Optimized Unsupervised Image Classification Based on Neutrosophic Set Theory

Optimized Unsupervised Image Classification Based on Neutrosophic Set Theory

A. E. Amin

Department of Computer Science Mansoura University,

Mansoura 35516, Egypt

Abstract: In this paper, a new technique is used to an unsupervised learning image classification based on integration between neutrosophic sets and optimization linear programming. Neutrosophic sets are used to segment the image into three main components namely objects (O), edges (E), and Background (B). The neutrosophic image

components O, E, B are corresponding to the

neutrosophic sets components T , I , F . The components of neutrosophic image valued in 0,1 are representing the

association intensities degree of pixel for each image components. Netrosophic image components are contributed to solving one of the important problems in image classification known as "overlapping" within cluster. While, the problem of overlapping between clusters is solved by using optimization linear programming.

Key words: Neutrosophic set, image classification, linear programming optimization

1. INTRODUCTION

   Since several decades, the world is witnessing a remarkable development in the science of computer vision. The principle of computer vision is based on deal with the images and methods of treatment. Hence the interest of researchers in the computer vision with image processing, which is concerned, in essence, on the methods and many different algorithms. Among these algorithms are image classification algorithms.

   Classification is the field devoted to the study of methods designed to categorize data into distinct classes. This categorization can be divided to distinct labeling of the data (supervised learning [1]), division of the data into classes (unsupervised learning [2]), selection of the most significant features of the data (feature selection [3]), or a combination of more than one of these tasks [4].

   Unsupervised image classification (UIC) starts by partitioning the image data into groups (or clusters). The classes in UIC are unknown, according to similarity measure, groups of image data can be compared with reference to data by an analyst [5]. UIC can be categorized into two main groups namely Hierarchical [6] and Partitional [7] algorithms.

   In hierarchical clustering algorithms (HCA) a sequence of clustering with each clustering being a partition of the data set are showing as a tree [8]. HCA is characterized by two advantages, first the number of classes does not need be

   specified a priori and the others they are independent of the initial condition. However, HCA is suffers from be a static algorithm and its inability to solve the overlapping clusters problem [9]. HCA are divided according to the clusters construction methods or according to the similarity measure. For methods construct the clusters by recursively partitioning the instances in either a top-down or bottom-up fashion. These methods can be subdivided as agglomerative [10] and divisive [11] methods. Whereas, the merging or division of clusters is performed according to some similarity measure, chosen so as to optimize some criterion (such as a sum of squares). The hierarchical clustering methods could be further divided according to the manner that the similarity measure is calculated [12].

   On the other hand, partitional clustering algorithms (PCA) are based on image data set segmentation into a specified number of clusters. PCA can be treated as an optimization problem as a result of reliance on the square error function to minimize certain criteria. Both HCA and PCA algorithms are participate in advantages and drawbacks. There are two categories from PCA namely Iterative [13] and non-iterative [14] algorithms. K-means algorithm [15] is the most widely used in iterative partitional algorithms. The basic idea for k-means algorithm is to find a clustering structure that minimizes a certain error criterion which measures the distance of each instance to its representative value. The most well-known criterion is the Sum of Squared Error (SSE) [16], may be globally optimized by exhaustively enumerating all partitions, which is very time- consuming, or by giving an approximate solution using heuristics. Another partitioning algorithm, which attempts to minimize the SSE is the K-medoids [17] or partition around medoids (PAM) [18].

   Lillesand and Kiefer [19] presented a non-iterative approach to unsupervised clustering with a strong dependence on the image texture. Researches [20-21] have shown that the iterative algorithms are more efficient than its counterpart non-iterative, where it does not rely too much on data points order.

   There are other unsupervised classifications methods are used recently represented in Density-based Methods [22] which assume that the points that belong to each cluster are drawn from a specific probability distribution. Model- based Clustering Methods [23], these methods attempt to optimize the fit between the given data and some mathematical models. Unlike conventional clustering, which identifies groups of objects; model-based clustering

   methods also find characteristic descriptions for each group, where each group represents a concept or class. The most frequently used induction methods are decision trees

   [24] and neural networks [25]. Grid-based Methods [26], these methods partition the space into a finite number of cells that form a grid structure on which all of the operations for clustering are performed. The main advantage of the approach is its fast processing time [27]. Soft-computing Methods, In addition to neural networks, there are some methods that belong to soft computing methods such as Fuzzy Clustering [28], Evolutionary Approaches for Clustering [29] and Simulated Annealing for Clustering [30].

   In this paper, a new an unsupervised image classification technique is used based on neutrosophic sets [31] and optimization linear programming [32]. Neutrosophic set

   Ns S considered a part from neutrosophy theory which

   interested by studies the origin, nature and scope of neutralities, as well as their interactions with different ideational spectra. The idea of neutrosophy theory depends

   on event or entity, where between an idea A and its opposite Anti A , there is a continuum power

2. GENERAL FRAMEWORK

   This paper presents a novel system to image clustering namely Optimization neutrosophic image classification system (ONsICS). As shown in figure 1, ONsICS consists of two techniques are neutrosophic image processing and optimization image clustering. Neutrosophic image processing is used to convert gray image to enhanced binary image (EBI) based on object, edge and background of image components. Each image can be represented as neutrosophic components (T, I, F) and stored the extracted image components feature as a vector in database. All similar image features are gathered together in a one category by using neutrosophic image clustering (NsIC) technique. Image clusters are optimized by using linear programming to solve image overlapping problem as shown in figure 1.

3. NEUTROSOPHIC IMAGE PROCESSING:

   Let Iimg be a Universe of discourse represents image and Icomp is a

   set of Iimg represents image components (as object, edge,

   background) which is composed by bright pixels. Aim of the neutrosophic image domain NS Dis transferring

   spectrum of neutralities

   Neut A

   . Truth value (T),

   image Iimg to neutrosophic domain by describing the pixel

   indeterminacy value (I) and falsehood value (F) are

   by three membersip sets T , I and F as

   represented neutrosophic components referring to neutrosophy, neutrosophic logic, neutrosophic set, neutrosophic probability, neutrosophic statistics [33]. In neutrosophic set, the indeterminacy is quantified explicitly

   PNs

   (T , I , F) [43]. The pixel can represents as:

   PNs i, j T i, j, I i, j, Fi, j where,

   and the truth-membership, indeterminacy-membership and falsity-membership are independent. The neutrosophic set is a generalization of an intuitionistic set [34], classical set [35], fuzzy set [36], paraconsistent set [37], dialetheist set

   T i, jis the probability belonging to white pixels set. It is defined as:

   T i, j g i, j gmin

   [38], paradoxist set [39], and tautological set [40].

   Linear programming is constrained optimization, where the

   constraints and the objective function are all linear. It is

   gmax

   1 iw 2

   • gmin

     j w 2

     called "programming" because the goal of the calculations

     g i, j

     gm, n

     help you choose a "program" of action [41]. The linear

     2

     programming model, for neutrosophic image classification

     w w miw

     n j w 2

     problem, involves on two main parts called constraints and objective function. Constraints are describing the query images as lower and upper weights for neutrosophic query image components. On neutrosophic image clustering

     Where, g i, j is the local mean value of pixels of the

     window.

     I i, jis indeterminate set. It is defined as:

     i, j

     classification to be maximized a linear objective function means that categorization of similar images in clusters with

     I i, j

     min

     max min

     out overlapping within or between clusters.

     The rest of the paper is organized as follows: section 2 presents general framework for proposed technique. neutrosophic image processing is given in section 3. feature extraction for neutrosophic image is presented in section 4. Neutrosophic image cluster is illustrates in section 5. Section 6 illustrates accuracy evaluation for the technique. Section 7 presents experimental results to illustrate the efficiency of the algorithm. Section 8 concludes the paper, and outlines future research.

     i, j absgi, j gi, j

     Where, i, jis the absolute value of difference between intensity gi, j and its local mean value gi, j.

     F i, jis non white pixels set. It is defined as:

     Fi, j 1 T i, j

     1. Neutrosophic Image Entropy:

     Neutrosophic image entropy [44] is defined as the

     EnNs EnT EnI EnF

     max T

     summation of the entropies of three sets T , I , and F , which is employed to evaluate the distribution of the elements in the neutrosophic domain:

     EnT

     pT iln pT i

     imin T

     Fig. 1: Optimization image clustering flowchart.

     EnI

     EnF

     max I

     pI

     imin I

     max F

     pF

     imin F

     iln pI i

     iln pF i

4. NEUTROSOPHIC IMAGE FEATURE

   EXTRACTION:

   Image feature Extraction is the first step to image retrieval system. Neutrosophic image NS Im is divided into three

   matrices are represented as images called object, edge and background. Each image is consisting of matrix representing the probability white pixel values for object component and probability of non white pixel values for

   where EnT , EnI , and EnF are the entropies of the sets T

   background component while the intermediate matrix

   , I and F , respectively. pT i, pI i, and pF iare the

   expresses the probability of the boundary between the

   white and non-white pixels. The combinations of pixel

   probabilities of elements in T , I and F, respectively, whose values equal to i .

   brightness value in NS Imcomponents are calculated by using a widely method namely Gray Level Co-occurrence Matrix (GLCM) []. The spatially related in various directions with reference to distance and angular relationships for co-occurring pairs of pixels is one of the most important advantages for GLCM calculations.

   The feature extraction for NS Imcomponents by GLCM is based on pixel and its next neighbor pixel. The Contrast, Energy, Homogeneity and Correlation are the parameters

   of GLCM which calculated by:

   can be computed by:

   ik

   c

   c x e

   1

   m1

   m n c k i

   Contrast i j2 I

   comp

   i, j,

   where

   j 1

   xk e j

   i 1 j 1

   0 i m , 0 j n

   where ei is the cluster center and can be computed by:

   (m, n)is image dim entions

   ik x

   n

   m

   k

   m

   n

   2

   e k 1 , 1 i c

   n

   i

   m

   ik

   Energy

   Icomp i, j

   i 1 j 1

   ik

   k 1

   m n I i, j

   The mean and the variance of

   ik for the cluster are

   Homogeneity comp

   i 1 j 1 1 i j

   computed as:

   ni

   ni 2

   m n i j Icomp i, j x y

   ik

   ik i

   Correlation

   i 1 j 1

   x y

   , where

   k 1 ,

   2 k 1

   n

   i

   i i

   .

   x , y aremeanof probability matrix.

   i

   n

   x , y are s tan dard deviations of probability matrix.

   Where, the fuzzy partition matrix

5. NEUTROSOPHIC IMAGE CLUSTERING:

   is i

   , i 2

   ,…,

   ini

   comp

   i1

   Image clustering can classify similar images into the same group. Let image data set be Im I i ,i 1,2,…, n ,

   As shown in figure 2, NS Imclustering method based on fuzzy c-means is used.

   and

   i

   I

   comp

   be an image in a d-dimensional space. Image

   1. Neutrosophic image clusters enhancement:

      clustering problem is to find image

      The indeterminacy of image pixel

      Icomp i, j is

      category Clim Clim ,Clim ,…, Clim , which satisfies:

      1 2 n

      m

      Im Climi i1

      i

      Clim for i 1,2,…, m

      determined by its intensity value I i, j. Strength of the correlation between neutosophic image components T and F with I are influenced by the distribution of the pixels and the entorpy of I.

      The set I Im 0,1 may represent not only indeterminacy

      i

      2

      Clim Clim j for i, j 1,2,…, m, i j Among clustering methods, the fuzzy c-means algorithm is widely used. An objective function for a clustering method is important to define. The objective function of fuzzy c- means is defined as:

      but also vagueness, uncertainty, imprecision, error, etc. So, the overlapping problem will appear within and between neutrosophic image cluster as shown in figure 3.

      k

      Threshold processing will solve the overlapping problem within and between neutrosophic image clusters by determine the mysterious region between background and

      n

      J m U , E

      ik

      m x

      • ei

      objects. In gray level domain, mean operation for

      c

      k 1 k 1

      Where m is constant, and m 1. Cluster i is expressed

      as e i 1,2,…, c. The membership between sample k

      image ImGL is defined as:

      2

      2

      1

      iw j w

      Im i, j

      mod m, n

      i

      and cluster is expressed as;

      GL w w

      ImGL

      miw 2 n j w 2

      ik

      i 1,2,…, c, , where,

      k 1,2,…, n

      Where w is the size of the window, m, nis the location of the pixel centered the window. A mean operation

      for P , P is defined as:/p>

      Ns Ns

      c

      ik 0,1, i, k;

      ik

      i1

      1, k

      P

      Ns

      PT , I , F

      F

      F I

      T T

      T

      I

      I

      F

      F i, j 1

      I

      iw 2

      j w 2

      T i, j 1

      iw 2

      j w 2

      T

      m, n

      F m, n

      2 2

      w w miw n j w

      2

      w w miw

      n j w 2

      Fig. 2: flowchart of neutrosophic image clustering.

      Fig. 3: Overlapping within and between neutrosophic image cluster.

      I i, j 1 H i, j H min

      H max H min

      of m neutronsophic image clusters (NsIC) from which the most preferred cluster is to be selected to including the query image. Each NsIC is assessed on n different

      Where

      H i, jis the homogeneity value of T

      at (i, j) .

      components as imcomp ,imcomp ,…, imcomp . The

      1

      2

      n

      w is the local window size.

      After mean operation subset T became more

      evaluation of the cluster Cli with respect to the component of imcomp is a neutrosophic set. The neutrosophic

      homogeneous after removing the noise. By using a simple

      threshold method can be segnent the subset T accurately.

      j

      index I ij is such that the larger

      I ij

      the higher a hesitation

   2. Optimization neutrosophic image clustering:

   margin of NsIC Cli with respect to the components of

   Optimization N ImClassification (ONsIC) method

   imcomp whose intensity is given by T . The NsIC matrix is

   S

   presents a new method to determine the best category to include the query images, where the characteristics of clusters are represented by neutrosophic sets. Suppose that

   a set of clusters Cl Cl1 ,Cl2 ,…, Clm which consists

   j ij

   given in the following form:

   	I1	I 2		I n
   Cl1	T11, I11, F11	T12, I12, F12		T1n , I1n , F1n
   Cl2	T21, I21, F21	T22, I22, F22		T2n , I2n , F2n
   Clm	Tm1, Im1, Fm1	Tm2 , Im2 , Fm2		Tmn , Imn , Fmn

   comp

   comp

   comp

   NS IC

   Where the characteristics of NsIC Cli are given by:

   Obviously,

   0 K l

   • K u 2 for all

   Cli

   Cl and

   ij

   ij

   I1 I 2

   I 3

   imcomp im . The degrees to the alternative

   Cl comp , comp ,…., comp ,

   where,1 i m j

   i T , I , F T , I , F T , I , F

   i1 i1 i1

   i 2 i 2 i 2

   in in

   in

   NsIC Cl satisfies and does not satisfy the can be measured

   NS IC(Cli ) can present by another form as: i

   Cl Cl ,K l , K u , Cl ,K l , K u ,…, Cl ,K l , K u , where

   i

   1

   i1

   i1

   2

   i 2

   i 2

   n

   in

   in

   by the evaluation function (E). The evaluation function

   K l , K u

   is closed NsIC interval computed by:

   i

   i

   ECl of alternative NsIC Cl can be expressed as:

   ij ij

   l u

   Tij Iij

   1 Fij Iij

   Tij Iij

   1 Fij Iij

   Kij , Kij min ,

   , max ,

   2

   2 2

   2

   i

   ij

   ij

   ik

   ik

   ip

   ip

   iQ

   iQ

   ECl K l , K u K l , K u …. K l , K u K l , K u

   Where, for each query images are

   0 wl

   wu 1.

   i

   minK l , K l ,…, K l , minK u , K u ,…, K u K l , K u j j

   maxminK l , K l ,…, K l , K l , maxminK u , K u ,…, K u , K u

   ij ik ip

   ij ik ip

   iQ iQ

   Then the suitability degree of alternative NsIC Cl

   ij ik

   l u

   ip iQ

   ij ik

   ip iQ

   satisfies the query images components can be measured by:

   KCl , KCl

   1 T Cl

   2 1 A Tl ,Tu

   where,1 i m , and and denote the minimum

   RCl maxW Cl 2

   i

   i , S Tl ,Tu

   c iq

   iq

   i i

   i

   2

   c iq iq

   2

   and maximum operator of neutrosophic set respectively.

   The optimal weights value can be computed via the

   The score of alternative NsIC by:

   Sc Cli can be evaluated

   following programming:

   K l K u

   1 ij ij

   S Cl 2T u T l

   RCl m 2K u K l 2 2 w j

   c i Cli

   Cli

   TCl ICl 1 FCl ICl

   TCl ICl 1 FCl ICl

   i ij ij

   j 1

   2 *

   2 max i i , i i min i i , i i

   2

   2

   2 2

   An accuracy function Ac is used to evaluate the degree of

   Subject to the conditions:

   wl w j wu

   accuracy of neutrosophic elements as follows:

   A Cl 1 T l T u

   j * j

   c i 2

   1

   Cli

   Cli

   TCl ICl 1 FCl ICl

   TCl ICl 1 FCl ICl

   where,

   j 1,2,…, n

   2

   2

   2

   min i i , i i max i i , i i

   2 2

6. Accuracy assessment:

   A neutrosophic image is a combination of various

   The larger value of Ac Cli represents the more degree of accuracy of an element Cli in the neutrosophic set. Based

   components built on the intensity of pixels. The unsupervised classification applied to neutrosophic image sets result in clusters. The comparison between

   on the score function Sc and the accuracy function Ac

   degree of suitability that the corresponding cluster

   the

   Cli

   neutrosophic image components and clusters indicates the accuracy of the classification. Confusion matrices are used to assess this accuracy as they compare the relationship

   satisfies the query images component can be measured by:

   2 1 Ac ECli

   between the neutrosophic image components as the

   reference images and the corresponding results of the

   W ECli Sc ECli

   2

   unsupervised classification technique.

   A resulting cluster by unsupervised classification is not

   The coefficients in W ECli have been chosen so thatW ECl 0,1. The larger value ofW ECl ,

   automatically labeled nor identified as corresponding to specific image. So, image is investigated with respect to all

   i i clusters, and the cluster containing most of image

   the more suitability to which the alternative NsIC

   Cli satisfies to query image components, where

   1 i m.

   Assume that there is a query images wants to choose an alternative NsIC which satisfies the components

   components closest to the mean of that image is considered as its corresponding to a specific cluster. Based on confusion matrix, the accuracy is then expressed in terms of the kappa statistic (k) where the difference between the clustering accuracy and the chance agreement between the images and the clusters is calculated [45].

   of QI j , where, 1 j n , each QI j

   degre of components T, I, F . Where,

   have a different

   It results in a value between 0 and 1 for each classification, where 0 indicates that the clustering is no better than grouping the image by chance:

   0 T 1 ,

   c

   N yij

   y y

   c

   0 I 1,

   k i1 i1

   c

   0 F 1,

   N 2

   y

   i

   j

   i

   i1

   • y j

   and 0 T I F 3

   Where:

   T,.I, and F

   are the degrees of membership (object),

   c : is the number of clusters 1 i c

   N : is the total number of images in the image database

   j

   indeterminacy (edge) and non-membership (background) of classified

   the images

   imcomp imcomp

   to the vague concept

   i : refers to the images corresponding to cluster i

   importance of criterion respectively.

   j j

   The weight of query image components lies in closed interval wl , wu where,

   2

   2

   2

   2

   j j j j , j j j j

   wl , wu min T I , 1 F I max T I , 1 F I

   yij :

   yi :

   y j :

   is the number of images in row i and column j in the confusion matrix

   is the number of images in row i in the confusion matrix

   is the number of images in column j in the confusion matrix

   j j

7. EXPERIMENTAL AND RESULTS:

   The WBIIS image database [46] is used to evaluate the ONsIC system. WBIIS image database consists of 10,000 generic images with variety size distributed on 10

   categories of Africa, Beaches, Building, Buses, Dinosaurs, Elephants, Flowers, Foods, Horses and Natural. Several test images were used during experimentation and result as shown in figure (4).

   Fig. 4: Database categories

   The experimental results are discussed under 4 heading:

   1. Transform image to neutrosophic image.

      Figure 5 illustrates three steps to transformation from RGB images to neutrosophic images. First step convert the RGB image to Gray image. A gray image is a simple kind of image that contains one domain, and each pixel in the

      image can be represented by an integer. Second step, an input image is mapped to T and F by the S-function [47] and indeterminacy domain I by homogeneity [48]. There

      are 12 features (4 parameter of GLCM 3 image

      component (T, I and F)) stored in features vectors database for the further processes.

      Fig. 5: Neutrosophic image transform steps.

   2. Neutrosophic image clustering.

      The major advantage of neutrosophic image clustering (NsIC) method based on fuzzy c-means can deal with indeterminate intensity regions effectively and accurately. Another advantage of NsIC is that it can smooth the complex background; therefore, it can prevent the object

      region from connecting with the false foregrounds. Besides the above two major advantages, NsIC finds more accurate object boundaries. Table 1 illustrate example of image clusters.

      Table 1: Example of image clusters.

      Fig. 6: Enhanced within neutrosophic image.

   3. Neutrosophic image cluster enhanced.

      l u Tij Iij

      1 Fij Iij Tij Iij 1 Fij Iij

      KCl , KCl min , , max ,

      2

      2

      2

      The window size 5 5is used for computing the

      ij ij 2

      standard deviation and discontinuity of pixels intensity. Figure 6 illustrates the homogeneity image in domain I.

      The result is:

      L ,U 0.7,0.725 0.57,0.63 0.275,0.3 0.3,0.6

      Clij

      Clij

      0.6,0.675 0.825,0.925 0.25,0.75 0.35,0.75

      Otsu's method [49] is utilized a simple thresholding

      method. The global t optimum threshold is finds that

      minimizes the overlap variance of the background and

      Scoring and accuracy evaluation matrices Sc , Ac are

      computed by:

      c

      Cl

      Cl

      objects within cluster and the overlap variance between

      images between clusters by the following equation:

      S 2 U

      ij

      L

      Clij

      , Ac

      0.5 L

      ij

      U

      Clij

      t c1 t1 t c2 t 2 t

      The result is:

      Where, t is the sum of variances of the two clusters as a t threshold function. i t and ci t are the variance

      0.05 0.12

      S

      c 0.15 0.2

      0.05 0.6

      ,

      1 0.8

      0.7125

      A

      c 0.6375

      0.6

      0.875

      0.2875

      0.5

      0.45

      0.55

      and probability of class i , i=1,2 respectively. Threshold t that results in the minimization of (t) separates the two

      Weights of NsC W NsCl

      are computed by:

      ij

      classes as the foreground and background, respectively.

      W NsCl

      S NsC

      2 1 Ac

      Nscij

   4. Optimization neutrosophic image clustering.

   ij

   The result is:

   c ij 2

   ONsIC technique is applied on neutrosophic set of image

   W N Cl 0.141

   0.186

   0.354

   0.085

   clusters (NsIC)

   i

   Cl

   Ti ,Ii ,Fi

   , i 1,2,…, m

   with

   S ij

   0.159

   0.022

   0.75

   0.415

   ij

   evaluations as neutrosophic components. For example, there are two clusters each cluster including four images components (T, I, F) are represent as:

   The coefficients of the linear programming problem are computed by sum each column inW NsC :

   I

   1

   comp

   2

   I

   comp

   3

   I

   comp

   4

   I

   comp

   W NsClij 0.3

   0.208

   0.396

   0.5

   NS C Cl1

   0.8,0.6,015 0.68,0.46,0.2 0.45,0.1,0.5 0.5,0.4,0.8

   Four neutrosophic query images

   Cl2 0.4,0.8,0.45 0.75,0.9,0.05 1,0.5,1 0.5,0.6.0.9

   QIj , j 1,2,…, n are given as:

   Evaluation of neutrosophic component for NsC

   Tj ,Ij ,Fj

   ENsC K l , K u is computed by from:

   ij ij ij

   imcomp

   imcomp

   imcomp

   imcomp 1

   QI4

   Q1 , Q2 , Q3 , Q4

   w* 0.275

   T,I,F

   0.25,0.3,0.25 0.35,0.6,0.41 0.32,0.55,0.67 0.64,0.98,0.57

   w2 0.475

   For simplifying computation, the neutrosophic set may be written as:

   *

   w3 0.44

   Q

   im comp

   T	0.25	0.35	0.32	0.64
   I	0.3	0.6	0.55	0.98
   F	0.25	0.41	0.67	0.57

   1

   im comp

   Q

   2

   im comp

   Q

   13

   im comp

   Q

   14

   *

   *

   w4 0.81

   Then the degree of suitability that the corresponding cluster

   Cli satisfies the query images can be measured by the

   Weights of query image are computed by formula:

   following function:

   K l K u

   m 1 ij ij

   T I 1 F I

   T I 1 F I

   RCl 2K u K l 2 2 w j

   wl , wu min j j , j j , max j j , j j

   i ij ij

   2 *

   2

   2

   2

   2

   j j

   j 1

   The result is:

   The result is:

   wi , wu 0.275,0.525 0.475,0.595 0.435,0.44 0.705,0.81

   1 0.7125

   1 0.6

   j j R Cl

   0.05 2

   0.275 0.12 2

   0.475

   2

   Thus the linear programming now can be set as:

   1 2

   n n

   0.052 1 0.5 0.44 0.62 1 0.45 0.81 0.21380375

   max

   w Aij

   2

   2

   j 1 i1

   2

   1 0.6375

   2

   1 0.875

   R Cl2 0.15

   0.275 0.2

   2

   0.475

   2

   Subject to:

   l j u

   12 1 0.5 0.44 0.82 1 0.55 0.81 0.61180625

   The result is:

   wj w

   wj

   2

   2

   Maximize :

   subject to :

   0.3w1 0.208w2 0.396w3 0.5w4

   0.275 w1 0.525

   0.475 w2 0.595

   Therefore we can see that the alternative cluster Cl2 is the best choice. And the optimal ranking order of the

   alternatives is given by Cl Cl .

   0.435 w3 0.44 2 1

   0.705 w4 0.81

   *

   The linear programming can easily solve to get w j :

8. Performance results

   Table 2 is show the confusion matrix for NsIC technique, the ten clusters found that represent a good approximation of the ten categories of database image.

   Table 2: Confusion matrix for NsIC.

   The experiment results have proved beyond any doubt that NsIcC technique can be of high-accuracy techniques.

   Where, the accuracy of similar images was gathered in different clusters exceeded 90%, as shown in figure 7.

   1.1

   1

   0.9

   0.8

   Kappa statistic (k)

   0.7

   0.6

   0.5

   0.4

   0.3

   0.2

   0.1

   1 2 3 4 5 6 7 8 9 10

   No. of clusters

   Fig. 7. The accuracy assessment (k) versus different clusters.

   	Recall Precision

   0.98

   0.96

   0.94

   0.92

   Measurs

   0.9

   0.88

   0.86

   0.84

   0.82

   0.8

   0.78

   1 2 3 4 5 6 7 8 9 10

   Cluster number

   Fig. 8. The recall and precision measures versus different clusters.

   The retrieval performance is defined by the precision and recall of the retrieved images [50]. As shown in figure 8 the beach category has been identified with high recall 98%. That is mean in this category the high number of similar images are gathered with very few errors. Whereas the low recall 80.8318% has been identified for house category. That is meaning there are misclassified for some images in this category as a result of the weakness of I set that is working to determine the threshold used in image enhanced. The precision value ranging from 86.9195% to 97.9798% where the highest value belonging to the cluster 8 and lowest for the cluster 9. This means there is a cluster 8 has more than an image that dos not belong to texture category and vice versa in cluster 5.

   The experiments were carried out on a large-scale over the WBIIS image database to measure the effectiveness of NSIC. The results focused on recall and precision measures and accuracy assessment as a criterion for comparison between NsIC technique and other technique namely G. Ciocca et al (2013).

   G. Ciocca et al [51] proposed a new technique for unsupervised image classification based on purely semantic representation not on statistical method. This technique is

   composed of two parts namely supervised feature [52] and unsupervised classification. The idea of supervised features part is based on the hypothesis in the semantic realm regarding categories. That an image d belongs to category ci with certain probability will not determine whether the

   same image belongs to a category c j , but it will modify the probability that it be so. There are supervised feature descriptions are used to classify image regions into

   semantic concept classes such as Classemes [53],

   Prosemantic features [54], Object bank [55], CCA-based features [56], and Primitive features [57]. The second part unsupervised classification algorithm such as k-means based on the output of a limited number of classifiers can be embedded into a feature space by a clustering algorithm.

   G. Ciocca et al is used prosemantic features as supervised feature and k-means algorithm as unsupervised classifier and applied on image database. The results of recall, precision and accuracy assessment indicates that the NsIC technique achieved a high performance compared with the other technique as shown in figures 9, 10 and 11.

   NsIC Other

   1.1

   1

   0.9

   0.8

   Recall

   0.7

   0.6

   0.5

   0.4

   0.3

   0.2

   0.1

   1 2 3 4 5 6 7 8 9 10

   No. of clusters

   Fig. 9. The recall for compared two techniques.

   In general, it is obvious that the NsIC technique has achieved a better precision, recall and accuracy assessment

   (k) at various numbers of clustered images than the other compared technique.

9. CONCLUSION:

This paper presents a new technique to unsupervised classification for images based on neutrosophis sets ad

optimization linear programming. Neutrosophic theory was used to transform the gray image to neutrosophic image components (O, E, B). Indeterminacy set (E) was worked on determine the objects boundaries with high precision. Determining the boundaries of objects accurately blunted the effect of the overlapping problem within the cluster.

1.1

1

0.9

0.8

Precision

0.7

0.6

0.5

0.4

0.3

0.2

0.1

1 2 3 4 5 6 7 8 9 10

No. of cluster

Fig. 10. The precision for compared two techniques.

1.1

1

0.9

0.8

Kappa statistic (k)

0.7

0.6

0.5

0.4

0.3

0.2

0.1

1 2 3 4 5 6 7 8 9 10

No. of clusters

Fig. 11. The k statistic for compared two techniques.

Neutrosophic image clustering method based on fuzzy c- means is used. Neutrosophic image clustering has been enhanced by using the -mean operation which helped on

solve the overlapping problem between clusters. Optimization neutrosophic image clustering is achieved by using the weight coefficient between image clusters and

images category as an object function in linear programming problem. Whereas, the constraints of linear programming problem are the weight limits for query images.

Practical results conducted on neutrosophic image clustering technique has proved its efficiency where it was to obtain the high performance rate in the accuracy of the resulting clusters as well as high values of recall and precision measures.

REFERENCE:

1. Lu, Dengsheng, and Qihao Weng. "A survey of image classification methods and techniques for improving classification performance.", International journal of Remote sensing 28.5 (2007): 823-870.

2. Lee, Te-Won, and Michael S. Lewicki. "Unsupervised image classification, segmentation, and enhancement using ICA mixture models." Image Processing, IEEE Transactions on 11, no. 3 (2002): 270-279.

3. Guyon, Isabelle, and André Elisseeff. "An introduction to variable and feature selection." The Journal of Machine Learning Research 3 (2003): 1157-1182.

4. Saeys, Yvan, Iñaki Inza, and Pedro Larrañaga. "A review of feature selection techniques in bioinformatics." bioinformatics 23.19 (2007): 2507-2517.

5. Omran, Mahamed GH, Andries Petrus Engelbrecht, and Ayed Salman. "Differential evolution methods for unsupervised image classification." Evolutionary Computation, 2005. The 2005 IEEE Congress on. Vol. 2. IEEE, 2005.

6. Deng, Jia, Alexander C. Berg, Kai Li, and Li Fei-Fei. "What does classifying more than 10,000 image categories tell us?" In Computer VisionECCV 2010, pp. 71-84. Springer Berlin Heidelberg, 2010.

7. Yang, Shulin, Liefeng Bo, Jue Wang, and Linda G. Shapiro. "Unsupervised Template Learning for Fine-Grained Object Recognition." In NIPS, pp. 3131-3139. 2012.

8. Murtagh, Fionn, and Pedro Contreras. "Algorithms for hierarchical clustering: an overview." Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery 2.1 (2012): 86-97.

9. Elavarasi, S. Anitha, J. Akilandeswari, and B. Sathiyabhama. "A survey on partition clustering algorithms." learning 1.1 (2011).

10. Meila, Marina, and David Heckerman. "An experimental comparison of several clustering and initialization methods." arXiv preprint arXiv:1301.7401 (2013).

11. Gaidon, Adrien, Zaid Harchaoui, and Cordelia Schmid. "Recognizing activities with cluster-trees of tracklets." BMVC. 2012.

12. Jain, A.K. Murty, M.N. and Flynn, P.J. Data Clustering: A Survey. ACM Computing Surveys, Vol. 31, No. 3, September 1999.

13. Dong, Weisheng, et al. "Sparsity-based image denoising via dictionary learning and structural clustering." Computer Vision and Pattern Recognition (CVPR), 2011 IEEE Conference on. IEEE, 2011.

14. Hanauer, Matthias, and Andreas Koehn. "Perturbative treatment of triple excitations in internally contracted multireference coupled cluster theory." The Journal of chemical physics 136.20 (2012): 204107.

15. Zhao, Z. L., Bo Liu, and Wei Li. "Image clustering based on extreme K-means algorithm." IEIT Journal of Adaptive & Dynamic Computing 2012.1 (2012): 12-16.

16. Celebi, M. Emre, Hassan A. Kingravi, and Patricio A. Vela. "A comparative study of efficient initialization methods for the k-means clustering algorithm." Expert Systems with Applications 40.1 (2013): 200-210.

17. Singh, Shalini S., and N. C. Chauhan. "K-means v/s K-medoids: A Comparative Study." National Conference on Recent Trends in Engineering & Technology. 2011.

18. Kaufman, L. and Rousseeuw, P.J., 1987, Clustering by Means of Medoids, InY. Dodge, editor, Statistical Data Analysis, based on the L1 Norm, pp. 405-416, Elsevier/North Holland, Amsterdam.

19. Mirik, Mustafa, and R. James Ansley. "Comparison of ground- measured and image-classified mesquite (Prosopis glandulosa)

    canopy cover." Rangeland Ecology & Management 65.1 (2012): 85-

    95.

20. Bringmann, Björn, Siegfried Nijssen, and Albrecht Zimmermann. "Pattern-based classification: a unifying perspective." arXiv preprint arXiv:1111.6191 (2011).

21. Voisin, Aurélie, et al. "Classification of very high resolution SAR images of urban areas." (2011).

22. Kriegel, Hans Peter, et al. "Density based clustering." Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery 1.3 (2011): 231-240.

23. Bouveyron, Charles, and Camille Brunet. "Simultaneous model- based clustering and visualization in the Fisher discriminative subspace." Statistics and Computing 22.1 (2012): 301-324.

24. Barros, Rodrigo Coelho, et al. "A survey of evolutionary algorithms for decision-tree induction." Systems, Man, and Cybernetics, Part C: Applications and Reviews, IEE Transactions on 42.3 (2012): 291- 312.

25. Krizhevsky, Alex, Ilya Sutskever, and Geoffrey E. Hinton. "ImageNet Classification with Deep Convolutional Neural Networks." NIPS. Vol. 1. No. 2. 2012.

26. Willems, Thomas F., et al. "Algorithms and tools for high-throughput geometry-based analysis of crystalline porous materials." Microporous and Mesoporous Materials 149.1 (2012): 134-141.

27. Han, J. and Kamber, M. Data Mining: Concepts and Techniques. Morgan Kaufmann Publishers, 2001.

28. Izakian, Hesam, and Ajith Abraham. "Fuzzy C-means and fuzzy swarm for fuzzy clustering problem." Expert Systems with Applications 38.3 (2011): 1835-1838.

29. Zhou, Aimin, et al. "Multiobjective evolutionary algorithms: A survey of the state of the art." Swarm and Evolutionary Computation 1.1 (2011): 32-49.

30. Dowsland, Kathryn A., and Jonathan M. Thompson. "Simulated annealing." Handbook of Natural Computing. Springer Berlin Heidelberg, 2012. 1623-1655.

31. Maji, Pabitra Kumar. "Neutrosophic soft set." Annals of Fuzzy Mathematics and Informatics 5.1 (2013): 2287-623.

32. Hromkovic, Juraj. Algorithmics for hard problems: introduction to combinatorial optimization, randomization, approximation, and heuristics. Springer-Verlag, 2010.

33. Smarandache, Florentin. Introduction to Neutrosophic Measure, Neutrosophic Integral, and Neutrosophic Probability. 2013.

34. Jiang, Yuncheng, Yong Tang, and Qimai Chen. "An adjustable approach to intuitionistic fuzzy soft sets based decision making." Applied Mathematical Modelling 35.2 (2011): 824-836.

35. Temme, Nico M. Special functions: An introduction to the classical functions of mathematical physics. John Wiley & Sons, 2011.

36. Fiss, Peer C. "Building better causal theories: A fuzzy set approach to typologies in organization research." Academy of Management Journal 54.2 (2011): 393-420.

37. Weber, Zach. "Transfinite numbers in paraconsistent set theory." The Review of Symbolic Logic 3.01 (2010): 71-92.

38. Smarandache, Florentin. "Neutrosophic Logic-A Generalization of the Intuitionistic Fuzzy Logic." Multispace & Multistructure. Neutrosophic Transdisciplinarity (100 Collected Papers of Science) 4 (2010): 396.

39. Mohan, J., V. Krishnaveni, and Yanhui Guo. "A neutrosophic approach of MRI denoising." Image Information Processing (ICIIP), 2011 International Conference on. IEEE, 2011.

40. Faber, Carel, and Rahul Pandharipande. "Tautological and non- tautological cohomology of the moduli space of curves." arXiv preprint arXiv:1101.5489 (2011).

41. Kumar, Amit, Jagdeep Kaur, and Pushpinder Singh. "A new method for solving fully fuzzy linear programming problems." Applied Mathematical Modelling 35.2 (2011): 817-823.

42. Maji, Pabitra Kumar. "Neutrosophic soft set." Annals of Fuzzy Mathematics and Informatics 5.1 (2013): 2287-623.

43. Zhang, Ming, Ling Zhang, and H. D. Cheng. "A neutrosophic approach to image segmentation based on watershed method." Signal Processing 90.5 (2010): 1510-1517.

44. Majumdar, Pinaki, and Syamal Kumar Samant. "On similarity and entropy of neutrosophic sets." Journal of Intelligent and Fuzzy Systems (2013).

45. Lillesand, T.M. and Kiefer, R.W. 2000. Remote sensing and image interpretation. John Wiley & Sons, Inc., New York, NY, USA.

46. Jia Li and James Z. Wang, “Real-time Computerized Annotation of Pictures,'' IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 30, no. 6, pp. 985-1002, 2008.

47. Zhang, Ming, Ling Zhang, and H. D. Cheng. "A neutrosophic approach to image segmentation based on watershed method." Signal Processing 90.5 (2010): 1510-1517.

48. Truc, Phan Tran Ho, et al. "Homogeneity-and density distance-driven active contours for medical image segmentation." Computers in biology and medicine 41.5 (2011): 292-301.

49. Farrahi Moghaddam, Reza, and Mohamed Cheriet. "AdOtsu: An adaptive and parameterless generalization of Otsu's method for document image binarization." Pattern Recognition 45.6 (2012): 2419-2431.

50. Viitaniemi, Ville, & Laaksonen, Jorma (2007). Evaluating the performance in automatic image annotation: Example case by adaptive fusion of global image features. Signal Processing: Image Communication, 22(6), 557568.

51. G. Ciocca et al., On the use of supervised features for unsupervised image categorization: An evaluation, Comput. Vis. Image Understand. (2014), http://dx.doi.org/10.1016/j.cviu.2014.01.010

52. Martínez Sotoca, José, and Filiberto Pla. "Supervised feature selection by clustering using conditional mutual information-based distances." Pattern Recognition 43.6 (2010): 2068-2081.

53. Torresani, Lorenzo, Martin Szummer, and Andrew Fitzgibbon. "Efficient object category recognition using classemes." Computer VisionECCV 2010. Springer Berlin Heidelberg, 2010. 776-789.

54. Ciocca, Gianluigi, et al. "Halfway through the semantic gap: prosemantic features for image retrieval." Information Sciences 181.22 (2011): 4943-4958.

55. Li, Li-Jia, et al. "Object Bank: A High-Level Image Representation for Scene Classification & Semantic Feature Sparsification." NIPS. Vol. 2. No. 3. 2010.

56. Golugula, Abhishek, et al. "Supervised Regularized Canonical Correlation Analysis: integrating histologic and proteomic measurements for predicting biochemical recurrence following prostate surgery." BMC bioinformatics 12.1 (2011): 483.

57. Rizoiu, Marian-Andrei, Julien Velcin, and Stéphane Lallich. "Unsupervised feature construction for improving data representation and semantics." Journal of Intelligent Information Systems 40.3 (2013): 501-527