Enhanced Spatial Fuzzy C-Means Algorithm for Medical Image Segmentation Myeongsu Kang1, Jaeyoung Kim1, Cheol–Hong Kim2, and Jong–Myon Kim1,* Department of Electrical, Electronic, and Computer Engineering, University of Ulsan, Ulsan, South Korea 2 School of Electronics and Computer Engineering, Chonnam National University, Gwangju, South Korea
[email protected],
[email protected],
[email protected],
[email protected]
1
Abstract - Image segmentation is an essential process in image analysis and is mainly used for automatic object recognition. Fuzzy c-means (FCM) is one of the most common methodologies used in clustering analysis for image segmentation. FCM clustering measures the common Euclidean distance between image pixels based on the assumption that each pixel has equal importance. However, image pixels are highly correlated, and thus it is necessary to exploit the spatial information to further improve clustering quality and correct misclassified pixels from noisy regions. To deal with this issue, this paper proposes an enhanced spatial FCM (ESFCM) that takes into account the influence of neighboring pixels on the center pixel by assigning weight to the neighbors by utilizing both pixel intensities and locations. In addition, the proposed ESFCM is robust to impulsive noise due to calculating the membership function of FCM using the vector median in a spatial domain. Experimental results indicate that the proposed ESFCM outperforms other FCM clustering algorithms using spatial information such as spatial fuzzy c-means and spatial fuzzy c-means modified in terms of clustering quality. In addition, the proposed ESFCM is more robust to impulsive noise than the other FCM clustering algorithms.1 Keywords: Enhanced spatial fuzzy c-means, fuzzy c-means, medical image segmentation, vector median
1
Introduction
Image segmentation is an important analysis step for computer-aided diagnosis and therapy [1, 2]. This process separates an image into distinct classes such as brain tumors, edema, and necrotic tissues, enabling early detection of abnormal changes in tissues and organs by quantifying tissue volumes. The field of medicine has become an attractive domain for the application of fuzzy set theory. Fuzzy sets were introduced in 1965 by Lotfi Zadeh to merge mathematical modeling with human knowledge in the engineering sciences [3]. Fuzzy models and algorithms for This work was supported by a National Research Foundataion of Korea (NRF) grant funded by the Korean government (MEST) (No. NRF-2013R1A2A2A05004566). * Corresponding author.
pattern recognition are widely used in advanced information technology [4]. One of the most well-known methodologies in clustering analysis is fuzzy c-means (FCM) clustering which was proposed by Dunn et al. in 1974 and extended by Bezdek in 1981 [5]. The standard FCM utilizes the Euclidean distance between pixels for computing memberships in order to segment an image based on the assumption that each pixel has equal importance; this affects performance degradation of clustering in cases in which neighboring pixels have strong correlation such as magnetic resonance (MR) images [6]. Likewise, the conventional FCM fails to segment images corrupted by noise despite the fact that it performs well on noise-free images. To address these drawbacks, many improved FCM clustering approaches that incorporate local spatial contextual information in images have been proposed. This is useful for reducing noise distortion and intensity inhomogeneity on image segmentation [7, 8]. The modified variant of FCM called spatial FCM clustering was proposed by Chuang et al.; it utilizes spatial information in the FCM membership function and is less sensitive to noise [9]. In spatial FCM, however, an equal weighting factor is given to the adjacent pixels in the predefined window, which results in inaccurate segmentation. To address this problem, Chaudhry et al. utilized weight depending on the contribution of the pixel and that is determined by using the Euclidean distance between neighboring pixels in a predefined window [10]. This method reduced the effects of noise compared to spatial FCM clustering and resulted in the formation of more homogeneous clustering than that of spatial FCM. To further enhance the clustering performance compared to these modified variants of FCM, this paper proposes an enhanced spatial FCM (ESFCM) using weight based on both pixel intensities and pixel locations. Furthermore, this study utilizes the vector median for computing the membership function of the proposed ESFCM, which decreases the number of misclassified pixels due to the impulsive noise inherent in an image. The rest of this paper is organized as follows. Section 2 briefly introduces the standard FCM, and Section 3 proposes a new variant of FCM. Section 4 validates the effectiveness of the proposed enhanced spatial FCM. Finally, Section 5 concludes this paper.
2
Standard fuzzy c-means
3
FCM [5] is one of the most well-known methodologies in clustering analysis. Clustering is the process of portioning an image into regions (or classes) such that each region is homogeneous and none of the unions of two adjacent regions is homogeneous. FCM clustering is an iterative algorithmbased clustering technique that produces an optimal number of c partitions, with centroids V={v1, v2, …, vc} which are exemplars, and radii which define these c partitions. Suppose the unlabeled dataset X={x1, x2, …, xn} is the pixel intensity, where n is the number of image pixels whose memberships are to be determined. The FCM clustering process partitions the dataset X into c clusters. The objective function of the standard FCM is defined as follows:
c
n
J m U , V uikm xk , vi ,
(1)
i 1 k 1
where d(xk, vi) represents the Euclidean distance between pixel xk and centroid vi, and uik represents the fuzzy membership of the kth pixel with respect to cluster i with the
Enhanced spatial fuzzy c-means
In spite of the fact that the image pixels are highly correlated and the spatial relationship of neighboring pixels is an important characteristic for image segmentation, the standard FCM clustering does not fully utilize this spatial information. To deal with this drawback, authors in [9, 10] used the spatial information of neighbors in a predefined window and increased the probability that a pixel in the predefined window belongs to the same cluster if its neighboring pixels belong to a certain cluster. Moreover, the spatial information is helpful for reducing the number of misclassified pixels due to noisy components in an image. To improve clustering performance compared to these modified FCM algorithms, this paper proposes an ESFCM that exploits more spatial information. Unlike standard and modified FCM algorithms, this study finds the vector median of neighbors falling into a predefined window (3×3 window in this study) around xk and utilizes the vector median for computing the membership of pixel xk in order to correct misclassified pixels from noisy regions. Consequently, the membership, uik, in the proposed ESFCM is defined as follows:
c
constraint
uikm 1 , and the degree of fuzzification m≥1.
1
i 1
The data point xk belongs to a specific cluster i which is given by the membership value uik of the data point to that cluster. Local minimization of the objective function Jm(U, V) is accomplished by repeatedly adjusting the values of uik and vi according to the following equations: 1
1 m 1 2 c d xk , vi uik 2 . j 1 d xk , v j
(2)
1 m 1 2 c d VM xk , vi uik 2 , j 1 d VM xk , v j
(4)
where VM(xk) is the vector median of the window around pixel xk. To find the vector median, this study utilizes the cumulative distances criterion and determines the lowestranked input vector as the vector median. Let Nk={x1, x2, …, xn} be a set of neighbors centered on the pixel xk in the predefined window, where n is the dimension of the window. For every vector, xi, in the window, the cumulative distances to all the other vectors using a norm metric are computed, resulting in
n
vi
uikm xk
k 0 n
k 0
, 1 i c.
(3)
uikm
j 1
As Jm(U, V) is iteratively minimized, vi becomes more stable. The pixel clustering iterations are terminated when the
termination measurement max vi(t ) vi(t 1) 1i c
n
is satisfied,
where vi(t) are the new centroids for 1≤i≤c, vi(t-1) are the previous centroids for 1≤i≤c, and ε is a predefined termination threshold. The output of the FCM algorithm is the cluster centroids V and the fuzzy partition matrix UC×N.
2
Li xi x j , i 1, 2,3,..., n, i j. 2
(5)
The vector median, VM(xk), is associated with the input vector yielding the minimum accumulated distance. In general, MR images include impulsive noise, which is independent and uncorrelated with the image pixels. In addition, this noise is randomly distributed over an image. Thus, uncertainty is widely presented due to impulsive noise, which results in low clustering performance. The proposed membership function reduces the misclassified pixels due to the impulsive noise by exploiting the vector median instead
of the center pixel in the window. In addition, the ESFCM utilizes a neighbor-weighting coefficient, pik, to further improve the segmentation performance, which is defined as follows:
,
n uij d L xk , L x j pik d 2 xk , x j j 1
(6)
where L(xk) and L(xj) are the locations of pixels xk and xj in the window, respectively. Likewise, d(L(xk), L(xj)) is the Euclidean distance, which can be expressed by
2 k x jx 2 k y j y , if L xk k x , k y , L x j j x , j y .
d L xk , L x j
(7)
The neighbor-weighting coefficient using both pixel intensity and location information of a pixel for a cluster leads to a higher probability if the majority of its neighborhood belongs to the same clusters. In other words, the greater number of neighbors in the same cluster, the higher is the probability that the center pixel is in that cluster. The weighted coefficient function, f(pik), is incorporated into the membership function of the standard FCM, and a new membership function, wik, is defined as follows:
wik
uik f
1
c
u jk f j 1
m 1 1
pik
m 1
p jk
.
(8)
Using the new membership function, wik, the centroid values, vi, of the proposed ESFCM are computed such that
partitions U 1/ c when m→∞. In contrast, when m→1, this reduces to hard c-means and terminal partitions become more and more crisp. In the method of Bezdek [5], the authors experimentally determined the optimal interval for the degree of fuzzification and found it to range from 1.1 to 5. In this study, we selected the value of m as 2 so as to have an optimal balance of speed and accuracy for all of the FCMbased clustering algorithms. The termination threshold ε controls the duration of iteration as well as the optimal terminal partition of the fuzzy clustering. Bezdek [5] experimentally determined the optimal interval for the termination threshold and found it to range from 0.01 to 0.0001. In this study, we selected the termination threshold value to be 0.001. The initialization of the centroid of a cluster is also important in FCM clustering because it is a searching technique that yields local maxima, thus greatly reducing the performance of clustering. In addition, when clustering is initialized from a different starting point, different solutions are found for the same terminal partition. In this study, the centroids were initialized by assigning the number of clusters (denoted as c), with points uniformly distributed according to the gray image (intensities ranging from 0 to 255).
4.2
Segmentation results
This paper evaluated the correctness of the segmentation using real brain scans with ground truth given by expert segmentations obtained from the Internet Brain Segmentation Repository (IBSR) website (http://www.cma.mgh.harvard.edu/ibsr/). A brain scan given is composed of a variable number of slices. We processed the slices individually. Figs. 1(a)–(b) show an example of the slices and the manual labeling provided by the IBSR site. A comparison of the segmentation results obtained by applying four clustering algorithms on a T1-weighted MR phantom is shown in Fig. 1. The segmentation results attained for two slices using four methods are shown in Figs. 1(c)-(f). In addition, the quantitative comparison scores corresponding to Fig. 1(a) for gray matter (GM) and white matter (WM) are given in Table 1, and comparison scores are computed as follows:
n
vi
wikm xk k 1 n
wikm
, 1 i c.
si , j
(9)
k 1
4 4.1
Performance evaluation Parameter setup
Initialization for the degree of fuzzification m is very important in FCM. FCM clustering produces terminal
Ai , j Aref , j Ai , j Aref , j
,
(10)
where Ai,j represents the set of pixels belonging to the jth class found by the ith algorithm, and Aref,j represents the set of pixels belonging to the jth class in the reference segmented image. From Fig. 1 and Table 1, it can be seen that the proposed ESFCM outperforms the conventional algorithms.
Fig. 1. Comparison of the segmentation results on a simulated brain MR image. (a) Original T1-weighted image, (b) manual class labeling of gray matter (GM) and white matter (WM) slice regions ; results obtained with (c) standard FCM, (d) spatial FCM [9], (e) spatial FCM modified [10], (f) the proposed ESFCM
Fig. 2. (a) Synthetic circle image with small gradient noise, and resulting images of clustering using (b) standard FCM, (c) spatial FCM [9], (d) spatial FCM modified [10] and (e) the proposed ESFCM Table 1 : Comparison scores of four segmentation approaches with Fig. 1(a)
which is obtained from the IBSR website and Figs. 2(b)-(e) show the clustering results of four clustering methodologies. As shown in Fig. 2, the clustering result of the proposed ESFCM is superior to those of other algorithms.
Method
GM
WM
Standard FCM
0.6223
0.7870
5
Spatial FCM [9]
0.6478
0.7887
Spatial FCM modified [10]
0.6614
0.7833
Proposed ESFCM
0.7167
0.8014
FCM is one of the most well-known clustering algorithms. However, it uses the common Euclidean distance based on the assumption that each pixel has equal importance, resulting in clustering performance degradation since image pixels are highly correlated. To address this issue, this paper proposed ESFCM, which takes into account the influence of the neighboring pixels on the center pixel. In addition, this study utilized the vector median to reduce misclassified pixels due to the impulsive noise inherent in an image. Experimental results indicate that the proposed ESFCM significantly outperforms other FCM-based clustering algorithms.
We also simulated the effect of noise with target images for segmentation. In the conventional FCM, a noisy pixel can be wrongly classified due to its abnormal feature data. However, the proposed ESFCM can greatly reduce the effect of noise by incorporating spatial information. Fig. 2(a) depicts a synthetic circle image with small gradient noise
Conclusions
6
References
[1] S. Shen, W. Sandham, M. Grant, and A. Sterr, “MRI fuzzy segmentation of brain tissue using neighborhood attraction with neural-network optimization,” IEEE Transactions on Information Technology in Biomedicine, vol. 9, no. 3, pp. 459–467, 2005. [2] K. S. Fu and J. K. Mu, “A survey on image segmentation,” Pattern Recognition, vol. 13, pp. 3–16, 1981. [3] L. A. Zadeh, “Fuzzy sets,” Information Control, vol. 8, pp. 338–353, 1965. [4] J. C. Bezdek, J. Keller, R. Krisnapuram, and N. R. Pal, “Fuzzy models and algorithms for pattern recognition and image processing.” Springer, 1st edition, 2005. [5] J. C. Bedzdek, “Pattern recognition with fuzzy objective function algorithms.” Pleum Press, New York, 1981. [6] S. R. Kannan, S. Ramathilagam, R. Devi, and E. Hines, “Strong fuzzy c-means in medical image data analysis,” Journal of Systems and Software, vol. 85, no. 11, pp. 2425– 2438, 2012. [7] C. Qiu, J. Xiao, L. Yu, L. Han, and M. N. Iqbal, “A modified interval type-2 fuzzy c-means algorithm with application in MR image segmentation,” Pattern Recognition, vol. 34, no. 12, pp. 1329–1338, 2013. [8] M. Gong, Y. Liang, J. Shi, W. Ma, and J. Ma, “Fuzzy cmeans clustering with local information and kernel metric for image segmentation,” IEEE Transactions on Image Processing, vol. 22, no. 2, pp. 573–584, 2013. [9] K. –S. Chuang, H. –L. Tzeng, S. Chen, J. Wu, and T. –J. Chen, “Fuzzy c-means clustering with spatial information for image segmentation,” Computerized Medical Imaging and Graphics, vol. 30, no. 1, pp. 9–15, 2006. [10] A. Chaudhry, M. Hassan, A. Khan, J. Y. Kim, and T. A. Tuan, “Image clustering using improved spatial fuzzy cmeans,” Proceedings of the 6th International Conference on Ubiquitous Information Management and Communication, Kuala Lumpur, Malaysia, February 2012, pp. 1–7.
