Fuzzy c-means clustering with spatial information for image segmentation
Keh-Shih Chuang
a,*, Hong-Long Tzeng
a,b, Sharon Chen
a, Jay Wu
a,b, Tzong-Jer Chen
c aDepartment of Nuclear Science, National Tsing-Hua University, Hsinchu 30013 TaiwanbHealth Physics Division, Institute of Nuclear Energy Research, Atomic Energy Council, Taiwan cDepartment of Medical Imaging Technology, Shu-Zen College of Medicine and Management, Kaohsiung, Taiwan
Received 20 August 2004; revised 26 August 2005; accepted 6 September 2005
Abstract
A conventional FCM algorithm does not fully utilize the spatial information in the image. In this paper, we present a fuzzy c-means (FCM) algorithm that incorporates spatial information into the membership function for clustering. The spatial function is the summation of the membership function in the neighborhood of each pixel under consideration. The advantages of the new method are the following: (1) it yields regions more homogeneous than those of other methods, (2) it reduces the spurious blobs, (3) it removes noisy spots, and (4) it is less sensitive to noise than other techniques. This technique is a powerful method for noisy image segmentation and works for both single and multiple-feature data with spatial information.
q2005 Published by Elsevier Ltd.
Keywords: Fuzzy c-means; Spatial information; Image segmentation; Clustering
1. Introduction
Many neurological conditions alter the shape, volume, and distribution of brain tissue; magnetic resonance imaging (MRI) is the preferred imaging modality for examining these conditions. Reliable measurement of these alterations can be performed by using image segmentation. Several investigators have developed methods to automate such measurements by segmentation [1–4]. However, some of these methods do not exploit the multispectral information of the MRI signal.
Fuzzy c-means (FCM) clustering[1,5,6]is an unsuper-vised technique that has been successfully applied to feature analysis, clustering, and classifier designs in fields such as astronomy, geology, medical imaging, target recognition, and image segmentation. An image can be represented in various feature spaces, and the FCM algorithm classifies the image by grouping similar data points in the feature space into clusters. This clustering is achieved by iteratively minimizing a cost function that is dependent on the distance of the pixels to the cluster centers in the feature domain.
The pixels on an image are highly correlated, i.e. the pixels in the immediate neighborhood possess nearly the same feature data. Therefore, the spatial relationship of neighboring pixels is an important characteristic that can be of great aid in imaging segmentation. General boundary-detection techniques have taken advantage of this spatial information for image segmentation. However, the conven-tional FCM algorithm does not fully utilize this spatial information. Pedrycz and Waletzky [7] took advantage of the available classified information and actively applied it as part of their optimization procedures. Ahmed et al. [8]
modified the objective function of the standard FCM algorithm to allow the labels in the immediate neighborhood of a pixel to influence its labeling. The modified FCM algorithm improved the results of conventional FCM methods on noisy images. However, the way in which they incorporate the neighboring information limits their application to single-feature inputs.
The aim of this study is to introduce a new segmentation method for FCM clustering. In a standard FCM technique, a noisy pixel is wrongly classified because of its abnormal feature data. Our new method incorporates spatial infor-mation, and the membership weighting of each cluster is altered after the cluster distribution in the neighborhood is considered. This scheme greatly reduces the effect of noise and biases the algorithm toward homogeneous clustering.
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* Corresponding author. Tel.: C886 3 574 2681; fax: C886 3 571 8649. E-mail address: kschuang@mx.nthu.edu.tw (K.-S. Chuang).
2. Method
2.1. FCM clustering
The FCM algorithm assigns pixels to each category by using fuzzy memberships. Let XZ(x1, x2,.,xN) denotes an
image with N pixels to be partitioned into c clusters, where xirepresents multispectral (features) data. The algorithm is
an iterative optimization that minimizes the cost function defined as follows: J ZX N jZ1 Xc iZ1 umijkxjKvik 2; (1)
where uij represents the membership of pixel xjin the ith
cluster, viis the ith cluster center, k$k is a norm metric, and
m is a constant. The parameter m controls the fuzziness of the resulting partition, and mZ2 is used in this study.
The cost function is minimized when pixels close to the centroid of their clusters are assigned high membership values, and low membership values are assigned to pixels with data far from the centroid. The membership function represents the probability that a pixel belongs to a specific cluster. In the FCM algorithm, the probability is dependent solely on the distance between the pixel and each individual
cluster center in the feature domain. The membership functions and cluster centers are updated by the following:
uijZ 1 Pc kZ1 kxjKvik kxjKvkk 2=ðmK1Þ; (2) and viZ PN jZ1 umijxj PN jZ1 um ij : (3)
Starting with an initial guess for each cluster center, the FCM converges to a solution for vi representing the local
minimum or a saddle point of the cost function. Convergence can be detected by comparing the changes in the membership function or the cluster center at two successive iteration steps.
2.2. Spatial FCM
One of the important characteristics of an image is that neighboring pixels are highly correlated. In other words, these neighboring pixels possess similar feature values, and
the probability that they belong to the same cluster is great. This spatial relationship is important in clustering, but it is not utilized in a standard FCM algorithm. To exploit the spatial information, a spatial function is defined as
hijZ X
k2NBðxjÞ
uik; (4)
where NB(xj) represents a square window centered on pixel
xj in the spatial domain. A 5!5 window was used
throughout this work. Just like the membership function, the spatial function hijrepresents the probability that pixel xj
belongs to ith cluster. The spatial function of a pixel for a cluster is large if the majority of its neighborhood belongs to the same clusters. The spatial function is incorporated into membership function as follows:
uij0 Z upijhqij Pc kZ1 upkjhqkj ; (5)
where p and q are parameters to control the relative importance of both functions. In a homogenous region, the spatial functions simply fortify the original membership, and the clustering result remains unchanged. However, for a noisy pixel, this formula reduces the weighting of a noisy
cluster by the labels of its neighboring pixels. As a result, misclassified pixels from noisy regions or spurious blobs can easily be corrected. The spatial FCM with parameter p and q is denoted sFCMp,q. Note that sFCM1,0is identical to
the conventional FCM.
The clustering is a two-pass process at each iteration. The first pass is the same as that in standard FCM to calculate the membership function in the spectral domain. In the second pass, the membership information of each pixel is mapped to the spatial domain, and the spatial function is computed from that. The FCM iteration proceeds with the new membership that is incorporated with the spatial function. The iteration is stopped when the maximum difference between two cluster centers at two successive iterations is less than a threshold (Z0.02). After the convergence, defuzzification is applied to assign each pixel to a specific cluster for which the membership is maximal.
2.3. Cluster validity functions
Two types of cluster validity functions, fuzzy partition and feature structure, are often used to evaluate the performance of clustering in different clustering methods. The representative functions for the fuzzy partition are partition coefficient Vpc[9] and partition entropy Vpe[10].
They are defined as follows: VpcZ PN j Pc i u2 ij N (6) and VpeZ KP N j Pc i ½uijlog uij N (7)
The idea of these validity functions is that the partition with less fuzziness means better performance. As a result, the best clustering is achieved when the value Vpc is
maximal or Vpeis minimal.
Disadvantages of Vpcand Vpeare that they measure only
the fuzzy partition and lack a direct connection to the featuring property. Other validity functions based on the feature structure are available[11,12]. For example, Xie and Beni[11]defined the validity function as
VxbZ KP N j Pc i uijjjxjKvijj 2 N minisk jjvkKvijj 2 : (8)
A good clustering result generates samples that are compacted within one cluster and samples that are separated
between different clusters. Minimizing Vxb is expected to
lead to a good clustering.
2.4. Image data
In this study, we used one T1-weighted image and one T2-weighted MRI image from the same patient. To demonstrate the effect of noise on the segmentation, both images were added with a uniform random noise of magnitude between (K50, 50). The images were divided into six clusters: gray matter (GM), white matter (WM), cerebrospinal fluid, fat, bone, and air. In addition, two synthetic images with four gray levels are generated to serve as ‘ground truth’ images for segmentation evaluation. The images were then corrupted by using a uniform noise between (Kn, n), where nZ3.5!(pixel value)1/2.
3. Results and discussion
Fig. 1(a) and (b) show the T1- and T2-weighted images used for the study, respectively. Fig. 2(a) shows the segmentation results obtained by using a standard FCM algorithm and Fig. 2(b) and (c) show the results of the FCM incorporated into the spatial information with parameters (pZ1, qZ1) and (pZ0, qZ2), respectively.
The conventional FCM successfully classifies the MRI images into six clusters. However, spurious blobs of GM appear inside the WM cluster. The spatial function modifies the membership function of a pixel according to the membership statistics of its neighborhood. Such neighbor-ing effect biases the solution toward piecewise-homo-geneous labeling. Both sFCM techniques reduce the number of spurious blobs, and the segmented images are more homogeneous. The sFCM algorithm with a higher q parameter shows a better smoothing effect. The possible disadvantages of using higher spatial weighting are the blurring of some of the finer details. However, this is difficult to judge from the results.
Fig. 1(c) and 1(d) show the T1 and T2 images added with a uniform noise. Fig. 3 shows the segmented results of a noisy MRI image by using the three FCM techniques. As can be seen, the standard FCM technique misclassifies GM and WM at numerous places because the added noise changes the location of GM and WM pixels in the feature space and causes the misclassification of these noisy pixels. Because no similar cluster is present in the neighborhood, the weight of the noisy cluster is greatly reduced with sFCM. Furthermore, the membership of the correct cluster
is enhanced by the cluster distribution in the neighboring pixels. As a result, both sFCM techniques effectively correct the misclassification caused by the noise.
Fig. 4shows the segmentation results of MRI images in the feature domain for all FCM techniques. The cluster centered on the noisy image was different from that of the original noiseless image with standard FCM clustering. The sFCM techniques successfully corrected the misclassified pixels and kept the cluster centers unaffected by noise.
Fig. 5(a) and (b) depict the corrupted four-level simulated T1 and T2 images, and Fig. 5(c)–(e) show the clustering results of the FCM techniques. As can be seen, the clustering results of our sFCM algorithms were superior to those obtained by using conventional FCM. The classification error of the FCM was mostly the salt-and-pepper type. These spots were corrected with the sFCM by using the spatial contextual information.
Table 1tabulates the validity functions used to evaluate the performance of FCM clustering for six images. In most cases, the validity functions based on the fuzzy partition were better for the sFCM than the conventional FCM. The sFCM1,1technique showed the best clustering results. The
improvement was even better for noisy images. For
Fig. 4. The calculated centroids (5) of cluster in the feature domain (where T1 and T2 values are coordinates) for (a) FCM on the original image; (b) FCM on the noisy image; (c) sFCM1,1on the noisy image; and (d) sFCM0,2on the noisy image.
example, the reduction in Vpe with the sFCM1,1, as
compared with the conventional technique, was 50% for original images and 64.6% for noisy data.
Paired-sample t tests were performed on these data sets to test the significance of any difference in the measures.
Table 2lists statistical results. For Vpe(Vpc), the sFCM1,1is
significantly greater (smaller) than conventional FCM and sFCM0,2. However, the validity function based on feature
structure showed different results: Vxb increased for the
proposed FCM, which was understandable. Vxb measured
the compactness in the feature domain. Conventional FCM achieves partition by minimizing the metric difference in the feature domain, and its Vxb is minimized. The sFCM
modifies the partition on the basis of spatial distribution and causes deterioration of the compactness in the feature domain and subsequently the increase in Vxb.
Other variations of spatial function are available. For example, it can be expressed as
hijZ X k2NBðxjÞ gik (9) where gikZ 1; if uikRulkfor l Z 1; c 0; otherwise : (
The calculation of this membership has less fuzziness than others and causes a loss in the details.
This technique is applied only to image data with spatial information. The spatial function can be
Fig. 5. Simulated (a) T1 and (b) T2 image corrupted by additive uniform noise. The gray levels are 50 (UL), 100 (UR), 150 (LL), and 200 (LR) in the T1 image and 50 (UL), 150 (UR), 100 (LL), and 200 (LR) in the T2 image. Clustering resulted using (c) FCM; (d) sFCM1,1; and (e) sFCM0,2.
Table 1
The clustering results of six images using various FCM techniques
Images Tech-niques Vpc Vpe Vxb(!10 K3 ) Original MRI images FCM 0.888 0.234 0.918 sFCM1,1 0.924 0.117 2.46 sFCM0,2 0.899 0.160 3.38
Noise added MRI images FCM 0.779 0.414 1.22 sFCM1,1 0.909 0.147 3.14 sFCM0,2 0.884 0.191 4.14 Synthesized image uniform noise (K35,35) FCM 0.718 0.571 0.620 sFCM1,1 0.915 0.133 0.748 sFCM0,2 0.911 0.146 0.789 Synthesized image uniform noise (Kn, n) FCM 0.740 0.529 0.563 sFCM1,1 0.920 0.122 0.688 sFCM0,2 0.916 0.133 0.728 Synthesized image SNRZ5 FCM 0.685 0.615 0.793 sFCM1,1 0.840 0.271 1.032 sFCM0,2 0.832 0.301 1.182 Synthesized image SNRZ10 FCM 0.857 0.317 0.292 sFCM1,1 0.959 0.032 0.333 sFCM0,2 0.952 0.044 0.355
The MRI images and the simulated four-level T1 and T2 images are the same as what is shown inFigs. 1 and 5.
incorporated with other FCM techniques without a great deal of modifications. For example, the feature-weight learning FCM technique [13] assigns various weights to different features to improve the performance of clustering. The spatial function can be estimated at each iteration and incorporated into the membership function. Owing to the incorporation of spatial infor-mation, the new FCM technique is less sensitive than other methods to noise.
4. Summary
FCM clustering is an unsupervised clustering technique applied to segment images into clusters with similar spectral properties. It utilizes the distance between pixels and cluster centers in the spectral domain to compute the membership function. The pixels on an image are highly correlated, and this spatial information is an important characteristic that can be used to aid their labeling. However, the spatial relationship between pixels is seldom utilized in FCM.
In this paper, we proposed a spatial FCM that incorporates the spatial information into the membership function to improve the segmentation results. The membership functions of the neighbors centered on a pixel in the spatial domain are enumerated to obtain the cluster distribution statistics. These statistics are trans-formed into a weighting function and incorporated into the membership function. This neighboring effect reduces the number of spurious blobs and biases the solution toward piecewise homogeneous labeling. The new method was tested on MRI images and evaluated by using various cluster validity functions. Preliminary results showed that the effect of noise in segmentation was considerably less with the new algorithm than with the conventional FCM.
Acknowledgements
This work is supported in part by a research grant of NSC92-2218-E007-012 from the National Science Council, Taiwan.
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Table 2
Statistical results of the paired-sample t-test on the differences in the three performance index
(X:Y) W SW T p-value Vpc (sFCM1,1: FCM) 0.133 0.0586 5.57 0.001!p!0.0025 (sFCM1,1: sFCM0,2) 0.0122 0.01 2.99 0.025!p!0.01 Vpe (sFCM1,1: FCM) K0.309 0.115 K6.58 p!0.001 (sFCM1,1: sFCM0,2) K0.0255 0.0156 K4.00 py0.005 Vxb (sFCM1,1: FCM) 0.666 0.836 1.95 pO0.05 (sFCM1,1: sFCM0,2) K0.362 0.466 K1.90 pO0.05 H0: mxZmy, H1: mxOmy, WiZXiKYi, T Z W= SW= ffiffiffin p
