems (pp. 681–687).

PT

Ezghari, S., Belghini, N., Zahi, A., & Zarghili, A. (2017). Fuzzy similarity-based classification method for gender recognition using 3D facial images. International Journal of Biometrics, 9(4), 253–278.

CE

Gilbert, J. R., Moler, C., & Schreiber, R. (1992). Sparse matrices in MATLAB: Design and implementation. SIAM Journal on Matrix Analysis and Applications, 13(1), 333–356. Glowacz, A. (2015). DC motor fault analysis with the use of acoustic signals, Coiflet wavelet transform, and Knearest neighbor classifier. Archives of Acoustics, 40(3), 321–327.

AC

Glowacz, A., Glowacz, W., Glowacz, Z., Kozik, J., Gutten, M., Korenciak, D., … Carletti, E. (2017). Fault Diagnosis of Three Phase Induction Motor Using Current Signal, MSAF-Ratio15 and Selected Classifiers. Archives of Metallurgy and Materials, 62(4), 2413–2419. Glowacz, A., & Glowacz, Z. (2017). Diagnosis of stator faults of the single-phase induction motor using acoustic signals. Applied Acoustics, 117, 20–27. Gu, N., Wang, D., Fan, M., & Meng, D. (2014). A kernel-based sparsity preserving method for semi-supervised classification. Neurocomputing, 139, 345–356. https://doi.org/10.1016/j.neucom.2014.02.022 Hackbusch, W. (1999). A sparse matrix arithmetic based on-matrices. Part I: Introduction to-matrices. Computing, 62(2), 89–108.

ACCEPTED MANUSCRIPT

Hernandez-Matamoros, A., Bonarini, A., Escamilla-Hernandez, E., Nakano-Miyatake, M., & Perez-Meana, H. (2016). Facial expression recognition with automatic segmentation of face regions using a fuzzy based classification approach. Knowledge-Based Systems, 110, 1–14. Huang, G.-B., Zhu, Q.-Y., & Siew, C.-K. (2006). Extreme learning machine: theory and applications. Neurocomputing, 70(1–3), 489–501.

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Kang, C., Liao, S., Xiang, S., & Pan, C. (2014). Kernel sparse representation with pixel-level and region-level local feature kernels for face recognition. Neurocomputing, 133, 141–152. https://doi.org/10.1016/j.neucom.2013.11.022 Keller, J. M., & Gray, M. R. (1985). A Fuzzy K-Nearest Neighbor Algorithm. IEEE Transactions on Systems, Man and Cybernetics, SMC-15(4), 580–585. https://doi.org/10.1109/TSMC.1985.6313426 Lei, Z., Yang, M., & Feng, X. (2011). Sparse Representation or Collaborative Representation: Which Helps Face Recognition? In Computer vision (ICCV), 2011 IEEE international conference on (pp. 471–478). Liao, T. W., Celmins, A. K., & Hammell, R. J. (2003). A fuzzy c-means variant for the generation of fuzzy term sets. Fuzzy Sets and Systems, 135(2), 241–257.

AN US

Liu, S., Li, L., Peng, Y., Qiu, G., & Lei, T. (2017). Improved sparse representation method for image classification. IET Computer Vision, 11(4), 319–330. Melin, P., Castillo, O., Gonzalez, C. I., Castro, J. R., & Mendoza, O. (2016). General Type-2 fuzzy edge detectors applied to face recognition systems. In Fuzzy Information Processing Society (NAFIPS), 2016 Annual Conference of the North American (pp. 1–6).

M

Polyakova, A., & Lipinskiy, L. (2017). A study of fuzzy logic ensemble system performance on face recognition problem. In IOP Conference Series: Materials Science and Engineering (Vol. 173, p. 12013). Turk, M., & Pentland, A. (1991). Eigenfaces for Recognition. Journal of Cognitive Neuroscience. https://doi.org/10.1162/jocn.1991.3.1.71

ED

Vishwakarma, V. P. (2015). Illumination normalization using fuzzy filter in DCT domain for face recognition. International Journal of Machine Learning and Cybernetics, 6(1), 17–34.

PT

Vishwakarma, V. P., Pandey, S., & Gupta, M. N. (2010). Fuzzy based pixel wise information extraction for face recognition. International Journal of Engineering and Technology, 2(1), 117.

CE

Wang, J. J.-Y., Huang, J. Z., Sun, Y., & Gao, X. (2015). Feature selection and multi-kernel learning for adaptive graph regularized nonnegative matrix factorization. Expert Systems with Applications, 42(3), 1278–1286. Wang, M., Hu, Z., Sun, Z., & Zhao, S. (2018). Kernel collaboration representation-based manifold regularized model for unconstrained face recognition. Signal, Image and Video Processing, 12(5), 925–932.

AC

Wang, S. (1994). Generating fuzzy membership functions: a monotonic neural network model. Fuzzy Sets and Systems, 61(1), 71–81. Wang, Y., Wang, M., Chen, Y., & Zhu, Q. (2014). A novel virtual samples-based sparse representation method for face recognition. Optik, 125(15), 3908–3912. https://doi.org/10.1016/j.ijleo.2014.01.161 Wright, J., Yang, A. Y., Ganesh, A., Sastry, S. S., & Ma, Y. (2009). Robust face recognition via sparse representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 31(2), 210–227. https://doi.org/10.1109/TPAMI.2008.79 Wu, D. (2012). On the fundamental differences between interval type-2 and type-1 fuzzy logic controllers. IEEE Transactions on Fuzzy Systems, 20(5), 832–848.

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Xu, Y., & Zhu, Q. (2013). A simple and fast representation-based face recognition method. Neural Computing and Applications, 22(7–8), 1543–1549. Yadav, S., & Vishwakarma, V. P. (2016). Interval type-2 fuzzy based pixel wise information extraction: An improved approach to face recognition. In Computational Techniques in Information and Communication Technologies (ICCTICT), 2016 International Conference on (pp. 409–414).

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Yadav, S., & Vishwakarma, V. P. (2016). Interval type-2 fuzzy based pixel wise information extraction: An improved approach to face recognition. In 2016 International Conference on Computational Techniques in Information and Communication Technologies, ICCTICT 2016 - Proceedings. https://doi.org/10.1109/ICCTICT.2016.7514616 Yadav, S., & Vishwakarma, V. P. (in press). (2018). A New Interval Type 2 Fuzzy based Pixel Wise Information Extraction for Face Recognition. International Journal of Applied Pattern Recognition. Yin, J., Liu, Z., Jin, Z., & Yang, W. (2012). Kernel sparse representation based classification. Neurocomputing, 77(1), 120–128. https://doi.org/10.1016/j.neucom.2011.08.018

AN US

Yong, X., Zhong, Z., Jian, Y., You, J., & Zhang, D. (2016). A New Discriminative Sparse Representation Method for Robust Face Recognition via l₂ Regularization. IEEE Trans Neural Netw Learn Syst, PP(99), 1–10. Yu, K., Ji, L., & Zhang, X. (2002). Kernel nearest-neighbor algorithm. Neural Processing Letters, 15(2), 147–156. Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353.

Zeng, S., Gou, J., & Deng, L. (2017). An antinoise sparse representation method for robust face recognition via joint l 1 and l 2 regularization. Expert Systems with Applications, 82, 1–9.

M

Zhang, Z., Xu, Y., Yang, J., Li, X., & Zhang, D. (2015). A survey of sparse representation: algorithms and applications. IEEE Access, 3, 490–530.

ED

Zhu, N., & Li, S. (2014). A Kernel-based sparse representation method for face recognition. Neural Computing and Applications, 24(3–4), 845–852.

AC

CE

PT

Zong, W., Huang, G.-B., & Chen, Y. (2013). Weighted extreme learning machine for imbalance learning. Neurocomputing, 101, 229–242.