affine graph regularized sparse coding for robust face

International Journal of Computing and ICT Research, Vol. 9, Issue 1, June 2015 12

Affine Graph Regularized Sparse Coding for Robust face recognition

Mohsen Nikpoura2,

Department of computer and electrical engineering,Babol university of technology and

Department of computer and electrical engineering, Mazandaran Inistitute of technology

Email address: [email protected]

Mohammad Reza Karamia

Department of computer and electrical engineering, Mazandaran Inistitute of technology

Reza Ghaderic

Department of nuclear engineering, Shahid Beheshti university of technology

________________________________________________________________________________________

Abstract

Sparse coding is an unsupervised method in which learns sets of over-complete bases to represent data such as

image, video and etc. Because of the performance of this method in bag of visual words for image

representation, this technique has attracted increasing interest. But in the cases where we have some similar

images from the different classes, using the sparse coding method they may be classified into the same class

and degrade classification performance. In this paper, we propose an Affine Graph Regularized Sparse Coding

(AGRSC) approach for resolving this problem. Specifically, the objective function of sparse coding is

incorporated to make the new representations robust to the similar manifold. Experiments on two well-known

face datasets show that AGRSC can significantly outperform state-of-the- art methods in classification.

KeyWords: Sparse coding, Manifold learning, Graph regularized, Affinity, image representation, image

classification

________________________________________________________________________________________

IJCIR Reference Format:

Mohsen Nikpoura, Mohammad Reza Karamia and Reza Ghaderic. Affine Graph Regularized Sparse Coding

for Robust face recognition. International Journal of Computing and ICT Research, Vol. 9, Issue 1 pp 12 - 24.

http://ijcir.mak.ac.ug/volume9-issue1/article2.pdf

2 Author’s Address: Mohsen Nikpoura, Department of computer and electrical engineering,Babol university of technology and

Department of computer and electrical engineering, Mazandaran Inistitute of technology.

Email address: [email protected]

Mohammad Reza Karamia, Department of computer and electrical engineering, Mazandaran Inistitute of technology, Reza Ghaderic, Department of nuclear engineering, Shahid Beheshti university of technology

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© International Journal of Computing and ICT Research 2015. International Journal of Computing and ICT Research, ISSN 1818-1139 (Print), ISSN 1996-1065 (Online), Vol. 9, Issue 1,pp. 12 - 24,

June 2015

http://ijcir.mak.ac.ug/volume9-issue1/article2.pdf


1. Introduction

Image classification is an important task in image processing and computer vision studies. Sparse coding is a

method which can represent images using a few active coefficients(Long, M. et. Al. 2013). For this reason,

interpreting and manipulating the sparse representations are easy and facilitates efficient content-based image

indexing and retrieval (ZHANG, J. et. al. 2014). The area of sparse coding methods have been increased

every day in many fields such as pattern recognition, machine learning and signal processing (LEE , H. et. al.

2006 & MAIRAL, J. et. al. 2009), face recognition (WRIGHT, J. et. al. 2009 & GAO , S. et. al. 2010) and

image classification (YANG, J. et. al. 2009 & ZHENG , M. et. al. 2011) in recent years. One of the most

important goals of sparse coding is the maximum signal fidelity preservation and simultaneously improving

the quality of the sparse representation. For achieving this goal, many works have been proposed to modify

the sparsity constraint. LIU , Y. N. et. al. (2010) for improving the sparse coding method, have added a

nonnegative constraint to the objective function of basis sparse coding method. PTUCHA , R. et. al. (2014)

proposed a Linear extension of Graph Embedding K-means-based Singular Value Decomposition (LGE-

KSVD) to address both issues of computational intensity and coefficient contamination. In (GAO , S. et. al.

2010) the authors have improved the sparse coding method by adding a Laplacian term. In (JIANG , X. et.

al. 2014) the authors have proposed a sparse and dense hybrid representation (SDR) framework to alleviate

the problems of SRC. YANG , M.et al. (2011) for improving the signal fidelity proposed a robust sparse

coding. However, In the case of the similar images, they may be transformed into identical visual words of the

codebook and encoded with the same representations. The dictionary learned from the images cannot

effectively encode manifold structure of the images in this case, and accordingly the similar images from

different classes may classify in the same class. This similarity will greatly challenge the robustness of

existing sparse coding algorithms for image classification problems. The similar images are lying on a

manifold structure and the images from different classes are lying on the different manifold structures (LU , Z.

et. al. 2011). It has been shown that if the geometrical structure is exploited and the local invariance is

considered, the learning performance can be significantly enhanced. Recently, many literatures have focused

on manifold learning problems where represent the samples from the different manifold structures. To

preserve the geometrical information of the data, ZHENG , M. et. al. (2007) proposed to extract a good

feature representation through which the manifold structure of data are spotted.

With the prescience by recent progress in sparse coding and manifold learning, we propose a novel Affine

Regularized Sparse Coding (AGRSC) algorithm to construct robust sparse representations for classifying

similar images accurately. Specifically, the objective function of sparse coding has been incorporated this

criterion to make the new representations of the similar images far from each other. Moreover, for improving

the objective function with more discriminating power in data representation, we also incorporate the graph

Laplacian term of coefficients (ZHENG , M. et. al. 2011) in our objective function. The experimental results

verify the effectiveness of our AGRSC approach.

The rest of this paper is as follows: In Section 2, some related works are introduced. The sparse coding and

graph regularized sparse coding is then describeded in Section 3. Section 4 is included the proposed method.

The experiment setup and results are drawn in section 5 and finally, some conclusions are given in Section 6.

2. Related Work

In this section, we discuss some prior papers in sparse coding and manifold learning area. In recent years,

sparse coding has been widely used in many fields in computer vision. LEE , H. et. al. (2006) proposed a

feature sign search method. This method reduces the non-differentiable problem to an unconstrained quadratic

programming (QP). This problem can be solved by the optimization process rapidly. Our work also uses their

method to solve the proposed AGRSC optimization problem. For adapting the dictionary to achieve sparse

representation, QUANZ , B. et. al. (2012) proposed a K-SVD method to learn the dictionary using orthogonal

matching pursuit or basis pursuit. Adding nonnegative term to the sparse constraint is a method to improve the

quality of sparse representations (LIU , Y. et. al. 2010). Also the other methods such as graph regularization

(GAO , S. et. al. 2010 & ZHENG , M. et. al. 2011) and using weighted ℓ2-norm constraint (v et. al. 2010) are

introduced for improving the sparse representation. In the machine learning literature, manifold learning (LU ,

Z. eet. Al. 2011), which aims to transfer knowledge between the similar data sampled from different classes,

has also attracted extensive research interest. ZHENG , M. et. al. (2011) proposed a graph based algorithm,


called Graph regularized Sparse Coding (GraphSC), to give a sparse representations that consider the local

manifold structure of the data well. By using graph Laplacian as a smooth operator, the obtained sparse

representations vary smoothly along the geodesics of the data manifold. Our work in addition to the affinity

constraint, incorporates the graph Laplacian term of coefficients (ZHENG , M. et. al. 2011) in the objective

function, which can discover more discriminating representations for image classification.

3. Preliminaries

This section introduces sparse coding and affine graph regularized sparse coding which are applicable in this

paper.

3.1. Sparse Coding

Assume a data matrix Y = [ , . . . , ] with m data points sampled in the n-dimensional feature

space. Let = [ , . . . , ] be the dictionary matrix where each column represents a basis

vector in the dictionary, and X = [ , . . . , ] be the coding matrix where each column is a

sparse representation for a data point . Sparse coding is a method in which learn a dictionary and its

corresponding sparse codes such that initial data can be well approximated (MAIRAL , J. et. al. 2009).

Assuming the reconstruction error for a data point follows a zero-mean Gaussian distribution with isotropic

covariance, while taking a Laplace prior for the coding coefficients and a uniform prior for the basis vectors,

then the maximum a posterior estimate of Ф and X given Y is reduced to:

(1)

In the above equation is a parameter for regularizing the level of sparsity of the obtained codes and the

approximation of initial data. The objective function in Equ.(1) is not convex in Ф and X, therefore solving the

above equation is not easy in this case. But it is convex in either Ф or X. Therefore, solving this problem is

done by alternatively optimizing Ф while fixing X and vice versa. As a result, the above mentioned problem

can be split into two reduced least squares problems: an ℓ1-regularized and an ℓ2-constrained, both of which

can be solved efficiently by existing optimization software (LEE , H. 2006 & MAIRAL, J. et. al. 2009).

3.2. Graph Regularized Sparse Coding

(ZHENG , M. et. al. (2011) have proposed a method called Graph Regularized Sparse Coding (GraphSC)

method, which considers the manifold assumption to make the basis vectors with respect to the intrinsic

geometric structure underlying the input data. This method assumes that if two data points and are close

in the intrinsic geometry of data distribution, then their codes and are also close. Consider a set of n-

dimensional data points { , . . . , }, GraphSC constructs a p-nearest neighbor graph G with n vertices each

representing a data point. Let W be the weight matrix of G, if is among the p-nearest neighbor of , =

1; otherwise, = 0. and graph Laplacian L = D−W. A

reasonable criterion for preserving the geometric structure in graph G is to minimize:

(2)

By replacing the result into Equation (1) the GraphSC (LIU , Y. N. 2010) is obtained:

(3)

In equation (3) is a parameter for regularizing the weight between sparsity of the obtained codes and

preserving the geometrical structure.

4. The proposed method: Affine Graph Regularized Sparse Coding

In this section, we present the Affine Graph Regularized Sparse Coding (AGRSC) algorithm for robust image

representation, which extends GraphSC by taking into account the affinity constraints on the samples.

4.1. Problem Definition


In linear sparse coding, a collection of m atoms are given that form the columns of the

overcomplete dictionary matrix Ф. With a L0-minimization problem, the sparse codes of a feature vector

can be determined:

(4)

Where the function is defined as . In the proposed ARGSC method the main technical

difficulty is the proper interpretation of the function in the manifold setting, where the

atoms are now points in M and Ф now denotes the set of atoms and because of the

nonlinearity property in this case, it is no longer possible to create a matrix with atoms. Moving to the more

general manifold setting, we have forsaken the vector space structure in .

In the linear sparse coding, each point is considered as a vector whose definition requires a reference point.

However, in AGRSC setting, each point cannot be considered as a vector and therefore, must be considered as

a point. This particular viewpoint is the main source of differences between linear and AGRSC sparse coding.

In this paper, a new method is proposed to modify the usual notion of sparsity by adding an affine constraint

to reduce the feature vectors dimension on a manifold. A vector is defined as an affine sparse vector if it can

be written as follows: (HAZEWINKEL, M. , 2001):

; (5)

For more perception of the proposed method, see figure1.

There are three feature vectors in the figure which s are demonstrated by red circles corresponding to

feature vector1, s are demonstrated by blue Celtic cross corresponding to feature vector2 and s are

demonstrated by green squares corresponding to feature vector3.

As can be shown from the figure, the point X is corresponding to feature vector1, but if we want to represent

this point by sparsity constraint, it may be represented by the points from the other feature vector points in

addition.

Figure1. The affine constraint

If the manifold constraint ( is interferences, it can represent the point more accurately. When the

feature vectors are from the similar images but different classes and the manifolds are overlying such as in

face and action recognition applications, the efficiency of this constraint solely could be decreased. In this

scene the affine constraint can enforce selecting the points that are on a subspace with low dimension. Look

back into figure1. The points may be selected in representing the point X but if

the affinity constraint is added to the objective function, the points which are on the

manifold with low dimension according to X point, are selected for its representation.

According to the above mentioned descriptions, we can add an affinity term to Equ.(1):

(6)

The constraint term is add to the main criterion as a lagrangian coefficient leads:


(7)

where is a parameter for tuning the affinity constraint. For tuning the parameters , and some

experiments has been done which can be seen in the next section. In figure 2 you can see the steps of the

proposed method. After preprocessing the input data the samples are clustered using k-means algorithm to

make the initial dictionary. Then the KSVD is applied for optimizing the dictionary. Finally the AGRSC

method is applied to extract the optimal coefficients and then classifying the test data based on the minimum

error.

Figure2. The diagram of Proposed method

4.2. Solution of AGRSC

We apply the feature-sign search algorithm (LEE , H. , 2006) to solve the optimization problem (Equ.(7)). For

solving non-differentiable problems in non-smooth optimization methods, a necessary condition for a

parameter vector to be a local minimum is that the zero-vector is an element of the sub-differential the set

containing all sub-gradients at the parameter vector (FLETCHER , R. ,1987).

Following FLETCHER , R. ,(1987) ,MAIRAL, J. et. al. (2009) and ZHENG , M. et. al. (2011), the

optimization of AGRSC has been divided into two steps: 1) ℓ1-regularized least squares problem; in this step

we learn the affine graph regularized sparse codes X with dictionary Ф fixed and 2) ℓ2-constrained least

squares problem; in this step we learn the dictionary Ф with affine graph regularized sparse codes X fixed.

The two above steps are repeated respectively until a stop criterion is satisfied.

The optimization problem in the first step can be solved by optimizing over each individually.

Because Equ.(7) with L1-regularization is non-differentiable when contains values of 0, for solving this

problem, the standard unconstrained optimization methods can not be applied. Several approaches have been

proposed to solve the problem of this form (AHARON , M. et. al. 2006 & BELKIN , M. et. al. 2001). In the

following, we introduce an optimization method based upon coordinate descent to solve this problem (LU , X.

2014). It is easy to see that the Equ.(7) is convex, thus the global minimum can be achieved.

We update each vector individually by holding all the other vectors constant. In order to solve the problem by

optimizing over each , we should rewrite the Equ.(7) in a vector form. The reconstruction error

can be rewritten as:

(8)

The Laplacian regularizer can be rewritten as :

(9)

Combining (7) , (8) , (9) the problem can be written as:

Input data &

Preprocessing

Clustering the samples using K-

means algorithm to make the initial

dictionary Applying the KSVD for

optimizing the dictionary atoms The test data Applying the AGRSC to

extract the optimal codes Classifying the data based

on the minimum error


(10)

When updating , the other vectors are fixed. Thus, we get the following optimization problem:

(11)

Where and is the j'th coefficient of

Following the feature-sign search algorithm proposed in (CAND'ES , E. et. al. 2006), the Equ.(11) can be

solved as follows. In order to solve the non-differentiable problem, we adopt a sub-gradient strategy, which

uses sub-gradients of at non-differentiable points. Firstly we define:

(12)

Then,

(13)

Recall that a necessary condition for a parameter vector to be a local minima in nonsmooth optimizations is

that the zero-vector is an element of the subdifferential, the set containing all subgradients at this parameter

vector [18].We define as the subdifferentiable value of the jth coefficient of . If then

the absolute value function is differentiable, therefore is given by sign( ). If then

the subdifferentiable value is set [-1,1]. So, the optimality conditions for achieving the optimal value

of is :

(14)

Then, we consider how to select the optimal sub-gradient when the optimality conditions are

violated, i.e., in the case that if . When we consider the first term in the

previous expression . Suppose that , this means that >0 regardless of the

sign of . In this case, in order to decrease , we will want to decrease . Since starts at zero,

the very first infinitesimal adjustment to will take it negative. Therefore, for our purposes we can let

. Similarly if then we let . To update suppose that

we have known the signs of the at the optimal value, then we can remove the L1-norm on by

replacing each term with either (if ) or - (if ) or 0 (if ). Thus,

equation (13) is converted to a standard unconstrained quadratic optimization problem (QP). In this case, the

problem can be solved by a linear system. The algorithmic procedure of learning affine graph regularized

sparse codes is described in the following:

for each , search for signs of

solve the reduced QP problem to get the optimal which minimizes the objective function

return the optimal coefficients matrix

In the algorithm, we maintain an active set for potentially nonzero

coefficients and their corresponding signs while updating each . Then, it systematically

searches for the optimal active set and coefficient signs which minimize the objective function (Equ.(9)) .In


each activate step, the algorithm uses the zero-value whose violation of the optimality condition

is largest.

The detailed algorithmic procedure of learning affine graph regularized sparse codes is stated in Algorithm 1.

Algorithm1: Learning Affine Graph Regularized Sparse codes

Input: Data set of n data points , the dictionary Ф, the graph laplacian matrix L, the parameters

.

For all I such that do

Initializing: and active set where denotes sign( ).

Activating: from zero coefficient of , select j=arg . Activate (add j to the active set) only

if it locally improves the objective function:

, then set

, then set

Feature sign: let we separate as some submatrix that contains only columns corresponding to the active set

as . Let and be subvectors of and p.

The resulting unconstrained QP is as follows:

Let , the optimal value of under the current active set is obtained as follows:

Where I is the identity matrix.

In the next step a discrete line search is performed on the line segment from to and check the objective

value at and all points where the sign of any coefficient changes. Then the point with lowest objective

value is replaced with . At last the zero coefficients of are removed from the active set and update

.

The optimality conditions:

Condition(1): nonzero coefficients have the optimality condition as:

If condition(1) is not established go back to step 4

Else

Check condition(2).

Condition(2): zero coefficients have the optimality condition as:

If condition(2) is not established go back to step 3

Else

Return as the solution.

End

4.3. Learning Dictionary

The learning dictionary with the sparse codes fixed is transformed to the following optimization

problem:

(16)

The solution for this problem has been well described by prior works (LEE , H. 2006 & MAIRAL, J. et. al.

2009 & ZHENG , M. et. al. 2011) and in this paper we don’t consider it.

5. Experiments


In this section, for evaluating the proposed AGRSC approach, some experiments for image classification has

been performed.

5.1. Data Preparation

ORL, Yale face database (see Figure4 and Figure5) are two well-known datasets widely used in computer

vision and pattern recognition researches.

ORL face dataset: This dataset contains of 400 images consisting 10 different images from 40 distinct persons.

The images of each persons, were taken at different conditions such as times, lighting, facial expressions such

as open / closed eyes, smiling / not smiling and facial details such as glasses / no glasses. The background of

the whole images was homogeneous and dark. The size of each image is 92x112 pixels. (Figure3)

Figure3. some example of the ORL dataset images

Yale face database The Yale Face Database contains of 165 images consisting 11 images from 15 different

persons in different conditions. The size of each image is 243*320. The conditions are consisting different

facial expression or configuration, center-light, with glasses, without glasses, left-light, right-light, normal,

happy, sad, sleepy, surprised, and wink. (see figure 4 )

Figure4. some example of the Yale face dataset images

5.2. Experimental Setup

For evaluation the proposed AGRSC approach, the results of this method on two defined dataset are compared

with two state-of-the-art basic approaches, Sparse Coding (SC) (ZHANG, J. et. al. 2014) and Graph

Regularized SC (GraphSC) (ZHENG , M. et. al. 2011) for image classification.

Each of the three methods can learn sparse representations for input data points. In particular, SC is a special

case of AGRSC with and GraphSC is a special case of AGRSC with = 0.

Following (ZHENG , M. et. al. 2011 & PAN , S. J. 2011) , SC, GraphSC, and AGRSC are performed on data

as an unsupervised dimensionality reduction procedure. For reducing dimension of data, before applying the

above algorithm, PCA is applied by keeping 98% information in the largest eigenvectors.

Under our experimental setup, we have tuned the optimal parameters for the target classifier using cross

validation. Therefore, we evaluate the three baseline methods on datasets by empirically searching the

parameter space for the optimal parameter settings, and report the best results of each method. For Sc and

Graphsc the parameters have been set according to ZHENG , M. et. al. (2007).

For the proposed AGRSC method, we set the trade-off parameters by searching. The parameter values

using ORL face dataset is shown in figure 5.


Recognition rate variations for gamma changes by setting Alpha=Beta=1

Recognition rate variations for alpha changes by setting Gamma=0.6 and Beta=1

Recognition rate variations for Beta changes by setting Gamma=0.6 and Alpha=48

Figure5.the parameters setting using ORL dataset

As can be seen from Figure 5, the parameters are set to 48, 0.1 and 0.6 respectively. It should be noted

that, the affinity constrained can be more successful when the sparsity is large enough because if the

coefficients has not enough sparsity, the coefficients may be selected from the hyperplane with higher

dimensions than data's original dimension. In this case if the affinity constraint is added to the objective

function, it can even worsen the performance according to the GraphSC method.

5.3. Experimental Results

For evaluating the proposed method, two experiments have been done. The first experiment has been done on

ORL face dataset for face recognition and the second one has been done on Yale face dataset for face

Expression recognition.

The classification accuracy of AGRSC and the two baseline methods on the two face image datasets ORL and

Yale face dataset is illustrated in Table 1 and Table 2 respectively. As mentioned before the ORL dataset

consists 40 classes of faces. Do to the lack of space in the table only 10 classes are depicted. Among the whole

dataset, class 4 and 6, class 8 and 10, class 14 and 17 , class 5 and 18 are very similar to each other. Therefore

we use these classes in addition to class 1 and 2 in the confusion matrix to show the superiority of the

proposed method in classifying face datasets in table 1. Also in Table2 the results for Yale face dataset are


shown. From the results we observe that AGRSC achieves much better performance than the two baseline

methods.

The average classification accuracies of AGRSC on the two datasets are 91% and 63.46%, respectively. The

performance improvements are 18% , 15.98 % and 3.1%,4.5%compared to baseline methods SC and

GraphSC, respectively. The summary of results is drawn in table3.

Tabel1. The recognition rate for ORl data between three methods Sc, GraphSc and AGRSC

Class18 Class17 Class14 Class10 Class8 Class6 Class5 Class4 Class2 Class1

0 2 1 2 0 0 1 2 0 92 Class1

0 0 0 0 2 0 0 0 98 0 Class2

1 0 2 3 5 22 1 65 0 1 Class4

20 13 0 0 2 0 61 2 0 2 Class5

1 0 1 0 0 68 1 28 1 0 Class6

6 8 2 14 69 0 0 0 1 0 Class8

1 7 1 69 19 0 2 0 0 1 Class10

1 15 71 3 0 2 3 3 0 2 Class14

3 69 8 7 2 3 7 0 0 1 Class17

68 2 1 1 2 4 21 0 0 1 Class18

SC recognition rate for the selected dataset from ORL


0 1 0 0 0 0 0 1 0 98 Class1

1 0 0 0 1 0 0 0 98 0 Class2

1 1 1 1 1 5 1 89 0 0 Class4

4 2 0 0 0 0 91 2 0 1 Class5

1 1 2 2 0 91 0 3 0 0 Class6

0 2 0 8 87 1 0 0 2 0 Class8

3 5 4 79 8 0 0 1 0 0 Class10

3 13 83 0 0 0 0 1 0 0 Class14

2 75 11 8 1 2 1 0 0 0 Class17

88 4 0 0 0 1 6 0 0 1 Class18

b) GraphSC recognition rate for the selected dataset from ORL


0 0 0 0 0 0 0 1 0 99 Class1

0 0 0 0 0 0 0 0 100 0 Class2

0 1 0 0 0 2 1 95 0 1 Class4

0 1 1 0 0 0 96 2 0 0 Class5

1 0 0 1 0 95 1 2 0 0 Class6

0 2 1 4 92 1 0 0 0 0 Class8

1 5 3 84 7 0 0 0 0 0 Class10

5 14 79 2 0 0 0 0 0 0 Class14

2 78 11 5 1 2 1 0 0 0 Class17

92 0 4 2 0 1 1 0 0 0 Class18

c) AGRSC recognition rate for the selected dataset from ORL.

We have noticed that our AGRSC approach outperforms all the first two baseline methods. Incorporating the

graph Laplacian term of coefficients in addition to affinity constraint into AGRSC, leads to improve the sparse

representations with more discriminating power to benefit the classification problems.

Tabel2. The recognition rate for Yale face dataset between three methods Sc, GraphSc and AGRSC

wink surprised sleepy sad right-

light normal

w/no

glasses

left-

light happy w/glasses

center-

light

6 0.6 0 4.2 4.2 21.8 21.2 6 4.2 7.8 23.6 center-

light


0 1.2 1.8 0 3 4.2 0 4.2 3 76.6 6 w/glasses

1.8 0 1.8 1.2 0 4.2 7.8 6 65.2 4.2 7.8 happy

2.4 3 0.6 1.2 1.8 21.2 15.7 45.1 4.2 3 1.8 left-light

0.6 1.8 4.2 4.8 3 23.6 25.1 7.2 3 3 23.6 w/no

glasses

4.2 3 3 1.2 4.8 23.6 21.8 10.9 4.2 0 23.3 normal

4.2 6.8 3 6 57.2 1.2 3 4.8 6 1.8 6 right-

light

6 12 30.8 35.6 7.2 0 0.6 6 1.8 0 0 sad

4.2 6 39.8 30 6.8 1.2 1.8 3 1.8 0 5.4 sleepy

3.6 60.4 6 12 6 0 0.6 4.2 4.2 1.2 1.8 surprised

70 1.2 4.8 9 6 1.2 1.2 1.8 3 1.8 0 wink

a) SC recognition rate for Yale face dataset


light normal

w/no

glasses

left-


center-

light

2.4 0 0 3 1.8 18.8 18 6 4.2 7.8 38 center-

light

0 0.6 1.8 0 1.2 1.8 1.2 3 1.8 82.6 6 w/glasses

4.8 0 0 1.2 0 1.2 4.2 3 74.8 3 7.8 happy

4.2 3 0.6 2.4 1.8 17.5 12 52.5 3 1.2 1.8 left-light

0.6 1.2 1.8 4.2 1.8 13.3 46.1 7.4 1.8 1.8 20 w/no

glasses

3 3 6 1.2 6 43 12.6 6 1.2 0 18 normal

1.8 3 1.2 4.2 68.8 1.2 1.8 4.8 6 1.2 6 right-

light

1.8 6 20 54.2 6 1.8 1.2 6 1.8 1.2 0 sad

0 1.2 53.6 29 4.8 1.2 1.2 3 0 0 6 sleepy

0 64.4 6.8 12 4.2 1.8 0 4.8 3 0.6 2.4 surprised

70.5 10.3 2.4 4.8 3.6 1.2 0.6 3.6 1.8 1.2 0 wink

b) GraphSC recognition rate for Yale face dataset


light normal

w/no

glasses

left-


center-

light

1.2 0 0 3 1.8 16.9 15.7 6 4.2 6.8 47.4 center-

light

0 1.2 1.8 1.2 0.6 0 1.2 1.8 1.8 83.4 6 w/glasses

3 0 0.6 1.2 0 1.2 4.2 3 78.2 1.8 6.8 happy

3 1.8 0.6 1.2 1.8 13.9 12 60.3 2.4 1.2 1.8 left-light

1.8 3 3 5.4 7.2 12 42.9 7.2 1.8 1.8 13.9 w/no

glasses

4.2 1.8 6 1.2 6 49.6 9.6 6 0.6 0 15 normal

1.2 3 1.8 4.2 72.4 0.6 2.4 4.8 4.2 1.2 4.2 right-

light

3 5.4 13.9 57.7 4.2 1.8 3 6 1.8 1.2 0 sad

0 1.2 68.1 17.5 1.2 1.2 1.8 3 0 1.2 4.8 sleepy

0.6 68.8 8.4 6 1.8 1.2 2.4 4.8 3 1.2 1.8 surprised

74.3 11.5 4.8 0 0.6 0 3 3 1.2 1.2 0 wink

c)AGRSC recognition rate for Yale face dataset.

Table3. the Mean Recognition rate for SC, GraphSc and AGRSC methods

Mean Recognition rate for

ORL

Mean Recognition rate for

Yale

Sc method 73% 47.48%


GraphSc method 87.9% 58.96%

AGRSC (Proposed) method 91% 63.46%

6. Conclusion

In this paper, a novel approach for robust image representation namely Affine Graph Regularized Sparse

Coding (AGRSC) has been proposed. In the proposed method, the well-defined graph regularized sparse

coding method has been improved by adding the affinity constraint. Using this term, until the sparsity is big

enough the manifold structure of features is better preserved. The results show that the recognition rate of the

proposed AGRSC comparing to the basic sparse coding approaches has the higher performance.

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