A Modified Intuitionistic Fuzzy Clustering Algorithm for Medical Image Segmentation

1 Introduction

In clinical diagnosis, medical imaging plays a crucial role which helps in understanding and analysis of the organs. Medical imaging is a technique to create an internal image of a human body for clinical or medical purpose. Medical images are obtained by using different modalities that includes X-ray, computed tomography (CT), positron emission tomography and magnetic resonance imaging (MRI). Segmentation is an indispensable step in medical image analysis which helps in identifying appropriate therapy for abnormal changes in tissues and organs. In the literature, there are many classical image processing techniques like thresholding, region growing and merging applied for medical image segmentation. However, these methods fail to give good segmentation results because the techniques work only on pixel attributes. Many researchers have applied clustering technique to solve medical image segmentation problems.

Cluster analysis involves classifying a collection of objects into homogeneous groups, such that objects in the same group are more similar when compared to objects present in other groups. Clustering algorithms are broadly classified into hierarchical and partitional clustering [20]. Hierarchical clustering algorithm generates a hierarchical tree of clusters called dendrogram which can be either divisive or agglomerative. The partitional clustering algorithm gives a single C partition of objects, with a predefined C number of clusters. There are two well-known partition clustering approaches: k-means and fuzzy C-means (FCM). In k-means clustering, data are divided into a number of clusters where data elements belong to exactly one cluster. The k-means clustering technique works well when data elements are not overlapped. To overcome the problem of overlapping, Bezdek [5] proposed FCM clustering algorithm. In FCM clustering, data elements are clustered based on the membership values assigned to each of them. In this technique, the data elements can belong to more than one cluster. In medical images, object definition is not always crisp and knowledge about objects in the image may be vague. To deal with vagueness, fuzzy logic, and fuzzy theory are ideally suited. Many researchers have developed fuzzy set theory based clustering methods to solve image segmentation problem [1], [11], [12], [21], [30], [33], [40]. Maji and Roy [27] proposed a texture based brain MRI image segmentation method. Texture features are selected based on maximum relevance and maximum significance. Further, rough fuzzy clustering has been applied for segmentation. Feng et al. [15] developed a modified FCM algorithm for MR image segmentation. This method incorporated nonlocal pixel information via Hausdroff distance metric. The objective and membership function of traditional FCM is modified with addition of Hausdroff distance function.

In medical images, uncertainty occurs in terms of vagueness in imprecise gray levels, object boundary and so on. While defining membership function, hesitation arises due to the presence of uncertainty in gray levels. Unfortunately, classical fuzzy clustering techniques fail to handle this hesitation. To handle this hesitation, Atanassov in [4] proposed another higher order fuzzy set called intuitionistic fuzzy set (IFS). IFS considers both membership (μ) and nonmembership (v) values. In traditional fuzzy set, the nonmembership degree is equal to the complement of the membership degree, but in IFS nonmembership the degree is less than or equal to the complement of the membership degree due to the index of intuitionism or hesitation degree. IFS found various applications in areas such as logic programming, medical image processing, decision making, etc. Recently, IFS based methods have been receiving great attention from researchers, and a lot of IFS based clustering techniques have been developed [8], [9], [22], [47], [53]. Chaira in [8] has developed a novel medical image segmentation method based on intuitionisitc fuzzy C-means algorithm. This method incorporates intuitionistic fuzzy entropy based objective function to the traditional FCM algorithm. Zhang et al. [53] proposed an IFS clustering method. This method creates intuitionistic fuzzy similarity matrix using similarity degree between two IFSs. Xu et al. [47] developed a clustering algorithm for IFS based on the concept of association matrix and equivalent association matrix. This method calculates association coefficient by considering hesitation degree. Chaira and Anand [9] developed a novel IFS approach for tumor detection in medical images. This method uses histogram thresholding to eliminate unwanted regions from clustered image. Further, the edge of the tumor is extracted. Zhao et al. [54] investigated the minimum spanning tree based intuitionistic fuzzy clustering (IFCM). Further, the same method was extended for interval-valued IFS (IVIFS). The IFCM is based on intuitionistic fuzzy equivalence matrices. As a result they need high computation time and they may fall into local optimal solution. To overcome these drawbacks, many researchers proposed variants of IFCM [42], [43], [48], [49], [55].

Xu et al. [49] developed an intuitionistic spectral clustering method. This method is based on eigen value analysis which takes less computation time and is not easy to fall into the local optimal solution. Wang et al. [42] developed an IFCM based on netting technique. The aforementioned method uses a new intuitionistic fuzzy similarity metric to construct an intuitionistic fuzzy similarity matrix. Experimentation results on numerical example reveals that this method takes less computation time than traditional IFCM. Wang et al. [43] developed a direct method based IFCM. This method clusters input data based on the intuitionistic fuzzy triangular product and square product. Zhao et al. [55] developed a new IFCM using λ cutting matrix of a similarity matrix. Further, the same method was extended for IVIFS. Lin [22] proposed a novel evolutionary kernel intuitionistic fuzzy C-means (EKIFCM) clustering algorithm. EKIFCM combines IFS with kernel fuzzy C-means and genetic algorithm. A genetic algorithm is used to select the parameters of EKIFCM. Chaudhuri [10] proposed an intuitionistic fuzzy possibilistic C-means clustering algorithm. This algorithm combines fuzzy possibilistic C-means with the IFS. Further, this method is extended to cluster interval valued IFSs. Huang et al. [17] proposed a novel hybrid model by combining neighborhood intuitionistic fuzzy C-means and genetic algorithm. This method uses neighborhood membership to reduce the noise/outlier influence. The optimal parameters are selected using genetic algorithm.

In the past decade, granular computing has emerged as a new research topic and has been widely applied for clustering, data mining and other fields. Granular computing is motivated by human problem solving strategies, and it deals with the information granules derived from underlying complex data. Recently, many researchers developed various clustering methods based on granular computing theory [2], [3], [13], [14], [23], [24], [25], [26], [28], [29], [32], [34], [35], [44], [46], [51]. Medical images are not equally and well illuminated. IFS deals with the uncertainty in the form of hesitation degree. So the results obtained using IFS based methods are better than the fuzzy methods. In recent days, IFS is being explored for more accurate segmentation results.

The existing IFS based method uses Sugeno and Terano’s [36] and Yager’s [50] complement generators to construct the IFS. If A = {x₁, x₂, …, x_n} is a finite set, then IFS A in a universe of discourse X is represented mathematically as A = {〈x, μ_A(x), v_A(x)〉 | x ∈ X}. The complement generators are defined as follows:

1. Sugeno and Terano’s negation [36]

   (1)cλ(x)=1 − μA(x)1+λμA(x);   λ>0

2. Yager’s negation [50]

   (2)cλ(x)=(1 − μA(x)λ)1λ;   λ>0

where λ is constant. Ref. [4] defines that IFS should satisfy the elementary condition of intuitionism, i.e. the index of intuitionism or hesitation degree which is obtained by subtracting the sum of the membership and nonmembership from 1 should be positive and less than or equal to 1. However, while considering the above complement generators, for certain values of λ, the index of intuitionism or hesitation degree becomes negative. Therefore, these complement generators fail to satisfy the elementary condition given by [4]. To overcome this problem, Bustince et al. [7] developed a new intuitionistic fuzzy generator which satisfies the elementary condition.

In soft clustering methods, the membership value is computed based on a distance function. So distance metric plays an important role. In the literature, many distance metrics are proposed for IFS [37], [38], [39]. Hung and Yang [18] developed similarity measures of IFSs based on Hausdorff distance. Experimental results show that Hausdorff distance is simple and works better than other distance metrics. Hence, in this paper, our motivation is to take advantage of IFS and Hausdorff distance to increase the segmentation accuracy. In this paper, we are proposing a medical image segmentation algorithm using modified intuitionistic fuzzy C-means.

In summary, the main contributions of this paper are the following:

• We presented a modified IFCM algorithm for medical image segmentation.

• To overcome the drawback of existing IFS generators, we used Bustince IFS generator.

• We used modified Hausdorff distance measure to compute the distance between cluster center and pixel value.

• We conducted experiments on standard MRI brain image dataset and our own created dataset of CT scan of lung tumor images, CT scan of liver images and MRI breast images.

• We validated the proposed method using four cluster validity functions.

The rest of the paper is organized as follows: Section 2 gives background information about IFS and Hausdorff distance. Section 3 presents the proposed method. The datasets used for experimentation and experimental results are presented in Section 4. Conclusion and future scope are presented in Section 5.

2 Background

3 Proposed Method

The proposed method consists of two steps: intuitionistic representation of image and IFCM.

3.2 Modified Intuitionistic Fuzzy Clustering

A modified IFCM algorithm for segmenting medical images is used in the present study. The proposed method clusters feature vectors by searching for local minima with the following objective function:

(20)min Jm=∑j=1p∑i=1Cμijmd(Aj, Vi)

where A = {A₁, A₂, …., A_p} are p IFSs each with n elements, C is the number of clusters and V = {V₁, V₂, …., V_C} are cluster centers. m is fuzzy coefficient value. μ_ij is membership value of the j^th pixel to i^th cluster. d is distance between cluster center (V_i) and IFS of pixel values (A). This membership value μ_ij satisfies the following condition:

(21)∑i=1Cμij=1, 1≤j≤p

(22)μij≥0, 1≤i≤C, 1≤j≤p

(23)∑j=1pμij>0, 1≤i≤C

If we solve optimization problem in equation (20) by employing Lagrange multiplier method [19], membership value becomes

(24)μij=1∑k=1c(d(Aj, Vi)d(Aj, Vk))2m − 1

Cluster center V_i which is associated with three values (μVi(xk)), υVi(xk) and πVi(xk)) is calculated using the following equations:

(25)μVi(xk)=∑j=1puijmμAj(xk)∑j=1puijm, 1≤i≤c, 1≤k≤n

(26)υVi(xk)=∑j=1puijmυAj(xk)∑j=1puijm, 1≤i≤c, 1≤k≤n

(27)πVi(xk)=∑j=1puijmπAj(xk)∑j=1puijm, 1≤i≤c, 1≤k≤n

The performance of clustering algorithm directly depends on distance metric. The existing IFCM methods use euclidean distance metric. This metric fails to give good results for nonspherical input data. To overcome this drawback, we used modified Hausdorff distance. Unlike existing Hausdorff distance metric, modified Hausdorff distance considers hesitation degree along with membership and nonmembership value. Modified Hausdorff distance is defined as follows:

(28)dH(Aj, Vi)=max{|μAj − μVi|, |υAj − υVi|, |πAj − πVi|}

In this paper, we used modified Hausdorff distance function to calculate the distance between cluster center and data points. We replace the distance function d in equations (20) and (24) by equation (28). After converting input image into intuitionistic fuzzy set representation, we initialize the cluster centers randomly and perform clustering algorithm. At each iteration, the cluster centers and membership value are updated using equations (24)–(27). Algorithm stops when {J(i) − J(i − 1)}< stopping criteria. To obtain the segmented image, pixels are assigned to a cluster based on maximum membership value. The individual steps of the proposed method are given in Algorithm 1.

4 Experimental Results

To demonstrate the effectiveness of proposed method, we performed experiments on medical images from different modalities including standard MRI Brain images and our own created datasets that include CT scan of lung tumor, CT scan of liver and MRI breast images.

We compared our proposed method (Alg6) with all five algorithms listed in Table 1. The algorithm uses two IFS generators and two distance metrics. For example, Alg1 uses Sugeno’s IFS generator and euclidean distance metric. In this paper, for all algorithms in comparison, we set fuzzy coefficient m to widely used value 2. All the cluster centers are initialized randomly. We set stopping criterion ε to 0.0001. All algorithms were simulated using matlab2013.

Performance of the proposed method was evaluated using cluster validity indices. Wide varieties of cluster validity indices are present in the literature [41]. In this study, we have used four well-known cluster validity functions: partition coefficient (V_pc), partition entropy (V_pe), Fukuyama-Sugeno function (V_fs) and Xie-Beni function (V_xb). Bezdek [5] proposed partition coefficient (V_pc) and partition entropy (V_pe), which uses only membership values to calculate the cluster validity value. Equations (29) and (30) show the V_pc and V_pe:

The value of V_pc varies between [1C, 1] where C indicates the number of clusters. The value of V_pe ranges between [0, log_aC] where C is the number of cluster and a is the base of the logarithm. When V_pc is maximum or V_pe is minimum, optimal clusters are achieved. The third cluster validity function used is Fukuyama-Sugeno (V_fs) [16] which is given by

where υ¯=1C∑i=1Cυi.V_fs uses both the membership information and input data. When V_fs value is minimum, better clustering results are achieved. The fourth function is Xie-Beni (V_xb) which was initially proposed by Xie and Beni (XB) [45] and modified by Pal and Bezdek [31]. It is defined as

In V_xb, the numerator indicates compactness of the fuzzy partition and the denominator indicates strength of the separation between clusters. When V_xb is minimum, better clustering results are achieved.

4.2 Experiments on Medical Images

From illustrations presented in the previous section, we can observe that our proposed method outperforms the existing methods. In this section, we conducted experiments specifically on medical images from different modalities including MRI brain, CT scan of lung, CT scan of liver and MRI breast images. The MRI image of brain chosen for the experiment is available in three bands: T1-weighted, proton density (pd)-weighted and T2-weighted. Normal brain images are obtained from Brain-web database [6]. In this paper, we use transversal slice map with slice thickness of 1 mm and size of 217×181 pixels. Further, we have created our own dataset of lung, liver and breast images. Our dataset consists of 50 different lung images, 30 different liver images and 50 MRI breast images.

Since Alg1 and Alg4, Alg2 and Alg5, and Alg3 and Alg6 use the same IFS generator, human perception fails to notice the difference in segmentation results. So we presented the results only for Alg1, Alg2 and Alg6. Figures 1–3 show the segmentation results of the Alg1, Alg2 and Alg6, respectively.

Figure 1:

Segmentation Result of Alg1 With Different λ Values on Brain Image With Cluster Number = 4 (A) Input Image (B) λ = 0.5, (C) λ = 1, (D) λ = 5.

Figure 2:

Segmentation Result of Alg2 With Different λ Values on Brain Image With Cluster Number = 4 (A) Input Image (B) λ = 0.5, (C) λ = 1, (D) λ = 5.

Figure 3:

Segmentation Result of Alg6 With Different λ Values on Brain Image With Cluster Number = 4 (A) Input Image (B) λ = 0.5, (C) λ = 1, (D) λ = 5.

We varied λ value from 0.1 to 30 with a uniform increment of 0.1. Since results are not varied in between values, we recorded the results only for those values which give different results. From experiments, we observe that there are no variations in the results when λ>30. Table 4 presents a performance comparison of the proposed method with other methods in terms of cluster validity functions for brain image. The proposed method achieves good results when λ = 30. Table 5 presents a performance comparison of lung images. For lung images, when λ = 10 the proposed method outperforms other methods. Table 6 presents a performance comparison of liver images. The proposed method outperforms other methods when λ = 25. Table 7 presents performance comparison of breast images. The proposed method outperforms other methods when λ = 5.

5 Conclusion and Future Scope

Fuzzy C-means is a well-known clustering algorithm which is widely applied for segmenting medical images. FCM is based on fuzzy set theory. But sometimes uncertainty arises due to manual error in defining membership function. IFS deals with this uncertainty by using hesitation degree. In this paper, we proposed a medical image segmentation algorithm based on IFS. We applied the proposed algorithm to segment different medical images, including brain, lungs, liver and breast images. Compared to FCM algorithm, our proposed algorithm considers uncertainty information captured by IFS to a better extent. We compared our proposed method with existing IFS based methods. Experimental results emphasize that our proposed method outperforms other methods in terms of cluster validity functions.

Granular computing is an emerging computing paradigm of information processing. The ability of rough context based clustering for granulation and rational reasoning has increased the robustness of granular computing. In future work, granular computing can be used to reduce the time complexity of the proposed system.