Studies on Image Segmentation Method Based On a New Symmetric Mixture Model with -K Means

Table of contents

1. INTRODUCTION

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), Jahne (1995), Cheng et al (2001), Mantas Paulinas and Audrius Usinskas (2007) and Shital Raut et al (2009) have discussed various image segmentation methods.

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., (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.

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 (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.

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., (1997). The probability density function of the pixel intensity is (1) The probability curve of new symmetric distribution is shown in Figure 1.

2. Its central moments are and

(2)

The kurtosis of the distribution is

where, K is number of regions , 0 ? i ? ? 1 are weights such that ? i ? = 1 and is as given in equation (1). i ? is the weight associated with ith region in the whole image.

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 ( 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.

3. ESTIMATION OF THE MODEL PARAMETERS BY EM ALGORITHM

4. FINITE MIXTURE OF NEW SYMMETRIC DISTRIBUTION

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 pixel

2 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 ( ) ( , )1K L f z s i i i s ? ? ? = ? ? = ? ? ? = ? ? ? log ( ) log ( , )1

1

N K L f z s i i i s i ? ? ? ? ? = ? ? ? ? = = ? ? , 2

( , , ; 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 ? ? ( ) ( ) ()( ) ( )1 1

;

( , ) log ( , ) log

K N l l i s i s i i s Q t z f z ? ? ? ? ? = = = + ??(6)

The updated equation of ? for ( l +1) th th estimate is ( )

1 ( ) 1 1 ( , ) N l l i i s s t z N ? ? + = = ?

Global Journal of Computer Science and Technology Volume XI Issue XVII Version I 52 2011

5. October

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.

6. The first step of the EM algorithm requires the estimation of the likelihood function of the sample observations. function of the sample is

The expectation of the log likelihood (7) The updated equation of ? at ( l +1) th iteration is where, =

The updated equation of

2 i ? at ( l +1) th iteration is(9)

where IV.

7. INITIALIZATION OF THE PARAMETERS BY K -MEANS

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.

The mixing parameters (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.

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., (1997) for new symmetric distribution .The initial estimate i ? is taken as

1 K i ? =

, where i = 1,2,...,K. V.

8. The parameters

9. SEGMENTATION ALGORITHM

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.

Step 1) Plot the histogram of the whole image.

Step 2) Obtain the initial estimates of the model parameters using K-Means algorithm and moment estimators as discussed in section 4

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

Step 4) Assign each pixel into the corresponding j th region (segment) according to the maximum likelihood of the j th component L j.

That is , VI.

10. EXPERIMENTAL RESULTS

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 ( , )( , ) 2 ( ,

) The initial estimates of the number of the regions K in each image are obtained and given in Table 1. From Table 1, we observe that the image HORSE has two segments, images TOWER and BIRD have three segments each and images 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) ( )2( ) ( 1) ( )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 ? + + = ?(8) ( ) ( ) ( ) ( ) ( )( ) 2 ( 1) 2 ( ) 2 2 2( )( 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)( 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/? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + + ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? ? ? + ( )2? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? = ? ? ? ? + ? ? ? ? ? ? ? ?2? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? 32.7780)(3) 2? ( )2? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? = ? ? ? ? + ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ?

The estimated probability density function of the pixel intensities of the image BOAT is

( )2 ( ) 2 2? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? = ? ? ? ? + ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? 2 73 (29.6973)(3) 2? ? ? ? ? ? ? ? ?

The estimated probability density function of the pixel intensities of the image TOWER is

( )2 ( ) 2 2? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? = ? ? ? ? + ? ? ? ? ? ? ? ?

11. PERFORMANCE EVALUTION

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 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.

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.

Using the estimated probability density functions of the images under consideration the retrieved images are obtained and are shown in Figure 4. From the Table 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.

12. VIII.

13. CONCLUSION

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.

Figure 1.
,Tolias Y.A. and Pamas S.M (1998), Brun L. (1998), Xu Y. et al (1998)). Among these methods model based image segmentation is more efficient since it preserves the neighborhood information and characterizes the features of the image region more accurately. Hence much emphasis is given for image segmentation based on finite Gaussian mixture model (Yamazaki et .al(1998) , Lie T. et al(1993), Zhang Z.H. et al (2003) and Nasios N. et al(2006)).
Figure 2.
In this section we derive the updated equations of the model parameters using Expectation Maximization (EM) algorithm. The likelihood function of the observations z1,z2,z3,?,zN drawn from an image is
Figure 3. i ? and the model parameters µ i , 2 i?
2are usually considered as known apriori. A commonly used method in initializing parameters is by drawing a random sample from the entire image Mclanchan G and Peel D
Figure 4. i µ and 2 i?
2are estimated by the method of moments as i z µ = and
Figure 5.
, S 2 is the sample variance.
Figure 6. 2011 October.
2011Research/Projects/CS/Vision/bsds/BSDS300/html). Global Journal of Computer Science and Technology Volume XI Issue XVII Version I 53 The images HORSE, MAN, BIRD, BOAT and TOWER are considered for image segmentation. The pixel intensities of the whole image are taken as feature. The pixel intensities of the image are assumed to follow a mixture of new symmetric distribution. That is, the image contains K regions and pixel intensities in each image region follow a new symmetric distribution with
Figure 7. Figure 2 .
2Figure 2.
Figure 8. Figure 2 :
2Figure 2 : Histograms Of The Images
Figure 9.
and BOAT have four segments each. The initial values of the model parameters i µ , 2 i ? and i ? for i = 1, 2,?,K for each image region are computed by the method given in section 3. Using these initial estimates and the updated equations of the EM Algorithm given in Section 3 the final estimates of the model parameters for each image are obtained and presented in tables 2.a, 2.b, 2.c, 2.c, 2.d ,and 2.e for different images.Table : 2.a Estimated Values of the Parameters for HORSE Image Number of Image Regions (K =2)
Figure 10.
Global Journal of Computer Science and Technology Volume XI Issue XVII Version I 55Using the estimated probability density function and image segmentation algorithm given in section 5, the image segmentation is done for the five images under consideration. The original and segmented images are shown in Figure3.
Figure 11. Figure 3 :
3Figure 3 : Original and Segmented Images ORIGINAL IMAGES SEGMENTED IMAGES
Figure 12. Figure 4 :
4Figure 4 : The Original and Retrieved Images ORIGINAL IMAGES RETRIEVED IMAGES
Figure 13. Table 1 :
1
IMAGE HORSE MAN BIRD BOAT TOWER
Estimate of K 2 4 3 4 3
Figure 14. Table :
:
Table : 2.d
Estimated Values of the Parameters for BOAT Image
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
Estimated Values of the Parameters for BIRD Image
Number of Image Regions (K =3)
Estimation of Initial Estimation of Final
Parameters Parameters Parameters by EM
Algorithm
Regions(i) Regions(i)
1 2 3 1 2 3
? i 1/3 1/3 1/3 0.13161 0.66786 0.20053
µ i 53.491 124.05 124.05 60.691 192.85 129.81
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
© 2011 Global Journals Inc. (US)
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
Figure 15. Table 3 :
3
PERFORMACE
IMAGES METHOD MEASURES
PRI GCE VOI
HORSE GMM NSMM-K 0.9283 0.1634 1. 8403 0.9142 0.1737 1.8643
MAN GMM NSMM-K 0.9342 0.1734 1.7875 0.9228 0.3107 1. 8389
BIRD GMM NSMM-K 0.9140 0.1352 1.7259 0.9106 0.1369 1. 7479
BOAT GMM NSMM-K 0.9174 0.6483 1.7542 0.9026 0.6485 1. 7882
TOWER GMM NSMM-K 0.9246 0.0981 1.7988 0.9102 0.1090 1. 8643
Figure 16. Table 4 :
4
IMAGE Quality Metrics FGMM FNSDMM Standard Limits
with K-Means
Average Difference 0.5011 0.44135 Close to 1
Maximum Distance 1.0000 1.0000 Close to 1
HORSE Image Fidelity Mean Square Error 1.0000 0.5011 1.0000 0.4414 Close to 1 Close to 0
Signal to Noise Ratio 5.6542 5.9301 As big as possible
Image Quality Index 1.0000 1.0000 Close to 1
Average Difference 0.4858 0.50021 Close to 1
Maximum Distance 1.0000 1.0000 Close to 1
MAN Image Fidelity Mean Square Error 1.0000 0.4995 1.0000 0.5079 Close to 1 Close to 0
Signal to Noise Ratio 5.6828 5.6251 As big as possible
Image Quality Index 1.0000 1.0000 Close to 1
Average Difference 0.4939 0.6573 Close to 1
Maximum Distance 1.0000 1.0000 Close to 1
BIRD Image Fidelity 1.0000 1.0000 Close to 1
Mean Square Error 0.8590 0.5050 Close to 0
Signal to Noise Ratio 5.6861 4.4842 As big as possible
Image Quality Index 1.000 1.0000 Close to 1
Average Difference 0.5039 0.6217 Close to 1
Maximum Distance 1.0000 1.0000 Close to 1
BOAT Image Fidelity Mean Square Error 1.0000 0.7931 1.0000 0.5070 Close to 1 Close to 0
Signal to Noise Ratio 5.6318 4.6573 As big as possible
Image Quality Index 1 1.0000 Close to 1
Average Difference 0.4936 0.6640 Close to 1
Maximum Distance 1.0000 1.0000 Close to 1
Image Fidelity 0.9999 0.9999 Close to 1
TOWER Mean Square Error 0.8788 0.5076 Close to 0
Signal to Noise Ratio 5.6870 4.4347 As big as possible
Image Quality Index 1.0000 1.0000 Close to 1
1
2
3
4

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Notes
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Studies on Image Segmentation Method Based On a New Symmetric Mixture Model with K -Means © 2011 Global Journals Inc. (US)
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Date: 2011-09-22