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CHAPTER 6

MODIFIED FUZZY TECHNIQUES BASED IMAGE SEGMENTATION

6.1 INTRODUCTION

Fuzzy logic based computational techniques are becoming increasingly important

in the medical image analysis arena. The significant reason behind the popularity

of fuzzy techniques is the high accuracy yielded by such techniques. Since

accuracy is one of the important factors for brain image segmentation

applications, they are highly preferred over other computational techniques.

Fuzzy C-Means (FCM) algorithm is one of the commonly used fuzzy techniques

for segmentation application. Even though FCM is accurate, the convergence

time period required by the algorithm is significantly high. This drawback has

reduced the usage of FCM technique for real-time applications such as the

medical applications where convergence time is also significant. In this research

work, this drawback is tackled by proposing two modifications in the

conventional FCM algorithm which guarantees quick convergence. The

modifications are not done in the algorithm but few pre-processing procedures are

implemented prior to the FCM algorithm which reduces the convergence time to

high extent. This reduction in the convergence time is achieved without

compromising the segmentation efficiency. Thus, the objective of this work is to

develop suitable fuzzy techniques for practical applications with the desired

performance measures.

6.2 PROPOSED METHODOLOGY OF FUZZY BASED SEGMENTATION

The proposed framework for the automated fuzzy based image segmentation

techniques is shown in Figure 6.1.

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Figure 6.1 Framework of the proposed fuzzy based segmentation system

The details of the image database, image pre-processing and feature extraction

has been dealt earlier in sections 3.2, 3.3 and 3.4. In this work, emphasis is given

to testing the algorithms with only the real-time images. Being an unsupervised

algorithm, the FCM technique can yield accurate results only if the abnormality

portion is sufficiently large. In the simulated images, the size of the abnormal

tumor portion is insignificant which is extremely difficult for any unsupervised

algorithm. The conventional FCM algorithm is discussed initially followed by the

extensive analysis on the two modifications of the FCM algorithm.

6.3 CONVENTIONAL FCM TECHNIQUE

The focus of this research is on the two modifications of the conventional FCM

algorithm. Since these modifications are based on the conventional FCM, it is

essential to implement the FCM approach before proceeding to the two

modifications.

Fuzzy C-means (FCM) is a method of clustering which allows one pixel to

belong to two or more clusters. The objective of the FCM algorithm is to partition

a finite collection of pixels into a collection of “C” fuzzy clusters with respect to

Comparative Analysis

Image Pre-processing

(skull removal)

Feature Extraction

(Pixel based)

FCM based image

segmentation

Modified FCM (I)

based image

segmentation

Modified FCM (II)

based image

segmentation

MR brain images

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some given criterion. Depending on the data and the application, different types

of similarity measures may be used to identify classes. Some examples that can

be used as similarity measures include distance, connectivity, and intensity. In

this work, distance is used as the similarity measure. Fuzzy C-means algorithm is

based on minimization of the following objective function:

c

i

n

j

ij

m

ij

c

i

ic dqJcccQJ1 1

2

1

21 ),...,,,( (6.1)

qij is the membership matrix with values between 0 and 1;

ci is the centroid of cluster i;

dij is the Euclidian distance between ith centroid (ci) and jth data point;

m є [1,∞] is weighting exponent (usually m=2).

Fuzzy partitioning is carried out through an iterative optimization of the objective

function shown in Equation (6.1), with the update of membership ijq and the

cluster centers ic given below. The entire algorithm can be summarized as

follows:

Step 1: The membership matrix Q=[qij] is initialized. The size of this matrix is

based on the number of rows and the number of columns in the input image. Each

pixel in the input image will have four membership values. The range of each

membership value is between 0 and 1. This initial membership matrix is

randomly initialized and these values are refined to determine the final

membership value.

Step 2: At tth number of iteration, the center vectors ic with ijq are calculated.

n

j

m

ij

n

j j

m

ij

i

q

xqc

1

1

(6.2)

The cluster centroid is the average intensity values of all the pixels in a particular

cluster. In the above expression, ‘i’ varies from 1 to 4 and hence four centroid

values are to be determined in this work. These centroid values are dependent on

the randomly initialized membership values. The pixel is assigned to the cluster

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based on the membership values. Also, ‘x’ corresponds to the input feature set

which consists of eight values for each pixel.

Step 3: The membership matrix Q for the tth step and (t+1)th step is then updated.

c

p

m

pj

ij

ij

d

dq

1

)1/(2

1 (6.3)

The membership matrix is further refined using Equation (6.3) based on the

cluster centroid values determined from Equation (6.2). In this expression, the

parameter ‘d’ is the distance between the input intensity values and the cluster

center. Thus, membership values are dependent on the centroid values. It is

actually the ratio of the distance between the input pixel and the cluster in

question to the distance between the intensity values and the entire cluster center.

Thus steps 2 and 3 are repeated recursively for the specified number of iterations.

Step 4: If || Q(t+1) - Q(t)||< r, then STOP; otherwise return to step 2.

When the membership matrix of the current iteration is almost equal to the

previous iteration, then the values are said to be stabilized. These are the final

membership matrix for any corresponding input image. Then, the pixel is

assigned to the cluster for which the membership value is maximum.

In the above algorithm, the entire image (all pixel feature values) is supplied

as input for image segmentation. The number of clusters used in this work is 4.

The images are of size 256×256 with 8 features for each pixel. Hence, the size of

the input dataset is 256×256×8 which is significantly high. Several parameters

such as the number of initial clusters, initial membership values and number of

iterations (or) error threshold value are randomly initialized. The error threshold

(r) used in this work is 0.01 and the number of iterations is 320. Also, the

parameter convergence is highly iterative in nature which consumes huge

computational time. This huge requirement for time period significantly limits the

practical applications of the FCM algorithm. This drawback of computational

complexity is eliminated in the modified FCM.

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6.4 MODIFIED FCM TECHNIQUES

The major drawback of the conventional FCM algorithm is the requirement of

high convergence time. In this work, two different versions of the conventional

FCM algorithm is proposed to tackle this problem. The modifications are not

actually performed in the algorithm but a pre-processing procedure is included

prior to the algorithm to reduce the convergence time. The ideology behind these

modifications is based on the size reduction of the input dataset which can impact

the convergence rate. This process is done without compromising the

segmentation efficiency. The procedural flow of these algorithms is given in

Figure 6.2.

Figure 6.2 Framework of the modified FCM algorithms

Two modified algorithm such as Modified FCM1 and Modified FCM2 are used

in this work. Both these techniques are different in the procedure of dataset

reduction. The detailed algorithms are discussed in the subsequent sections.

6.4.1 Modified FCM1 Technique

In this proposed approach, the computational complexity problem of the

conventional FCM is tackled by reducing the size of the input dataset. This

dimensionality reduction is achieved through a sequence of steps which involve

Distance metric

calculation between

the pixels

Grouping the pixels

into different

clusters based on the

distance measure

Representative

selection from each

cluster

Arranging the representative

pixels in vector form and

supplying as input to the

conventional FCM algorithm

FCM algorithm training using

the reduced input data set

(only the representative pixels)

Membership values sharing

between the representatives

and its cluster members

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the concept of distance metrics. Then, the reduced dataset is given as input to the

conventional FCM algorithm which converges much quickly than with the whole

dataset. Thus, this modified FCM yields accurate results within less time.

6.4.1.1 Algorithm of Modified FCM1

The algorithm of the proposed Modified FCM1 involves two phases: (1) Input

vector dimensionality reduction, (2) FCM algorithm and Membership value

assignment.

Phase 1: Input Vector Dimensionality Reduction

The size of the input dataset is reduced with the help of the Euclidean distance

measure.

Step 1: The initial data size used in this work is 256×256. The image is reshaped

to a size of 1×65536 for mathematical convenience. The Euclidean distance of the

first pixel with the remaining pixels is then calculated. The pixels whose distance

measure is in close proximity are grouped in one cluster. The same process is

repeated for the remaining pixels and all the distance measure values are

observed. Mathematically, this concept can be represented as follows:

jih

jhih xx

8

1

(6.4)

In the above equation, h corresponds to the input features since each input pixel

x is represented by eight features. Also, the two dimensional input dataset is

reshaped to single dimensional vector for convenience. In the above equation, i

varies from 1 to 65536 and j varies from 1 to 65536. Thus by changing the values

of i and j , the complete Euclidean distance metric can be calculated. Another

advantage is that the Euclidean distance between ix and jx is same as between

jx and ix .

ijji xxxx (6.5)

Theoretically, the number of operations used to calculate the entire set of

Euclidean distances is (n2-n) where ‘n’ is number of pixels in the input image.

Using Equation (6.5), the actual number of operations to be performed is

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only 1mn , m = 0, 1…n-2 which is significantly lesser than the theoretical

calculation.

Step 2: Based on the distance metric, the pixels are grouped into different

clusters. If the distance metric between the two pixels is less than a threshold ( ),

then the two pixels are grouped into the same cluster. Let .......,,, DCBA be the

set of clusters.

If

jih

jhih xx4

1

, then i, jA (6.6)

Equation (6.6) is repeated for all values of j with i as constant and the set of pixels

for cluster ‘A’ can be observed. This process is repeated for all values of i and

hence different clusters with distinct pixel elements can be determined. Thus, this

process has grouped the different input pixels into different clusters.

Selection of the threshold ( ) is the most critical factor of this algorithm. If

the threshold value is too high, the number of clusters formed will be less. This

will minimize the convergence time period to higher extent. But the segmentation

efficiency will be affected since pixels of different texture may get grouped under

the same cluster. If the threshold value is too small, then more number of clusters

will be formed which indirectly increases the convergence time period. But,

superior segmentation efficiency is guaranteed. Hence, an optimal value of

threshold must be selected which will be efficient in terms of both convergence

time period and segmentation efficiency.

Step 3: All the pixel elements in the clusters are not given as input to the

conventional FCM algorithm. Instead of supplying the complete set of clustered

pixel elements, only the representative pixel from each cluster is given as input to

the conventional FCM algorithm. This is the major difference between the

modified FCM and the conventional FCM algorithm where all the pixels are used

for segmentation. But, the segmentation efficiency will be affected if the

representative selection is not optimal. Statistical techniques such as mean,

median and mode are the commonly used method for representative selection

from a finite set.

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The pixel elements from each cluster are arranged in ascending order. The

selected pixel representative should not be biased towards both the extremes of

the cluster. Mode yields the most commonly occurring pixel value but the

probability of this value belonging to both the extreme values of the set is high.

Mean is the average of the complete set which may give a value in decimal point

representation. Again, quantization (round-off) is required which may affect the

accuracy.

In this work, median is used to determine the representative from the cluster

set. The center value is provided by the median which gives equal bias to the

values on both the sides of the set. If the number of elements in the cluster set is

even, any one of the center values may be chosen. Hence, representative selection

using median is the optimal method since the representative pixel highly shares

the characteristic features of its member pixels in the cluster.

Step 4: All the representative pixels are arranged in the vector format with its

textural features. The size of the input dataset with representative pixels is always

lesser than the size of the original input dataset. The extent of dimensionality

reduction depends on the selection of threshold value. For example, consider the

following input dataset (x) shown in Figure 6.3.

255 243 180 254 250 156 247

244 189 98 112 187 112 231

243 167 59 112 98 189 214

233 231 41 79 43 65 245

254 155 85 132 57 125 255

212 134 87 87 215 190 254

189 231 247 251 255 200 214

Figure 6.3 Sample dataset

The initial dataset size is 7×7 which consists of 49 pixels. Initially, let the

threshold value be zero, i.e, .0 Considering the first pixel, x (1,1), the dataset

size is reduced to 47 pixels (46+1 representative pixel) since 3 pixels are available

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with the intensity value 255. Let the threshold value be increased to 10. Again

considering the first pixel, x (1,1), the dataset size is reduced to 39 pixels (38+1)

since 11 pixels are available as neighbor to the first pixel value which is 255, i.e,

within the range of 245. Similarly, by considering the remaining pixels, the final

dataset size is reduced with limited number of pixels.

Hence, it is evident that the threshold value is inversely proportional to the size

of the dataset. In any case, the selection of representative values reduces the size

of the input dataset to higher extent. These representative values alone are used as

input to the conventional FCM algorithm for image segmentation.

Phase 2: Conventional FCM algorithm and membership value assignment

Step 5: The conventional FCM algorithm discussed under Section 6.3 is repeated

with the reduced dataset (representative pixels). The algorithm is implemented in

an iterative method with the updates of cluster centre and membership values. But

the cluster centre and membership value update equations are changed as follows:

c

k

m

kj

ij

ij

d

dq

1

)1/(2

1

;

n

j

m

ij

n

j j

m

ij

i

q

yqc

1

1

(6.7)

where jy = reduced dataset

ijd = ji yc

From Equation (6.7), it is evident that the number of iterations required for the

modified FCM is significantly lesser than the conventional FCM. Thus, the

modified FCM algorithm yields the membership values for the representative

pixels at a faster convergence rate.

Step 6: Since the whole image is considered for segmentation, the remaining

pixels (other than the representative pixels) also require membership values to

complete the process. To fulfill this requirement, the membership value of the

representative pixels is assigned to its cluster members. Since the representative

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and the cluster members are of same nature, the membership assignment

methodology does not affect the segmentation efficiency.

From the above procedural steps, it is evident that the modified FCM

converges very quickly than the conventional FCM algorithm. Besides being fast,

the accuracy of the proposed approach is also guaranteed. The time taken for the

extra procedure (distance metric calculation) is very less when compared with the

iterative time period of the conventional FCM algorithm. Thus, the proposed

approach proved to be time efficient and a better alternate for conventional FCM

algorithm. The implementation parameters are same as that of the conventional

FCM algorithm except for the requirement of number of iterations.

6.4.2 Modified FCM2 Technique

The second modified FCM algorithm proposed in this work is also based on the

same concept but with a change in the procedure of distance metric calculation. In

the first modification, the conventional Euclidean distance measure is used and in

the second the closest match between the pixels is determined using the distance

measures like ‘Matching’ and ‘Dice’. Calculation of these parameters is done

with the help of binary representations of the input pixel values. This

modification is done with an objective that a change in the distance metric

calculation can enhance the performance measures.

6.4.2.1 Algorithm of Modified FCM2

There are two phases in modified FCM2 algorithm: (1) Data Reduction and

Representative selection and (2) Conventional FCM algorithm and membership

value assignment.

Phase 1: Data reduction and Representative selection

Step 1: Initially, all the pixel intensity values are converted to binary

representation. Since the intensity value ranges from 0 to 255, 8 bits are used to

represent each pixel. For example, the intensity value ‘0’ is represented by

00000000 and ‘255’ is represented by ‘11111111’.

Step 2: The distance metrics between the first pixel and the rest of the pixels are

determined in a sequential manner. For example, let us assume that the distance

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between the following two pixels are to be estimated. Table 6.1 shows the sample

values for the two pixels.

Table 6.1 Sample input values

Decimal values Binary values

Pixel 1 93 0 1 0 1 1 1 0 1

Pixel 2 147 1 0 0 1 0 0 1 1

For the values in Table 6.1, the distance measures are estimated by forming

another table called as Response Table. Table 6.2 shows the general format for

forming the Response Table.

Table 6.2 General Response Table format

Subject 1 Subject 2

‘1’ ‘0’

‘1’ a b

‘0’ c d

Table 6.2 shows a 2×2 Response Table since only two subjects ‘1’ and ‘0’ are

involved in the binary representation. The values such as a, b, c and d must be

further estimated. Using these values, the distance measures such as ‘Matching’

and ‘Dice’ can be determined.

Step 3: In Table 6.2, ‘a’ corresponds to the number of times ‘11’ combination

occurred in the same bit position for the two input pixels, ‘b’ corresponds to the

number of times ‘10’ combination occurred in the same bit position, ‘c’

corresponds to the number of times ‘01’ combination occurred in the same bit

position and ‘d’ corresponds to the number of times ‘00’ combination occurred in

the same bit position. For Table 6.1, the values of ‘a’, ‘b’, ‘c’ and ‘d’ are 2, 3, 2

and 1 respectively. Using these values, the distance measures are calculated.

Step 4: Two distance measures are used in this work. Initially, the parameters

‘Matching’ (M) and ‘Dice’ (D) are estimated and further the distance is calculated

using (1-M) and (1-D) values. The parameter ‘Matching’ is estimated using the

following formula:

M = dcba

da

(6.8)

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This parameter is also called as ‘Matching Coefficient’ which involves the

attributes which has a perfect match in the bit positions (‘11’ and ‘00’

combinations). Hence, higher the value of M better is the similarity between the

two pixels. Another parameter ‘Dice’ is determined using the following formula:

D=cba

a

2

2 (6.9)

‘Dice’ corresponds to weighted distance measure for attributes with mutual

agreement (‘11’ combination). The final distance measure through Dice

Coefficient is determined by calculating (1-D). Hence, higher the value of D

better is the similarity between the two pixels.

Step 5: The final distance measure through ‘Matching’ is determined by

calculating (1-M) and the distance measure through ‘Dice’ is determined by

calculating (1-D). For the sample input values shown in Table 6.1, the parameter

‘M’ yields a value of 3/8 and ‘D’ yields a value of 4/9. Hence, the values of (1-M)

and (1-D) are 5/8 and 5/9 respectively. The distance measure values range from 0

to 1. If the distance measure values are low, then the similarity between the pixels

are high.

Step 6: The measure of closeness (or) similarity between the two pixels can be

determined by comparing these values with a specified threshold value. Since two

subjects (1 and 0) are involved in the binary representation, the threshold value is

set to 0.5 in this work. Higher value of threshold results in less number of

clusters. In this case, the probability of the non-neighboring pixels grouped under

the same cluster is high. This leads to inaccurate segmented results. On the other

hand, if the threshold value is too low, then the number of clusters increase which

results in increased computational complexity.

In this case, at one point of time, MFCM2 converges to FCM. Hence, an

optimum value of 0.5 is used as threshold value in this work. All the pixels whose

(1-M) and (1-D) values are lesser than 0.5 are grouped under the same cluster. For

example, the sample values shown in Table 6.1 do not belong to the same cluster

since their distance measure values are greater than the specified threshold value.

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Step 7: The same process is repeated for all the pixels and the pixels whose

distance measures are minimum are categorized to the same cluster. This process

is repeated until all the pixels belong to one of the clusters.

Step 8: The number of clusters is noted down and one representative from each

cluster is selected. Median is used to determine the representative pixel from each

cluster. The new dataset consists of pixels equal to the number of clusters which

is lesser than 65536 (original input dataset). Thus, the dataset is highly reduced

with the proposed methodology.

Phase 2: Conventional FCM algorithm and membership value assignment

Step 9: The conventional FCM algorithm discussed under Section 6.3 is repeated

with the reduced dataset (representative pixels). The algorithm is implemented in

an iterative method with the updates of cluster centre and membership values. But

the cluster centre and membership value update equations used are same as that of

Equation (6.7). It is evident that the number of iterations (convergence time)

required for the modified FCM2 is significantly lesser than the conventional

FCM. Thus, the modified FCM2 algorithm yields the membership values for the

representative pixels at a faster convergence rate.

Step 10: The membership assignment procedure is performed using earlier

procedure discussed in Phase 2 of the modified FCM1 algorithm.

From the above procedural steps, it is evident that the modified FCM2

converges very quickly than the conventional FCM algorithm. The segmentation

efficiency also will be verified with the experimental results. Thus, the second

modified FCM algorithm has been developed with an objective for application in

the real-time medical field.

6.5 EXPERIMENTAL RESULTS AND DISCUSSIONS

The experiments of the three FCM techniques are carried out on the Pentium

processor with speed 1.66 GHz and 1 GB RAM. The software used for the

implementation is MATLAB (version 7.0), developed by Math works Laboratory.

The experiments are carried out on the real-time dataset collected from the scan

center. The results of each technique are analyzed individually based on the

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performance measures. Finally, an extensive comparative analysis is performed to

highlight the optimal technique.

6.5.1 Results of conventional FCM

The performance of the conventional FCM is analyzed in terms of segmentation

efficiency and correspondence ratio. A brief analysis on the convergence rate is

also reported in this work. Initially, the qualitative analysis is presented in this

section followed by the quantitative analysis on segmentation efficiency. This

technique is applied on all the images but only few samples are shown in Figure

6.4.

(a) (b) (c) (d)

(a) (b) (c) (d)

(a) (b) (c) (d)

Figure 6.4 Sample FCM results: (a) Input images, (b) Clustered images, (c)

Tumor segment, (d) Tumor Phantom images.

In this work, the input image is clustered into four groups such as GM, WM, CSF

and abnormal tumor portion. Pseudo color is given to distinguish all the groups of

the clustered images. The phantom images can be compared with the clustered

images to check the clustering capability of the conventional FCM algorithm. An

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adaptive threshold is used to extract the tumor portion from the clustered image.

From the above Figures, it is evident that the clustering process is not convincing

for the input images. The qualitative analysis has shown that few tumor pixels are

missing and few non-tumorous pixels are misclassified as tumor tissues. The

quantitative analysis of the above three input images are given in Table 6.3.

Table 6.3 Quantitative analysis of conventional FCM

The segmentation efficiency and the correspondence ratio are not sufficient for

the conventional FCM algorithm. The efficiency results are almost same for all

the input images. It may be noted that the pre-processed input image is given as

input to the FCM algorithm. Further, an analysis in terms of convergence rate is

also performed for the conventional FCM algorithm.

The FCM algorithm is iterative in nature and the conventional FCM algorithm

require an average 730 CPU seconds for an input image of size 256×256×8. The

convergence time is different for input images of different sizes and hence the

convergence time drastically increases for large size dataset. The convergence is

based on the error tolerance value of 0.01.

6.5.2 Results of Modified FCM1

The experiments are also conducted using the Modified FCM1 algorithm. The

proposed approach is analyzed in terms of segmentation efficiency and

convergence rate. The same dataset is used for implementation and sample results

of the qualitative analysis are shown in Figure 6.5.

Input No. of

ground

truth

pixels

True

Positive

pixels

False

Positive

pixels

Segmentation

Efficiency (%)

CR

Image 1 1596 1261 200 79 0.72

Image 2 3836 3146 179 82 0.80

Image 3 5405 4108 74 76 0.75

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(a) (b) (c) (d)

(a) (b) (c) (d)

(a) (b) (c) (d)

Figure 6.5 Sample Modified FCM1 results: (a) Input images, (b) Clustered

images, (c) Tumor segment, (d) Tumor Phantom images.

The sample results are shown for the same images used in the conventional FCM

algorithm which can aid the comparative analysis. A visual observation of these

results has clearly revealed the superior nature of the proposed approach in terms

of segmentation efficiency. The segmented outputs are better than the

conventional FCM algorithm. One of the significant reasons is that the data

reduction method has yielded an optimal initial cluster center which has resulted

in the enhanced performance. The qualitative results are shown in Table 6.4.

Table 6.4 Quantitative analysis of Modified FCM1

Input No. of

ground

truth

pixels

True

Positive

pixels

False

Positive

pixels

Segmentation

Efficiency (%)

CR

Image 1 1596 1478 1071 92 0.59

Image 2 3836 3533 642 92 0.83

Image 3 5405 5226 668 96 0.90

111

The qualitative analysis has verified the fact that the Modified FCM1 is superior

to the conventional FCM algorithm in terms of the efficiency measures. The

number of True Positive (TP) pixels is significantly high which has improved the

segmentation efficiency. The correspondence ratio is almost similar to the values

of the conventional FCM algorithm even though slight variations occur for

independent images.

The convergence rate required for the Modified FCM1 algorithm is 32 CPU

seconds for an image size of 196×8. It may be noted that an input image with

65536 pixels has been reduced to 196 pixels using the data reduction method. The

required number of iterations is approximately 50 with an error tolerance value of

0.01. Thus, it can be seen that the modified FCM1 algorithm is superior to the

conventional FCM algorithm in terms of the performance measures. But, the

performance measures shown above are based on the optimal threshold value (0-

22) used in the data reduction procedure. The selection of threshold value is very

critical since the quality measures can significantly vary for different threshold

values. An analysis of the impact of threshold value on the performance measures

is discussed in the next section.

6.5.2.1 Effect of threshold values on the performance measures

The effect of threshold value on the amount of data reduction and the

performance measures is shown in Table 6.5.

Table 6.5 Impact of threshold value on quality measures

Technique Threshold

Range

Average No. of

representative

pixels

Average

Segmentation

efficiency (%)

Average

Convergence

time period

(CPU secs)

Modified

FCM1

algorithm

191-255 1-30 27 5

136-190 35- 82 48 10

91-135 85-145 65 14

51-90 148-180 77 20

23-50 181-195 89 24

0-22 196-208 96 32

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From the above table, it is evident that the size of the dataset is highly reduced for

a higher threshold value and vice-versa. But, the segmentation efficiency is highly

reduced for high threshold values. Hence, an optimal value of threshold (0-22) is

used in this work. Since the performance measures are more important than the

amount of data reduction, the threshold value which yields better segmentation

efficiency is selected as the optimal value. Thus, an input image of size

256×256×8 has been reduced to 196×8 pixels. The average values are shown here

since the performance measures are displayed for a range of input image size.

The time taken for clustering the pixels is also inversely proportional to the

threshold value. If the threshold value is maximum, then the number of grouping

operations required is very less which minimizes the time period. If the threshold

value is minimum, more number of clusters is to be formed which results in

increased clustering operations. These facts are evident from Table 6.5. In any

case, the maximum possible time period required is very less when compared

with the convergence time period required for conventional FCM.

6.5.3 Results of Modified FCM2

The qualitative result analysis of the Modified FCM2 algorithm is shown in

Figure 6.6.

(a) (b) (c) (d)

(a) (b) (c) (d)

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(a) (b) (c) (d)

Figure 6.6 Sample Modified FCM2 results: (a) Input images, (b) Clustered

images, (c) Tumor segment, (d) Tumor Phantom images.

The results have shown significant improvement in the efficiency measures of

the proposed approach. The number of FP pixels has been significantly reduced

which is verified with the quantitative analysis shown in Table 6.6.

Table 6.6 Quantitative analysis of Modified FCM2

Thus, the proposed Modified FCM2 algorithm is efficient in terms of

segmentation efficiency and the correspondence ratio measures. This has verified

the fact that the proposed approach is successful in identifying the tumor pixels

and non-tumor pixels simultaneously. The significant reason is that the data

reduction procedure has yielded an optimal value of initial parameters such as the

membership values and the cluster centers. The selection of initial parameters

plays a major role in determining the accuracy of the FCM approaches.

The convergence time required for the Modified FCM2 algorithm is 34 CPU

seconds for an input data of size 180×8. The required number of iterations

required for convergence is approximately 58 with an error tolerance of 0.01.

Thus, the original input data of size 256×256×8 is reduced to 180×8. The

threshold value used for this algorithm is 0.5 since the range of the threshold

Input No. of

ground

truth

pixels

True

Positive

pixels

False

Positive

pixels

Segmentation

Efficiency (%)

CR

Image 1 1596 1478 19 92 0.93

Image 2 3836 3533 21 92 0.92

Image 3 5405 5215 15 96 0.97

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value is between 0 and 1. Thus, the efficiency of the proposed approach in terms

of the quality measures is verified from the experimental results.

The testing process is also implemented with the simulated images.

Experimental results have verified the fact that the FCM algorithms are capable

of segmentation only if the tumor region is sufficiently large. The reason is that

the FCM algorithms are unsupervised in nature which lacks the assistance of

target values. Lack of target values and the insignificant size of the tumor region

in the simulated images are the causes for the inferior results of FCM algorithm

with these images. The performance measures are almost zero and the detailed

analysis is not reported in this work.

6.6 CONCLUSION

In this work, suitable alternates for the conventional FCM algorithm is proposed

for MR brain image segmentation. Two modified approaches are proposed in this

work which are found to be efficient than the conventional FCM in terms of

convergence rate and segmentation efficiency. Thus, the huge computational

complexity of the conventional FCM algorithm is tackled by these proposed

approaches. The convergence time of these approaches are reduced without

compromising the segmentation efficiency. Thus, this work has suggested few

solutions to overcome the drawbacks of conventional FCM algorithm. This work

also highlighted the optimal FCM technique for real-time applications such as

brain image analysis.