CHAPTER 6 MODIFIED FUZZY TECHNIQUES BASED IMAGE SEGMENTATION

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1 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

2 MR brain images 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 Comparative Analysis 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 96

3 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 i1 c n m 2 J( Q, c, c,..., c J q d (6.1) 1 2 c ) i i1 j1 q is the membership matrix with values between 0 and 1; c i is the centroid of cluster i; d is the Euclidian distance between i th centroid (c i ) and j th 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 cluster centers follows: q and the c i given below. The entire algorithm can be summarized as Step 1: The membership matrix Q=[q ] 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 t th number of iteration, the center vectors c i with q are calculated. c i n j 1 n q j1 m q x m j (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 97

4 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 t th step and (t+1) th step is then updated. 1 q 2 /( m1) (6.3) c d p 1 d pj 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 with 8 features for each pixel. Hence, the size of the input dataset is 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. 98

5 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. Distance metric calculation between the pixels Grouping the pixels into different clusters based on the distance measure 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) Representative selection from each cluster Membership values sharing between the representatives and its cluster members 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 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 99

6 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 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 The image is reshaped to a size of 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: 8 h1 x ih x jh i j (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 and j varies from 1 to 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 x i and x j and x i. x j is same as between x i x x x (6.5) j j i Theoretically, the number of operations used to calculate the entire set of Euclidean distances is (n 2 -n) where n is number of pixels in the input image. Using Equation (6.5), the actual number of operations to be performed is 100

7 only n m 1, 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 A, B, C, D... be the set of clusters. If 4 h1 x ih x jh i j, 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. 101

8 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 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 102

9 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: q where c k 1 1 d d kj 2 /( m1) ; c y j = reduced dataset d = c y i j i n j 1 n q j1 m q y m j (6.7) 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 103

10 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 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 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 and 255 is represented by 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 104

11 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 Pixel 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 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 = a d a b c d 105 (6.8)

12 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: 2a D= 2a b c (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. 106

13 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 (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 107

14 performance measures. Finally, an extensive comparative analysis is performed to highlight the optimal technique 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 108

15 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. Input Table 6.3 Quantitative analysis of conventional FCM No. of ground truth pixels True Positive pixels 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 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 Results of Modified FCM1 False Positive pixels 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. Segmentation Efficiency (%) Image Image Image CR 109

16 (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. Input Table 6.4 Quantitative analysis of Modified FCM1 No. of ground truth pixels True Positive pixels False Positive pixels Segmentation Efficiency (%) Image Image Image CR 110

17 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 It may be noted that an input image with 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 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 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. Technique Modified FCM1 algorithm Table 6.5 Impact of threshold value on quality measures Threshold Range Average No. of representative pixels Average Segmentation efficiency (%) Average Convergence time period (CPU secs)

18 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 has been reduced to 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 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) 112

19 (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. Input No. of ground truth pixels Table 6.6 Quantitative analysis of Modified FCM2 True Positive pixels False Positive pixels Segmentation Efficiency (%) Image Image Image 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 The required number of iterations required for convergence is approximately 58 with an error tolerance of Thus, the original input data of size is reduced to The threshold value used for this algorithm is 0.5 since the range of the threshold CR 113

20 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. 114

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