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\title{Studies on Image Segmentation Method Based On a New Symmetric Mixture Model with -K Means}
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             \author[1]{Dr. Seshashayee  M.}

             \author[2]{Srinivasa Rao  P.}

             \affil[1]{  GITAM UNiversity}

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\date{\small \em Received: 8 August 2011 Accepted: 7 September 2011 Published: 22 September 2011}

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\begin{abstract}
        


In this paper we propose an image segmentation algorithm based on new-symmetric mixture model. Here the pixel intensities of the whole image are characterized through a newsymmetric mixture distribution, such that the statistical characteristics of the image coincide with that of the new symmetric distribution. Using the K-Means algorithm the number of image regions and initial estimates of the model parameters for the EM algorithm are obtained. The segmentation algorithm is proposed by component maximum likelihood under Bayesian frame work. The efficiency of the proposed method is studied with the five images taken from the Berkeley image dataset and computing the values image segmentation measures like global consistency error, probabilistic rand index and variation of information. A comparative study of the proposed model with Gaussian mixture model reveals that the proposed method performs better. The efficiency of the proposed method with respect to the image retrieval is also studied.

\end{abstract}


\keywords{Image Segmentation, New Symmetric Mix- ture Model, Image Quality Metrics, Kmeans algorithm, EM algorithm.}

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\let\tabcellsep& 	 	 		 
\section[{INTRODUCTION}]{INTRODUCTION}\par
mage segmentation is a preprocessing step in image analysis and understanding. Much work has been reported in literature regarding image segmentation. Pal S.K. and Pal N.R (1993), \hyperref[b7]{Jahne (1995)}, \hyperref[b1]{Cheng et al (2001)}, Mantas Paulinas and Audrius Usinskas  {\ref (2007)} and Shital \hyperref[b19]{Raut et al (2009)} have discussed various image segmentation methods.\par
The image segmentation methods are usually classified into three categories namely (i) segmentation methods based on histogram, threshold and edge based techniques, (ii) model based image segmentation methods and (iii) image segmentation based on other methods like graph, saddle point, neural networks, fuzzy logic etc., \hyperref[b0]{(Caillol H. et al (1993)} In Gaussian mixture model the whole image is characterized by the collection of several image regions, where each region is characterized by a Gaussian distribution. That is the pixel intensities in each image region follow a Gaussian distribution. This Gaussian assumption serves well only when the pixel intensities in each image region are meso-kurtic and symmetric. But in some images like natural scenes the pixel intensities of the image region may not be meso-kurtic even though they are symmetric. Hence to have an accurate analysis of the images, it is needed to develop image segmentation methods based on Non-Gaussian mixture models.\par
In Non-Gaussian symmetric mixture models the kurtosis plays a dominant role. Based on the kurtosis the Non-Gaussian models can be classified into two categories platy-kurtic and lepto-kurtic. In general many of the natural scenes will have image regions having platy-kurtic nature. That is the kurtosis of the pixel intensities in the image regions is less than three. One such model available in literature is new-symmetric distribution given by Srinivasa Rao K. et al  {\ref (1997)}. The new-symmetric distribution is having kurtosis 2.52 and symmetric. So it is a platy-kurtic distribution. Hence to have an efficient image segmentation algorithm for images having platy-kurtic distributed pixel intensities in the image regions, we develop and analyze an image segmentation algorithm based on new-symmetric mixture model.\par
For developing the image segmentation algorithm we require the number of components in the image. This is obtained from K-means algorithm. The initial estimates of the model parameters are obtained from the moment estimates. The updated equations for estimating the model parameters through the EM algorithm are derived. The segmentation algorithm is also presented by taking component maximum likelihood. The efficiency of the proposed algorithm is studied through experimentation. intensity z = f(x , y) is a random variable, because of the fact that the brightness measured at a point in the image is influenced by various random factors like vision, lighting, moisture, environmental conditions etc,. To model the pixel intensities of the image region it is assumed that the pixel intensities of the region follow a new symmetric distribution given by Srinivasa Rao K. et al.,  {\ref (1997)}. The probability density function of the pixel intensity is \hyperref[b0]{(1)} The probability curve of new symmetric distribution is shown in Figure  {\ref 1}. 
\section[{Its central moments are and}]{Its central moments are and}\par
(2)\par
The kurtosis of the distribution is\par
where, K is number of regions , 0 ? i ? ? 1 are weights such that ? i ? = 1 and is as given in equation \hyperref[b0]{(1)}. i ? is the weight associated with ith region in the whole image.\par
In general the pixel intensities in the image regions are statistically correlated and these correlations can be reduced by spatial sampling (Lie.T and Sewehand. W( 1992 ) ) or spatial averaging  {\ref ( Kelly P.A. et al.( 1998 )} ) . After reduction of correlation the pixels are considered to be uncorrelated and independent. The mean pixel intensity of the whole image is . III. II. 
\section[{ESTIMATION OF THE MODEL PARAMETERS BY EM ALGORITHM}]{ESTIMATION OF THE MODEL PARAMETERS BY EM ALGORITHM} 
\section[{FINITE MIXTURE OF NEW SYMMETRIC DISTRIBUTION}]{FINITE MIXTURE OF NEW SYMMETRIC DISTRIBUTION}\par
In low level image analysis the entire image is considered as a union of several image regions. In each image region the image data is quantized by pixel intensities. For a given point (pixel) (x , y), the pixel2 2 2 1 2 2 ( , , ) , , , 0 3 2 z z e f Z Z µ µ ? ? µ ? µ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? = ?? < < ? ?? < < ? > Figure 1 : Probability curve of new symmetric distribution 2 3 1 ( ) ( ) 2 2 2 2 (3 / 2) n n n n n µ ? ? ? ? + Î?" + ? ? = ? ? ? ? ? ? 2 1 0 n µ + = 2.52 2 ? = 2 ( ) ( / ) , 1 K p z f z i i i i i ? µ ? = ? = 2 ( , , ) i f z µ ? ( ) 1 K E Z i i i ? µ = ? = . N ( ) 1 ( ) ( , ) l s L p z s ? ? = = ? . N 1 ( ) ( , )\textbf{1}K L f z s i i i s ? ? ? = ? ? = ? ? ? = ? ? ? log ( ) log ( , )\textbf{1}\par
1N K L f z s i i i s i ? ? ? ? ? = ? ? ? ? = = ? ? , 2\par
( , , ; 1, 2,..., )i i i i K ? µ ? ? = = 2 1 2 2 2 log ( ) log 3 2 1 1 s z i z s i i e i i N K L s i i µ ? µ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + = ? ? = = ( ) ( ) ; l Q ? ? [ ] ( ) log ( ) / l E L z ? ? ( ) ( ) (\textbf{)}( ) ( )\textbf{1 1}\par
;\par
( , ) log ( , ) logK N l l i s i s i i s Q t z f z ? ? ? ? ? = = = + ??\textbf{(6)}\par
The updated equation of ? for ( l +1) th th estimate is ( )1 ( ) 1 1 ( , ) N l l i i s s t z N ? ? + = = ?\par
Global Journal of Computer Science and Technology Volume XI Issue XVII Version I 52 2011 
\section[{October}]{October}\par
The entire image is a collection of regions which are characterized by new symmetric distribution. Here, it is assumed that the pixel intensities of the whole image follow a K -component mixture of new symmetric distribution and its probability density function is of the form. 
\section[{The first step of the EM algorithm requires the estimation of the likelihood function of the sample observations. function of the sample is}]{The first step of the EM algorithm requires the estimation of the likelihood function of the sample observations. function of the sample is}\par
The expectation of the log likelihood \hyperref[b8]{(7)} The updated equation of ? at ( l +1) th iteration is where, =\par
The updated equation of2 i ? at ( l +1) th iteration is\textbf{(9)}\par
where IV. 
\section[{INITIALIZATION OF THE PARAMETERS BY K -MEANS}]{INITIALIZATION OF THE PARAMETERS BY K -MEANS}\par
The efficiency of the EM algorithm in estimating the parameters is heavily dependent on the number of regions in the image. The number of mixture components initially taken for K -Means algorithm is by plotting the histogram of the pixel intensities of the whole image. The number of peaks in the histogram can be taken as the initial value of the number of regions K.\par
The mixing parameters   {\ref (2000)}. This method performs well if the sample size is large and its computational time is heavily increased. When the sample size is small, some small regions may not be sampled. To overcome this problem we use the K -Means algorithm to divide the whole image into various homogeneous regions. In K -Means algorithm the of the clusters are recomputed as soon as the pixel joins a cluster.\par
After determining the final values of K (number of regions) , we obtain the initial estimates of 2 , i i µ ? and i ? for the i th region using the segmented region pixel intensities with the method given by Srinivasa Rao et al.,  {\ref (1997)} for new symmetric distribution .The initial estimate i ? is taken as1 K i ? =\par
, where i = 1,2,...,K.  V. 
\section[{The parameters}]{The parameters} 
\section[{SEGMENTATION ALGORITHM}]{SEGMENTATION ALGORITHM}\par
In this section, we present the image segmentation algorithm. After refining the parameters the prime step in image segmentation is allocating the pixels to the segments of the image. This operation is performed by Segmentation Algorithm. The image segmentation algorithm consists of four steps.\par
Step 1) Plot the histogram of the whole image.\par
Step 2) Obtain the initial estimates of the model parameters using K-Means algorithm and moment estimators as discussed in section 4\par
Step 3) Obtain the refined estimates of the model parameters 2 , i i µ ? and i ? for i=1,2,...,K by using the EM algorithm with the updated equations\par
Step 4) Assign each pixel into the corresponding j th region (segment) according to the maximum likelihood of the j th component L j.\par
That is , VI. 
\section[{EXPERIMENTAL RESULTS}]{EXPERIMENTAL RESULTS}\par
To demonstrate the utility of the image segmentation algorithm developed in this chapter, an experiment is conducted with five images taken from Berkeley images dataset (http://www.eecs.berkeley .= ( ) ( ) ( ) ( ) 1 1 ( , ) 1 ( , ) l l N i i s K l l s i i s i f z N f z ? ? ? ? = = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ( ) ( ) 2( ) ( ) ( ) ( ) 2 2( ) ( ) 1 1 ( 1) ( ) 1 2 ( , )\textbf{( , ) 2 ( ,}\par
)   The initial estimates of the number of the regions K in each image are obtained and given in Table \hyperref[tab_0]{1}. From Table \hyperref[tab_0]{1}, we observe that the image HORSE has two segments, images TOWER and BIRD have three segments each and images  {\ref MAN}           l l N N i s i l l s i s i s l l s s i s i l i N l i s s z z t z t z z t z ? µ ? ? ? µ µ ? = = + = ? ? ? ? ? ? ? ? + ? ? ? = ? ? ? ( ) ( , ) l i s t z ? ( ) ( ) ( ) ( 1) ( )\textbf{2}( ) ( 1) ( )\textbf{2 1 ( , , ) ( , )}l l l i i s i i K l l l i i s i i i f z f z ? µ ? ? µ ? + + = ?\textbf{(8) ( ) ( ) ( ) ( ) ( )}( ) 2 ( 1) 2 ( ) 2 2 2( )\textbf{( 1) 1 ( 1) 2 (}) 1 1 2 ( ) ( , ) 2 2 ( , ) i l N i l l s i i s l l s i s l i N l i s s z t z z t z ? µ ? ? µ ? ? + + = + = ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? = ? ? ( ) ( ) ( ) ( 1) ( 1) 2 ( ) ( ) ( 1)\textbf{( 1) 2 1 ( , , ) ( , ) ( , , )}l l l i i s i i l i s K l l l i i s i i i f z t z f z ? µ ? ? ? µ ? + + + + = = ? 2 1 j 2 1 2 L (3 2 ) 2 max s j j s j j j k j z z e µ ? µ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? , , s z ?? < < ? , 0 j j µ ? ?? < < ? > edu/? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + + ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? ? ? + ( )\textbf{2}? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? = ? ? ? ? + ? ? ? ? ? ? ? ?\textbf{2}? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? 32.7780)(3) 2? ( )\textbf{2}? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? = ? ? ? ? + ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ?\par
The estimated probability density function of the pixel intensities of the image BOAT is  ( )\textbf{2 ( ) 2 2}? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? = ? ? ? ? + ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? 2 73 (29.6973)(3) 2? ? ? ? ? ? ? ? ?\par
The estimated probability density function of the pixel intensities of the image TOWER is    ( )\textbf{2 ( ) 2 2}? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? = ? ? ? ? + ? ? ? ? ? ? ? ? 
\section[{PERFORMANCE EVALUTION}]{PERFORMANCE EVALUTION}\par
After conducting the experiment with the image segmentation algorithm developed in this chapter, its performance is studied. The performance evaluation of the segmentation technique is carried by obtaining the three performance measures namely, (i) Probabilistic Rand Index (PRI), (ii) Variation Of Information (VOI) and (iii) Global Consistence Error (GCE). The performance of developed algorithm using finite new symmetric distribution mixture model (NSMM-K) is studied by computing the segmentation performance measures namely PRI, GCE, and VOI for the five images under study. The computed values of the performance measures for the developed algorithm and the earlier existing finite Gaussian mixture model(GMM) with K-Means algorithm are presented in Table \hyperref[tab_6]{3} for a comparative study. From table 3 it is observed that the PRI values of the proposed algorithm for the five images considered for experimentation are less than that of the values from the segmentation algorithm based on finite Gaussian mixture model with K-means. Similarly GCE and VOI values of the proposed algorithm are less than that of Finite Gaussian Mixture Model. This reveals that the proposed algorithm outperforms the existing algorithm based on the finite Gaussian mixture model. When the kurtosis parameter of each component of the model is zero, the model reduces to finite Gaussian mixture model and even in this case the algorithm performs well.\par
After developing the image segmentation method it is needed to verify the utility of segmentation in model building of the image for image retrieval.The performance evaluation of the retrieved image can be done by subjective image quality testing or by objective image quality testing. The objective image quality testing methods are often used since the numerical results of an objective measure allows a consistent comparison of different algorithms. There are several image quality measures available for performance evaluation of the image segmentation method. An extensive survey of quality measures is given by Eskicioglu A.M. and Fisher P.S. (1995). For the performance evaluation of the developed segmentation algorithm, we consider the image quality measures like average difference, maximum distance, image fidelity, mean square error, signal to noise ratio and image quality index.\par
Using the estimated probability density functions of the images under consideration the retrieved images are obtained and are shown in Figure \hyperref[fig_19]{4}.  From the Table \hyperref[tab_7]{4}, it is observed that all the image quality measures for the five images are meeting the standard criteria. This implies that using the proposed algorithm the images are retrieved accurately. A comparative study of proposed algorithm with that of algorithm based on Finite Gaussian Mixture Model reveals that the MSE of the proposed model is less than that of the finite Gaussian mixture model. Based on all other quality metrics also it is observed that the performance of the proposed model in retrieving the images is better than the finite Gaussian mixture model. 
\section[{VIII.}]{VIII.} 
\section[{CONCLUSION}]{CONCLUSION}\par
An image segmentation algorithm based on new symmetric mixture model with K-means is developed and evaluated. This algorithm is more suitable for the images having platy-kurtic image regions. The new symmetric mixture model is capable of characterizing several natural images with kurtosis close to 2.52. The updated equations of the model parameters are derived through EM algorithm under Bayesian framework. The estimated probability density function of the pixel intensities in the whole image is useful for the image retrieval. The experimental results revealed that the proposed method out performs the existing Gaussian mixture model in both image segmentation and image retrieval.\begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-2.png}
\caption{\label{fig_0}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-3.png}
\caption{\label{fig_1}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2}\includegraphics[]{image-4.png}
\caption{\label{fig_2}i ? and the model parameters µ i , 2 i?}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2}\includegraphics[]{image-5.png}
\caption{\label{fig_3}i µ and 2 i?}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-6.png}
\caption{\label{fig_4}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2011}\includegraphics[]{image-7.png}
\caption{\label{fig_5}2011 October.}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2}\includegraphics[]{image-8.png}
\caption{\label{fig_6}Figure 2 .}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2}\includegraphics[]{image-9.png}
\caption{\label{fig_7}Figure 2 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-10.png}
\caption{\label{fig_8}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-11.png}
\caption{\label{fig_17}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{3}\includegraphics[]{image-12.png}
\caption{\label{fig_18}Figure 3 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{4}\includegraphics[]{image-13.png}
\caption{\label{fig_19}Figure 4 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{1} \par 
\begin{longtable}{P{0.584375\textwidth}P{0.017708333333333333\textwidth}P{0.017708333333333333\textwidth}P{0.017708333333333333\textwidth}P{0.19479166666666664\textwidth}P{0.017708333333333333\textwidth}}
\multicolumn{4}{l}{IMAGE HORSE MAN BIRD}\tabcellsep \multicolumn{2}{l}{BOAT TOWER}\\
Estimate of K\tabcellsep 2\tabcellsep 4\tabcellsep 3\tabcellsep 4\tabcellsep 3\end{longtable} \par
 
\caption{\label{tab_0}Table 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{:} \par 
\begin{longtable}{P{0.7681175430844012\textwidth}P{0.0003756076005302696\textwidth}P{0.0003756076005302696\textwidth}P{0.03605832965090588\textwidth}P{0.013146266018559434\textwidth}P{0.010517012814847548\textwidth}P{0.020658418029164825\textwidth}P{0.0003756076005302696\textwidth}P{0.0003756076005302696\textwidth}}
\tabcellsep \tabcellsep \tabcellsep \tabcellsep \multicolumn{2}{l}{Table : 2.d}\\
\multicolumn{7}{l}{Estimated Values of the Parameters for BOAT Image}\\
\multicolumn{7}{l}{2.b Estimated Values of the Parameters for MAN Image Number of Image Regions (K =4) Parameters Estimation of Initial Parameters Estimation of Final Parameters by EM Algorithm Regions(i) Regions(i) 1 2 1 2 i ? 1/2 1/2 0.39702 0.60298 i µ 121.47 187.91 134.09 184.97 2 i ? 609.82 426.21 1302.8 561.41 Estimation of Initial Parameters Estimation of Final Parameters by EM Algorithm Regions(i) Regions(i) 1 2 3 4 1 2 3 4 1/4 1/4 1/4 1/4 0.24315 0.2306 0.34648 0.17977 63.5 20.234 184.29 106.38 64.541 23.197 183.65 103.01 190.98 165.05 547.54 361.45 497.03 214.15 509.25 1074.40 Number of Image Regions (K =4) Estimated Values of The Parameters For TOWER Image Param eters i ? i µ 2 i ? Number of Image Regions (K =3) Substituting the final estimates of the model shown in parameters, the probability density function of pixel intensities of each image are estimated. The estimated probability density function of the pixel intensities of the image HORSE is The estimated probability density function of the pixel intensities of the image MAN is Estimation of Initial Parameters Estimation of Final Parameters by EM Algorithm Parameters Regions(i) Regions(i) 1 2 3 4 1 2 3 4 i ? 1/4 1/4 1/4 1/4 0.2570 0.24231 0.28458 0.22741 i µ 34.98 216.5 81.146 131.13 41.008 212.7 81.062 128.11 2 i ? 374.1 657.54 259.39 387.02 636.2 699.25 785.09 881.93 Table : 2.e Parameters Estimation of Initial Parameters Estimation of Final Parameters by EM Algorithm Regions(i) Regions(i) 1 2 3 1 2 3 i ? 1/3 1/3 1/3 0.43267 0.051312 0.51602 i µ 55.663 223.75 107.79 60.79 193.31 104.42 2 i ? 276.53 1082.4 297.62 487.89 3140.4 404.79 ( ) ( ) 2 2 ( ) 1 2 2 134.09 1 2 36.0943 184.97 1 2 23.6941 134.09 36.0943 184.97 23.6941 2 2 (0.39702) , (36.0943)(3) 2 (0.60298) Table : 2.are (23.6941)(3) 2}\\
\multicolumn{7}{l}{Estimated Values of the Parameters for BIRD Image}\\
\tabcellsep \tabcellsep \tabcellsep \multicolumn{4}{l}{Number of Image Regions (K =3)}\\
\tabcellsep \tabcellsep \tabcellsep \multicolumn{3}{l}{Estimation of Initial}\tabcellsep Estimation of Final\\
\multicolumn{3}{l}{Parameters}\tabcellsep \tabcellsep Parameters\tabcellsep \tabcellsep Parameters by EM\\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep Algorithm\\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep Regions(i)\tabcellsep \tabcellsep Regions(i)\\
\tabcellsep \tabcellsep \tabcellsep 1\tabcellsep 2\tabcellsep 3\tabcellsep 1\tabcellsep 2\tabcellsep 3\\
\multicolumn{2}{l}{?}\tabcellsep i\tabcellsep 1/3\tabcellsep 1/3\tabcellsep \multicolumn{2}{l}{1/3 0.13161 0.66786 0.20053}\\
µ\tabcellsep \multicolumn{2}{l}{i}\tabcellsep \multicolumn{4}{l}{53.491 124.05 124.05 60.691 192.85 129.81}\\
\multicolumn{7}{l}{2 i The estimated probability density function of the pixel ? 535.4 513.93 513.93 857.07 86.799 1581.2}\\
intensities of the image BIRD is\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \\
© 2011 Global Journals Inc. (US)\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \end{longtable} \par
  {\small\itshape [Note: c Global Journal of Computer Science and Technology Volume XI Issue XVII Version I 54 2011 October different parameters. The number of segments in each of the five images considered for experimentation is determined by the histogram of pixel intensities. The histograms of the pixel intensities of the five images]} 
\caption{\label{tab_1}Table :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{3} \par 
\begin{longtable}{P{0.08865030674846625\textwidth}P{0.7248466257668711\textwidth}P{0.028680981595092022\textwidth}P{0.007822085889570552\textwidth}}
\tabcellsep \multicolumn{3}{l}{PERFORMACE}\\
\multicolumn{2}{l}{IMAGES METHOD}\tabcellsep MEASURES\\
\tabcellsep PRI\tabcellsep GCE\tabcellsep VOI\\
HORSE\tabcellsep \multicolumn{3}{l}{GMM NSMM-K 0.9283 0.1634 1. 8403 0.9142 0.1737 1.8643}\\
MAN\tabcellsep \multicolumn{3}{l}{GMM NSMM-K 0.9342 0.1734 1.7875 0.9228 0.3107 1. 8389}\\
BIRD\tabcellsep \multicolumn{3}{l}{GMM NSMM-K 0.9140 0.1352 1.7259 0.9106 0.1369 1. 7479}\\
BOAT\tabcellsep \multicolumn{3}{l}{GMM NSMM-K 0.9174 0.6483 1.7542 0.9026 0.6485 1. 7882}\\
TOWER\tabcellsep \multicolumn{3}{l}{GMM NSMM-K 0.9246 0.0981 1.7988 0.9102 0.1090 1. 8643}\end{longtable} \par
 
\caption{\label{tab_6}Table 3 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{4} \par 
\begin{longtable}{P{0.016857360793287566\textwidth}P{0.35205949656750574\textwidth}P{0.1173531655225019\textwidth}P{0.13161708619374524\textwidth}P{0.23211289092295959\textwidth}}
IMAGE\tabcellsep Quality Metrics\tabcellsep FGMM\tabcellsep FNSDMM\tabcellsep Standard Limits\\
\tabcellsep \tabcellsep \tabcellsep with K-Means\tabcellsep \\
\tabcellsep Average Difference\tabcellsep 0.5011\tabcellsep 0.44135\tabcellsep Close to 1\\
\tabcellsep Maximum Distance\tabcellsep 1.0000\tabcellsep 1.0000\tabcellsep Close to 1\\
HORSE\tabcellsep Image Fidelity Mean Square Error\tabcellsep 1.0000 0.5011\tabcellsep 1.0000 0.4414\tabcellsep Close to 1 Close to 0\\
\tabcellsep Signal to Noise Ratio\tabcellsep 5.6542\tabcellsep 5.9301\tabcellsep As big as possible\\
\tabcellsep Image Quality Index\tabcellsep 1.0000\tabcellsep 1.0000\tabcellsep Close to 1\\
\tabcellsep Average Difference\tabcellsep 0.4858\tabcellsep 0.50021\tabcellsep Close to 1\\
\tabcellsep Maximum Distance\tabcellsep 1.0000\tabcellsep 1.0000\tabcellsep Close to 1\\
MAN\tabcellsep Image Fidelity Mean Square Error\tabcellsep 1.0000 0.4995\tabcellsep 1.0000 0.5079\tabcellsep Close to 1 Close to 0\\
\tabcellsep Signal to Noise Ratio\tabcellsep 5.6828\tabcellsep 5.6251\tabcellsep As big as possible\\
\tabcellsep Image Quality Index\tabcellsep 1.0000\tabcellsep 1.0000\tabcellsep Close to 1\\
\tabcellsep Average Difference\tabcellsep 0.4939\tabcellsep 0.6573\tabcellsep Close to 1\\
\tabcellsep Maximum Distance\tabcellsep 1.0000\tabcellsep 1.0000\tabcellsep Close to 1\\
BIRD\tabcellsep Image Fidelity\tabcellsep 1.0000\tabcellsep 1.0000\tabcellsep Close to 1\\
\tabcellsep Mean Square Error\tabcellsep 0.8590\tabcellsep 0.5050\tabcellsep Close to 0\\
\tabcellsep Signal to Noise Ratio\tabcellsep 5.6861\tabcellsep 4.4842\tabcellsep As big as possible\\
\tabcellsep Image Quality Index\tabcellsep 1.000\tabcellsep 1.0000\tabcellsep Close to 1\\
\tabcellsep Average Difference\tabcellsep 0.5039\tabcellsep 0.6217\tabcellsep Close to 1\\
\tabcellsep Maximum Distance\tabcellsep 1.0000\tabcellsep 1.0000\tabcellsep Close to 1\\
BOAT\tabcellsep Image Fidelity Mean Square Error\tabcellsep 1.0000 0.7931\tabcellsep 1.0000 0.5070\tabcellsep Close to 1 Close to 0\\
\tabcellsep Signal to Noise Ratio\tabcellsep 5.6318\tabcellsep 4.6573\tabcellsep As big as possible\\
\tabcellsep Image Quality Index\tabcellsep 1\tabcellsep 1.0000\tabcellsep Close to 1\\
\tabcellsep Average Difference\tabcellsep 0.4936\tabcellsep 0.6640\tabcellsep Close to 1\\
\tabcellsep Maximum Distance\tabcellsep 1.0000\tabcellsep 1.0000\tabcellsep Close to 1\\
\tabcellsep Image Fidelity\tabcellsep 0.9999\tabcellsep 0.9999\tabcellsep Close to 1\\
TOWER\tabcellsep Mean Square Error\tabcellsep 0.8788\tabcellsep 0.5076\tabcellsep Close to 0\\
\tabcellsep Signal to Noise Ratio\tabcellsep 5.6870\tabcellsep 4.4347\tabcellsep As big as possible\\
\tabcellsep Image Quality Index\tabcellsep 1.0000\tabcellsep 1.0000\tabcellsep Close to 1\end{longtable} \par
 
\caption{\label{tab_7}Table 4 :}\end{figure}
 			\footnote{© 2011 Global Journals Inc. (US)} 			\footnote{Studies on Image Segmentation Method Based On a New Symmetric Mixture Model with K -Means © 2011 Global Journals Inc. (US)} 			\footnote{OctoberStudies on Image Segmentation Method Based On a New Symmetric Mixture Model with K -Means} 			\footnote{© 2011 Global Journals Inc. (US) Studies on Image Segmentation Method Based On a New Symmetric Mixture Model with K -Means} 		 		\backmatter  			  				\begin{bibitemlist}{1}
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