TJFS: Turkish Journal of Fuzzy Systems (eISSN: 1309–1190)

An Official Journal of Turkish Fuzzy Systems Association Vol.4, No.1, pp. 34-47, 2013.

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A Novel Fast Fuzzy C-Means Clustering Technique for Segmentation of Human-

Brain Magnetic Resonance Images

Abbas Biniaz1,2 1Computational Neuroscience Laboratory, Department of Biomedical Engineering, Faculty of Electrical Engineering, Sahand University of Technology, Tabriz, Iran.

2Faculty of Electrical Engineering, Islamic Azad University, Lamerd branch, Fars, Iran. E-mail: abbass_biniaz@yahoo.com

Mousa Shmasi

Assistant professor, Department of Biomedical Engineering, Faculty of Electrical Engineering, Sahand University of Technology, Tabriz, Iran.

E-mail: shamsi@sut.ac.ir

Ataollah Abbasi * Assistant professor, Computational Neuroscience Laboratory, Department of

Biomedical Engineering, Faculty of Electrical Engineering, Sahand University of Technology, Tabriz, Iran.

E-mail: ata.abbasi@sut.ac.ir, *Corresponding author Received: July 31, 2013 – Revised: November 11, 2013 – Accepted: November 11, 2013

Abstract

In medical applications all effectual agents in patient health must be fast, even medical algorithms such as clustering ones. In this paper an optimized technique is presented to decrease execution time and iterations of standard Fuzzy C-Means (FCM) alghorythm. New approach calculates cluster center in each iteration by new formula. Applying proposed method decreases the complexity of FCM algorithm. A type of averaging among cluster centers is applied in each iteration step however, membership function is a fuzzy coefficient. By proposed approach intensity of pixels in a cluster is averaged in each time step. Simulation results show that the proposed Fast FCM (FFCM) spends moderately half time of standard FCM and decreases iteration numbers. Moreover, to decrease the time of convergence considerably and decline the number of iterations significantly, cluster centroids are initialized by an algorithm. FCM and FFCM techniques are applied to segment magnetic resonance (MR) images. Accuracy of the proposed approach is significantly same as standard FCM. Applying fuzzy validity functions to quantitative assessment of FFCM in comparison with FCM verifies efficient performance of the proposed approach. Keywords: MR image Segmentation, Clustering, Fast Fuzzy C-Means, Cluster center initialization, Fuzzy membership.

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1. Introduction

Image Segmentation is the procedure of dividing an image into several homogenous regions based on different properties such as image gray levels, intensity gradient, textures, colors or tissue contrast. Most medical imaging modalities are noninvasive approaches and give the possibility of simply observing internal body parts. Magnetic resonance imaging (MRI) is a low-risk non-invasive method provides detailed anatomical images of tissues and other organs. MR imaging modalities facilitate in detecting tissue deformities and brain pathologies such as tumors, cancers, edema, enlarged ventricles, multiple sclerosis (MS), necrotic tissues, and pathological lesions (Balafar, Ramli et al. 2010; Sharma and Aggarwal 2010). MR image segmentation techniques facilitate in extracting diverse brain tissues such as gray matter (GM), white matter (WM), and cerebrospinal fluid (CSF) popularly (Sathya and Kayalvizhi 2011). However the principal difficulties affecting on segmentation procedure are: 1-Noise. 2-Partial-volume effect (a voxel contains several tissues). 3-Intensity non-uniformity (intensity smooth variation inside discrete uniform tissue classes). 4-Closeness in intensity of dissimilar soft tissues; and other image acquisition artifacts occurrence(Sharma and Aggarwal 2010; Bandhyopadhyay and Paul 2012). In the last decades numerous methods were used to segmentation of medical images. Artificial neural networks (ANNs) are powerful data modeling tool and are able to learn and extract complicated relations between input/output data by their distinctive topology (Ortiz, Gorriz et al. 2012). Watershed method is a well-known gradient-based algorithm which is plentifully utilized in image segmentation. It can extract tumors and other tissues by managing internal and external markers (Kaleem, Sanaullah et al. 2012). Deformable models are region based approaches which are extensively developed in MR image segmentation. These models are very robust to nuisance factors. Segmentation results are affected by the initial contour placement and this is deformable models constraint (Shah and Ross 2009; Jayadevappa, Kumar et al. 2011). The region growing is one of the region-based approaches. Starting by implanting a seed in the image, a connected region is extracted. In fact pixels in the neighborhood of a seed, based on homogeneity criterion are added to specify area. Therefore throughout the region growing process a connectedness region is formed. Region growing methods have high sensitivity to noise and cause to disconnectedness in segmented images (Lu, Jiang et al. 2003; Balafar, Ramli et al. 2010). Among other approaches fuzzy c-means clustering technique is an unsupervised method which has been well developed for MR image segmentation (Chuang, Tzeng et al. 2006). Due to fuzzy membership, FCM is capable of reservation more information about the original image compared to the other algorithms (Górriz, Ramírez et al. 2006). However, this method is not fast enough in medical applications and needs a lot of time to converge. There are several fast fuzzy c-means methods. One approach utilizes the gray level histogram instead the whole data of image to speed up the FCM algorithm (Yong, Chongxun et al. 2004). Some of them initialize cluster centers by an operator therefore they are supervised approaches (Balafar, RAMLI et al. 2010). Also a fast FCM algorithm is proposed for color image segmentation in (Le Capitaine and Frélicot 2011) which initializes cluster centers by the numerical methods. However, this paper is presented a Fast FCM (FFCM) algorithm with a novel rule to update cluster centers in each iteration step. A type of averaging among cluster centers is applied in each iteration step. however, membership function is

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a fuzzy quantity. By proposed approach intensity of pixels in a cluster is averaged in each time step. The new rule relatively preserves the quality of standard FCM and is a faster technique. Besides, the number of iterations in new algorithm is reduced comparatively. Furthermore, to additional increase FCM/FFCM convergence rate, the dist-max algorithm is used to initialize cluster centers (Ramathilagam, Pandiyarajan et al. 2011). The present work is organized as follows: In Section 2.1 the traditional FCM method is reviewed. In Section 2.2 using the dist-max algorithm center of clusters are initialized. Section 2.3 presents the proposed FFCM. Validation functions are expressed in Section 2.4. Results and discussion are presented in Section 3; and Section 4 summarizes conclusions of this paper.

2. Method

2.1. Standard FCM

The c-means families are well developed group of batch clustering type because they are “least square” models. Each cluster consists of one or more common characteristics depending on the dimension of input data. FCM developed in 1970s, assigns fuzzy memberships to each element of dataset instead of hard membership (Dunn 1973); Therefore in FCM each data point belongs to multiple clusters with different membership values. Let { }1 2, ,..., nX x x x= denote an input vector with n pixels which should be partitioned into c clusters (2≤ c≤n) and xj is feature value. FCM is an iterative optimization procedure which minimizes the following cost function (Chuang, Tzeng et al. 2006):

(1)

And

0 1iju≤ ≤ for 1 ,i c≤ ≤ 1 ,j n≤ ≤

1

0N

ijk

u n=

< <∑ for 1 ,i c≤ ≤

11,

c

iji

u=

=∑ for 0 .j n≤ ≤

Where n is data point numbers, m is the fuzzy fitness grade (m equals to 1 in hard clustering and more than 1 in fuzzy clustering), uij is the membership of pixel xj in

the i-th cluster, vi is the centroid of i-th cluster, and ||.|| is Euclidean norm. Since the cost function must be minimized, pixels which are close to their clusters center should have high membership values. Vice versa low membership values are assigned to pixels with data far from cluster center. In the other hand, the maximum distance between the cluster centroids leads to the optimum clustering. Membership function and cluster centers are updated by the following equations:

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(2)

(3)

Starting with an initial value for each cluster center, the FCM converges to a solution for vi representing the local minima or a saddle point of the cost function (Chuang, Tzeng et al. 2006). Convergence rate can be determined by comparing the differences between the membership function or cluster centers in two successive iterations. Convergence time depends on computing time of membership function and cluster centers in the iterations.

2.2. Initialize cluster centroids

In order to avoid the random initialization, the dist-max algorithm for the center initialization has been offered (Ramathilagam, Pandiyarajan et al. 2011). In FCM algorithm random initialization consumes more time to be converged. Cluster center initialization algorithm for FCM is as follows [14]: Step 1: Sorting mi’s in ascending order where and i=1,2,…n for input

vector { }1 2, ,..., nX x x x= which is p-dimensional data. Step 2: Relabeling and reorganizing the dataset matrix as { }' ' ' '

1 2, ,..., .nX x x x= Partition the data in to c groups and find /knc n c= , where knc is number of data points in k-th cluster; The number of cluster c is specified according to the nature of the dataset ( 1<c<n ). Step 3: Making a distance tables that show the distance between the elements within each group. (ie) if group 1 2, ,...,k k k

nk x x x = , the distance table in this group is:

Figure 1. distance matrix within each data group Step 4: Selecting the maximum distance from each distance table of groups. If k

ijd is maximum distance of k-th group, find the mean value Mk of the elements xi and xj then assign centroid of k-th cluster as Mk, and k=1,2,…c.

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2.3. Fast FCM algorithm

As mentioned, high speed algorithms such as clustering ones are required in medical applications. Moreover, standard FCM doesn’t have enough convergence speed especially in emergency conditions. FCM assigns c membership grades to every pixel. This means that each pixel have c membership for each cluster. Iteratively updating the cluster centers and membership grades, FCM moves the cluster centers to the right position within a data set. However, updating membership matrix with c × n member is a time consuming procedure. In FCM, centroids are updated by fuzzy memberships which need much time because cluster centers are selected as a fuzzy quantity. Whereas, cluster centers can be calculated by a hard membership. Hence to reduce time and amount of computations in FCM, a hard membership can be assigned to pixels for updating cluster centers in each iteration step. That means each data point is belonged to only one cluster for centroid updating. However, segmentation will be a fuzzy procedure. Applying hard membership, the new algorithm to update centroids is proposed as following: Step1: For p-dimensional input data, rearrange iju to d1×d2 matrix; where d1, d2 are input dimensions. Step2: Set new fuzzy membership as *

iju and label matrix as 1 2{ , ,... }cL L L L= ; where kL is label matrix of k-th cluster in current iteration. Step3: Set all data points which are correspond to kL label matrix as Ik. Step4: Define 1 2, ,...,

k

k k k kncI I I I= for k-th cluster, where knc is number of data points in k-

th cluster Step5: Update centroid of k-th cluster by equation 5:

1*

knck

jj

kk

Iv

nc==∑

(4)

The new fuzzy membership and cost function can be calculated by following equations:

(5)

(6)

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And 0 1iju≤ ≤ for 1 ,i c≤ ≤ 1 ,j n≤ ≤

10

N

ijk

u n=

< <∑ ,for 1 ,i c≤ ≤

11,

c

iji

u=

=∑ for 0 .j n≤ ≤

It will be shown the new FCM algorithm works faster than the standard one therefore it can be called Fast FCM (FFCM). The proposed FFCM algorithm by cluster center initializing can be summarized as following: Step 1: Select the number of clusters(c) and fuzziness value (m=2). Step 2: Initialize V*(0) by dist-max algorithm. Step 3: Update the new membership matrix U* by Eq.(6). Step 4: Update cluster center matrix by Eq.(5).

Step 5: Repeat steps 2–3 like standard FCM until ( 1) ( )t ti iv v ε+ − < , where ε is a small

positive constant.

2.4. Cluster validity functions

Mostly two types of validity functions are used to evaluate the performance of clustering: fuzzy partition and geometric structure. Partition coefficient (Vpc) and partition entropy (Vpe) are fuzzy partition functions and defined as following (Wang and Zhang 2007; XiaoLi, Ying et al. 2010):

(7)

(8)

In these equations less fuzziness shows better performance of the algorithm. As a result, the best clustering is achieved when Vpc has maximum value (close to one) or Vpe has minimum value (close to zero). These functions can only measure the fuzzy partition and don’t have a direct access to the intensity vector. This problem can be solved using validity functions based on the geometric structure. To optimum clustering in validity functions based on the geometric structure, samples within one partition should be compact and samples between different clusters should be separate (Xiao, Ho et al. 2010). To quantify the ratio of total variation within clusters, Vfs and Vxb are defined as following (Wang and Zhang 2007; XiaoLi, Ying et al. 2010):

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(9)

(10)

Where, vi vk and minimized Vfs or Vxb lead to optimal clustering.

3. Results and discussion

To verify the efficiency of the proposed FFCM in comparison to FCM they are evaluated on both synthetic and real clinical MR images. Experimental results show proposed fast FCM technique by new rule efficiently updates cluster centers. In all simulations cluster centers are initialized by dist-max algorithm. Figure 2 (a) is simulated T1-weighted image (Chuang, Tzeng et al. 2006). This image in Figure 2 (b) corrupted by additive Gaussian white noise ( 0.002σ = ). The gray levels in the T1-weighted image are 50 (UL), 100 (UR), 150 (LL), and 200 (LR). Each gray level represents a living tissue on MR image and at the end of the segmentation should be indicated as a separate cluster. As seen in Figure 2 (c) and (d) in the presence of additive Gaussian noise standard and fast FCM techniques have a closely similar performance.

Figure 2. (a) Segmentation results: (a) simulated T1-weighted image, (b) corrupted

by additive Gaussian noise; Clustering results using (c) FCM and (d) FFCM.

Figure 3 (a) displays the simulated brain MR phantom corrupted by additive Gaussian white noise ( 0.0001σ = ). Segmentation results using FCM and FFCM are shown in Figure 3 (b) and (c) respectively. Figure 3 demonstrates that FCM and FFCM with similar accuracy classify simulated brain MR image to their tissues including white matter (WM), gray matter (GM), and the cerebrospinal fluid (CSF). Figure 4 represents fuzzy clustering results by standard FCM and FFCM in first and second rows successively. Columns from left to right are background, CSF, GM, and WM. As can be seen fuzzy clustering results between FCM and FFCM are highly similar. However, iteration numbers and consumed time by FFCM are comparatively less than FCM.

a

b c

d

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Figure 3. (a) Simulated T1-weighted image corrupted by Gaussian white noise;

Segmentation results using (b) FCM, (c) FFCM.

Figure 4. Fuzzy clustering by FCM and FFCM; Columns from left to right are background, CSF, GM, and WM. First row is fuzzy clustering by FCM, and second

row is fuzzy clustering by FFCM.

To compare performance of standard and fast FCM algorithms, simulations were done on real T1-weighted image. Figure 5 (a) depicts T1-weighted image; (b) and (c) represent the segmentation results by standard and fast FCM respectively. In medical imaging a pixel might be a member of various tissues because of partial volume effect, noise interference or imaging acquisition error. Figure 6 represents fuzzy clustering by standard and fast FCM. The idea of using T1-weighted images in MR image segmentation is high contrast and resolution. T1-weighted MR images use the longitudinal component of magnetic resonance imaging.

a b c

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Figure 5. (a) T1-weighted real MR image. Segmented image using (b) FCM, (c)

FFCM.

Figure 6. Fuzzy clustering results on real MR image. Columns from left to right are

background, WM, GM, and CSF respectively. First and second rows are fuzzy clustering by FCM and FFCM respectively.

To scrutinize performance of standard and fast FCM approaches, simulations were done on several axial and coronal real MR images. Results of experimentations containing CSF, GM, WM, and segmented images are depicted in Figure 7. Standard and fast FCM clustering results are shown in odd and even rows respectively.

a b c

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Figure 7. FCM and FFCM clustering results. Odd rows are results of FCM and

even rows are result of FFCM. The 1st to 5th columns are original image, CSF, GM, WM, and segmented image respectively.

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Quantitative evaluation of FCM and proposed FFCM are represented in Table 1-4. Standard FCM2 is standard FCM initialized by the dist-max algorithm and fast FCM2 is fast FCM initialized by the dist-max algorithm. Table 1 and 2 represent partition coefficient and partition entropy respectively. Table 3 and 4 signify geometric structure Vxb and Vfs. In most cases, the validity functions for the standard and fast FCM are similar; and differences between two techniques are comparatively trivial.

Table 1. Partition coefficient (Vpc) for three images using FCM and FFCM techniques

method simulated 4 level MRI (σ=0.002)

simulated MRI (σ=0.0001)

Real MRI

Standard FCM .858 0.905 0.867 fast FCM 0.858 0.904 0.863 Standard FCM2 0.858 0.904 0.867 fast FCM2 0.859 0.903 0.863

Table 2. Partition entropy (Vpe) for three images using FCM and FFCM techniques method simulated 4 level MRI

(σ=0.002) simulated MRI (σ=0.0001)

Real MRI

Standard FCM 0.126 0.083 0.109 fast FCM 0.126 0.084 0.115 Standard FCM2 0.126 0.084 0.109 fast FCM2 0.126 0.085 0.109

Table 3. Geometric structure(Vxb) for three images using FCM and FFCM techniques method simulated 4 level MRI

(σ=0.002) simulated MRI (σ=0.0001)

Real MRI

Standard FCM 0.037 0.027 0.069 fast FCM 0.039 0.028 0.073 Standard FCM2 0.037 0.027 0.069 fast FCM2 0.039 0.028 0.074

Table 4. Geometric structure (Vfs×(-106)) for three images using FCM and FFCM techniques

metho simulated 4 level MRI (σ=0.002)

simulated MRI (σ=0.0001)

Real MRI

Standard FCM 169 140 195 fast FCM 162 138 189 Standard FCM2 169 140 195 fast FCM2 163 138 185

Elapsed time by standard and fast FCM algorithms is main difference between FCM and FFCM approaches. Furthermore, with using the dist-max algorithm for cluster center initialization elapsed time is decreased more. Without applying dist-mx algorithm for FCM and FFCM techniques, simulations repeated ten times on each image. Same initial cluster centers were selected for both approaches randomly. In all simulations FCM spent more time than FFCM also iteration numbers of FFCM were less than FCM. In Figure 8 and 9 simulation time and iteration numbers are

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shown. As can be seen simulation time of standard FCM2 in all cases has decreased significantly than standard FCM. Also, elapsed time by fast FCM2 is less than fast FCM.

Figure 8. Simulation time to convergence of FCM and FFCM in MR images.

159 9

6

2115

13

7

4642

1916

05

101520253035404550

Standard FCM Standard FCM2 fast FCM fast FCM2

iterations

simulated 4 level MRI (σ=0.002) simulated MRI (σ=0.0001) Real MRI(0)

Figure 9. Iterations numbers to convergence of FCM and FFCM in MR images.

4. Conclusions

Segmentation of medical images using fuzzy clustering methods has numerous applications. FCM is a common unsupervised clustering method in segmentation of medical images. FCM is not fast enough in medical applications particularly in emergency situations. This paper proposed a fast FCM algorithm by a new rule to update cluster centers. Number of iteration steps in the proposed FFCM algorithm significantly was decreased, and subsequently elapsed time was reduced; whereas the performance of proposed approach is similar to standard FCM algorithm. Furthermore, initial cluster centers were assigned to both FCM and FFCM approaches by the dist-max algorithm and as a result, elapsed time and iteration numbers were decreased. Quantitative assessment of FCM and FFCM algorithms was evaluated by traditional fuzzy validation functions. Experimental results prove efficient performance of proposed technique.

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