2016 Second International Conference on Cognitive Computing and Information Processing (CCIP)
A Picture Fuzzy Clustering Approach for Brain Tumor Segmentation S V Aruna Kumar
B S Harish, Member, IEEE
V N Manjunath Aradhya
Dept. of Information Science and Engg. Dept. of Information Science and Engg. Dept. of Master of Computer Applications JSS Science and Technology University JSS Science and Technology University JSS Science and Technology University Mysuru, India Mysuru, India Mysuru, India Email:
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Abstract—This paper presents a Picture Fuzzy Clustering (PFC) method for MRI brain image segmentation. The PFC is based on the Picture fuzzy set, which is the generalization of the traditional fuzzy set and intuitionistic fuzzy set. In traditional fuzzy set, the problem of uncertainty arises in defining the membership function. Intuitionistic fuzzy set handles this uncertainty by considering hesitation degree. However, intuitionistic fuzzy set fails to solve real time problems which require answers like yes, abstain, no and refusal. The picture fuzzy set solves these problems by considering refusal degree along with membership, neutral and nonmembership degree. Thus, the cluster centers in the PFC may converge to a desirable location than the cluster centers obtained using traditional Fuzzy C-Means (FCM) and Intuitionistic Fuzzy Clustering (IFC). Experimentation is carried out on the standard MRI brain image dataset. To assess the performance, the proposed method is compared with the existing FCM and IFC methods. Results show that the proposed method gives the better result. Keywords—Segmentation, Clustering, Fuzzy C Means, Intutionistic fuzzy set, Picture fuzzy clustering.
I.
I NTRODUCTION
Magnetic Resonance Imaging (MRI) is a commonly used modality in medicine, which helps physician to diagnosis and monitor various diseases. Segmentation has become a fundamental building block in medical image analysis. However, traditional segmentation algorithm fails to give good results in MRI brain image segmentation due to various types of uncertainty present that includes noise, intensity inhomogeneity and imprecise gray levels in the images. Fuzzy C-Means (FCM) clustering method has been widely used for medical image segmentation. FCM is soft clustering method, where image pixels are clustered based on the membership value assigned to each pixel. The membership is mainly decided by the distance measure between cluster center and pixel. In literature, many researchers developed variants of traditional Fuzzy CMeans (FCM) based clustering algorithms for medical image segmentation [1]–[4]. Ahmed et al.,[4] proposed a modified FCM called FCM S method, which considers the spatial information. Even though this method gives better results, it is computationally expensive. To overcome this problem, Chen and Zhang [5] proposed two variants of FCM S. This method initially computes mean and median by which the computation time is reduced. Lung and Kim [6] developed a generalized spatial FCM algorithm. This method considers both the pixels attributes and local spatial information.
Even though FCM handles uncertainty and gives better result compared to traditional segmentation methods, but it is sensitive to noise. To address this problem, Krishnapuram and Keller.,[7] proposed a Possibalistic C-Means (PCM) algorithm. However, PCM handles noise but this method is too sensitive to initialization. Ji et al., [8] developed a modified PCM clustering algorithm for brain MRI image segmentation. This method resolves the classification ambiguity by utilizing the local spatial information. Traditional FCM uses euclidean distance measure. But, the euclidean distance fails to give good results when data contains noise. Thus many researchers used kernel distance metric [9], [10]. Kannan et al.,[9] developed a robust kernel FCM based method for breast image segmentation. This method uses tangent kernel function as distance metric. Zhang and Chen [10] proposed a kernealized modified FCM method. This method incorporates the spatial penalty term into objective function and uses the Gausssian kernel as distance metric. In our previous work [11], we proposed a novel Robust Spatial Kernel FCM (RSKFCM) method for MRI brain image segmentation. RSKFCM incorporates the spatial information into conventional FCM memebrship function and uses kernel distance metric. The main drawback of the RSKFCM is it initializes the cluster centers randomly which leads to convergence of the objective function to local minima. To overcome this problem Kumar et al.,[12] proposed a evolutionary based RSKFCM method. In this method genetic algorithm is used to initialize the cluster centers. Another uncertainty arises in FCM in defining the membership function. This uncertainty is due to lack of knowledge or personal error. To handle this uncertainty, Atanassov [13] introduced higher order fuzzy set which is called Intuitionistic Fuzzy Set (IFS). IFS considers both membership and non membership values. Some early researchers developed FCM on IFS. Chaira., [14] developed an novel IFS clustering method for medical image segmentation. In this method to maximize the good points in the class, new objective function called intuitionistic fuzzy entropy is incorporated into objective function of conventional FCM. Zhang et al.,[15] proposed an intuitionistic fuzzy set clustering method. This method creates intuitionistic fuzzy similarity matrix using similarity degree between two intuitionisitc fuzzy sets. Xu et al.,[16] developed clustering algorithm for intuitionisitc fuzzy set based on the concept of association matrix and equivalent association matrix. This method calculates association coefficient by considering hesitation degree. Chaira and Anand [17] developed a novel IFS approach for tumor detection in
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medical images. This method uses histogram thresholding to eliminate unwanted regions from clustered image. Further, edge of the tumor is extracted. Chaudhuri [18] developed a Intuitionistic Fuzzy Possibilistic C Means (IFPCM) algorithm. IFPCM resolves problem regarding the membership value by generalizing membership and non membership with hesitancy degree. Cuong [19] has presented Picture Fuzzy Set (PFS) which is generalization of the traditional fuzzy set and IFS. PFS solves real time problems which requires answers like yes, abstain, no and refusal. Thong and Son., [20] proposed a new Picture Fuzzy Clustering (PFC) algorithm. Experimental results reveals that, PFC gives better clustering results. Inspired by a good performance of the PFC, in this paper we proposed a Picture Fuzzy Clustering (PFC) algorithm to segment MRI brain images. In fuzzy set based clustering algorithms, the membership value depends on the distance metric. The pixel have higher membership value, if the pixel intensity is closer to the cluster center value. Hence, the membership value is highly sensitive to noise. The euclidean distance metric fails to give good segmentation results on MRI brain images due to noise and intensity inhomogeneity present in the image. To handle noise and intensity inhomogeneity, in this paper we used picture euclidean distance function to calculate distance between cluster center and pixel. The rest of the paper is organized as follows: Section 2 discuss the background information regarding the conventional FCM, Intuitionistic Fuzzy Set (IFS), Intuitionistic Fuzzy C-Means (IFCM) and Picture Fuzzy Set (PFS). Section 3 discuss the proposed method. Section 4 evaluates the performance of the proposed method. Conclusion are drawn in section 5. II.
This iteration will stop when {J(i) − J(i − 1)} < ε,where ε is a termination criterion. B. Intuitionistic Fuzzy set In conventional fuzzy set, element belongingness is computed based on membership value. The non membership value is computed as complement of the membership value. But, in real time non membership value is less than or equal to complement of the membership value. To address this problem, Atanassov [13] proposed a higher order fuzzy set called Intuitionistic Fuzzy Set (IFS). IFS considers both membership and non membership value. Let Intuitionistic Fuzzy Set A in finite set X = {x1 , x2 , x3 , ......, xn } may be represented mathematically as: A˜ = {hx, µ ˜ (x) , v ˜ (x)i|x ∈ X} (3) A
where µA˜ (x) : X → [0, 1], vA˜ (x) : X → [0, 1] with the condition 0 ≤ µA˜ (x) + vA˜ (x) ≤ 1 ∀x ∈ X µA˜ (x) , vA˜ (x) denotes membership and non-membership degree respectively. Along with the membership and non membership values, IFS considers hesitation degree. The hesitation degree of an element x ∈ X is defined as: πA˜ (x) = 1 − µA˜ (x) − vA˜ (x) where, πA˜ (x) is hesitation degree and should satisfy the elementary condition of intuitionism i.e 0 ≤ πA˜ (x) = 1 − µA˜ (x) − vA˜ (x) ≤ 1. In literature, two fuzzy complements or IFS generators are used to construct IFS. They are as follows: 1)
BACKGROUND INFORMATION
Sugeno’s negation[21] cλ (x) =
A. Traditional Fuzzy c-means Traditional Fuzzy C-Means (FCM) is soft clustering algorithm, where data element are clustered based on the membership value. In FCM, the data elements can belong to more than one cluster. Let X = (x1 , x2 ,.......xN ) denotes an image with N pixels to be partitioned into c clusters, where xi denotes ith pixel. FCM is based on minimization of the following objective function: 2 c N X X um kx − v k (1) J= j i ij j=1 i=1 th
th
Where uij is the membership degree of xj pixel to i cluster, vi is the ith cluster center, k.k is a euclidean distance metric and m is a fuzzifier. Fuzzy partitioning is carried out through an iterative optimization of the objective function shown in equation 1, with the update of membership uij and the cluster center vi by, uij =
N P
vi =
j=1
1
2 Pc
xj −vi /(m − 1) k=1 xj −vk
um ij xj
N P j=1
i = 1, 2, . . . . . . .c um ij
(2)
A
2)
1−x ; 1 + λx
λ>0
(4)
Yager’s negation[22] cw (x) = 1 − xλ
λ1
;
λ>0
(5)
The fuzzy complement function is defined as: N (µ (x)) = g −1 (g (1) − g (µ (x)))
(6)
where g(.) is an increasing function and g → [0, 1]. Yager’s class can be generated by using the following function: g (x) = xλ
(7)
So, Yager’s intuitionistic fuzzy complement is written as: 1 λ>0 (8) N (x) = 1 − xλ λ ; Non-membership values are calculated from Yager’s intuitionistic fuzzy complement N (x). The IFS’s using Yager’s intuitionistic fuzzy complement becomes: 1 λ λ IF S ˜ Aλ = x, µA˜ (x) , 1 − µA˜ (x) |x ∈ X (9) Sugeno’s complement can be generated by using the following function: x (1 + λ) g (x) = (10) 1 + λx So, Sugeno’s intuitionistic fuzzy complement becomes: 1−x N (x) = ; λ>0 (11) 1 + λx
Non-membership values are calculated from Sugeno’s intuitionistic fuzzy complement N (x). IFS constructed using Sugeno’s fuzzy complement is as follows: 1 − µA˜ (x) S A˜IF |x ∈ X (12) = x, µ (x) , ˜ λ A 1 + λµA˜ (x) C. Intuitionistic Fuzzy C-Means
N X c X
u∗ij m d2 (xj , vi )
(13)
j=1 i=1
where xj is the j th data point, vi is the ith cluster center, d2 (xj , vi ) is the Intutionistic euclidean distance between ith cluster center and j th data point. uij ∗ is the Intutionistic membership function, which is calculated as uij ∗ = uij + πij where uij fuzzy membership function is given by: ! 1 c X d2ij m−1 uij = d2jk 1 − uij 1 + λuij
(15)
(16)
vi =
(17) u∗ij m
D. Picture Fuzzy Set Cuong [19] proposed a Picture Fuzzy Set (PFS), which is generalization of conventional fuzzy set and intuitionistic fuzzy set. A PFS is a non empty set X given by A = {hx, µA (x) , ηA (x) , γA (x)i|x ∈ X}
(18)
where µA (x) is the positive membership value of each element, ηA (x) is the neutral membership degree and γA (x) is the negative membership degree satisfying the constrains, 0 ≤ µA (x) + ηA (x) + γA (x) ≤ 1
(19)
The refusal degree of an element is calculated as: ξA (x) = 1 − (µA (x) + ηA (x) + γA (x))
1
(23)
(20)
In case ξA (x) = 0 PFS returns Intuitionistic fuzzy set. If ξA (x) = ηA (x) = 0 PFS returns to fuzzy set.
The refusal degree of the pixel is calculated as:
B. Picture Fuzzy Clustering In this section, a Picture Fuzzy Clustering (PFC) algorithm for MRI brain image segmentation is given. PFC algorithm clusters the image I (shown in equation 21) by searching for local minima of the following objective function: J=
N X c X
m
(uij (2 − ξij )) kxi − vj k
i=1 j=1
+
N X c X
ηij (log ηij + ξij ) (26)
i=1 j=1
where vj is the j th cluster center, uij is the positive membership degree, ηij is the neutral membership degree, ξij is the refusal degree of the element which satisfies the following constraints: uij + ηij + ξij ≤ 1 c X
(uij (2 − ξij )) = 1
j=1
III.
1
α
IαP F S = {(xij , uI (xij ) , ηI (xij ) , (1 − (uI (xij ) + ηI (xij )) ) α , ξI (xij ))} (24)
α
u∗ij m xj
j=1
α
γI (xij ) = (1 − (uI (xij ) + ηI (xij )) ) α
1
ξI (xij ) = 1−(uI (xij ) + ηI (xij ))−(1 − (uI (xij ) + ηI (xij )) ) α (25) where α is exponent, the value varies between 0 and 1.
The cluster centers vi are updated as: j=1 n P
In this paper, we used Yager’s fuzzy complement generator to calculate the negative membership value. Yager’s fuzzy complement generator is as follows:
Thus after applying Yager’s fuzzy complement generator the PFS image becomes:
πij is hesitation degree, which is given by: πij = 1 − uij −
(21)
where uI is the positive membership value, ηI is the neutral membership value, γI is the negative membership value, ξI is the refusal degree of the pixel. In image, each pixel is associated with intensity value. To convert intensity values into membership values we calculate normalized intensity level for each pixel. i.e: xij uI (xij ) = (22) L−1
(14)
k=1
n P
Picture fuzzy image is constructed from Yager’s fuzzy complement generator. Let us consider an image X = hx1 , x2 , .....xN i consisting of N pixels having intensity level between 0 and L − 1. The PFS representation of the image can be given as: I = {(xij , uI (xij ) , ηI (xij ) , γI (xij ) , ξI (xij ))}
Based on Intuitionistic Fuzzy set, Chaira., [14] developed a Intuitionistic Fuzzy c-Means (IFCM) algorithm. The main objective of the IFCM is to minimize the objective function shown in below equation: J=
A. Picture Fuzzy set representation of Image
P ROPOSED M ETHOD
The details of the proposed method is explained in this section.
c X j=1
ηij +
ξij c
=1
(27)
To determine the optimal solutions of the objective function shown in equation 26 Lagrangian method is employed. The optimal solutions of the systems for vj , uij , ηij , ξij are: PN m i=1 (uij (2 − ξij )) xi vj = P N m i=1 (uij (2 − ξij )) uij =
c P
(2 − ξij )
k=1
e−ξi j ηij = P c e−ξi k
1
kxi −vj k kxi −vk k
c
(28)
(29)
2 m−1
1X 1− ξik c
! (30)
k=1
k=1
α
1
ξij = 1 − (uij + ηij ) − (1 − (uij + ηij ) ) α
(31)
where i = 1., , , , N , k = 1., , , , , n, j = 1, .....c. In fuzzy set based clustering algorithms, the membership value depends on the distance metric. The pixel have higher membership value, if the pixel intensity is closer to the cluster center value. Hence, the membership value is highly sensitive to noise. The euclidean distance metric fails to give good segmentation results on MRI brain images due to noise and intensity inhomogeneity present in the image. To handle noise and intensity inhomogeneity, in this paper we used picture euclidean distance function to calculate distance between cluster center and pixel. Picture euclidean distance is given by: d(xi , vj ) = ((u(xi ) − u(vj )) + (η(xi ) − η(vj ))+ 1
(γ(xi ) − γ(vj ))) 2
(32)
Algorithm 1 describes the steps involved in the proposed method Algorithm 1: Proposed Method Data: Input image X having N pixels in d dimensions; Number of clusters (c); Fuzzifier m; α; Stopping criteria ε; Result: Positive membership value u, Neutral membership degree membership value η, refusal degree of an element ξ and Cluster centers v Convert input image into Picture fuzzy set representation; Initialize cluster centers vj ; repeat Calculate Picture euclidean distance between each pixel and cluster center using equation 32; Calculate uij according to equation 29; Calculate ηij according to equation 30; Calculate ξij according to equation 31; Calculate vj according to equation 28; Calculate objective function J value according to equation 26; until {J(i) − J(i − 1)} < ε;
IV.
E XPERIMENTAL R ESULTS
This section gives the details about the experimental environments such as dataset, parameter settings and performance results. The MRI image for the experimentation are obtained from Multimodal Brain Tumor image segmentation benchmark (BRATS) [23]. Dataset contains 65 multi contrast MR scans of low and high grade glioma patients and 65 comparable scans generated using tumor image simulation software. The dataset also contains the segmentation ground truth for all images. For all the algorithms in comparison, we used fuzzifier (m)= 2. The stopping threshold () for all the algorithms is that the largest difference between a cluster center and its updated value is smaller than 10−5 or the maximum iteration number of 100 has been achieved. We used the exponent α = {0.2, 0.4, 0.6, 0.8}. We implemented and simulated all the algorithms with MatlabR R2013a. To evaluate the performance quantitatively, Jaccard index (J) and Dice index (D) are used which is given in below equations: |Seg Im ∪ GT Im| J= (33) |Seg Im| + |GT Im| − |Seg Im ∩ GT Im| D=
2 ∗ |Seg Im ∩ GT Im| |Seg Im| + |GT Im|
(34)
Where Seg Im represents the segmented image and GT Im represent the ground truth of the segmentation image obtained from dataset. The values of J and D varies between 0 and 1. The segmented and ground truth images are similar if the values J and D tends to 1. In this paper, we varied the exponent α value from 0.2 to 0.8. Figure 1 shows the segmentation results obtained using the proposed method with different α values. Table I presents the Jaccard and Dice index values for different α values. From experimental results presented in Table I we found that when α = 0.8 gives better segmentation results. When α = 1 picture fuzzy set becomes Intutionistic fuzzy set. We compared our algorithm with traditional FCM and Intuitionistic Fuzzy Clustering (IFC). Table II compares the performance of the proposed method. From Table II, it is obvious that the proposed method obtained better clustering results. However, the performance of the proposed method depends on the value of α. When the value of α = 0.8 gives the better results. TABLE I: Jaccard and Dice index for different α values α value 0.2 0.4 0.6 0.8
Jaccard Index 0.65 0.72 0.80 0.84
Dice Index 0.68 0.70 0.79 0.83
TABLE II: Performance Comparison with other Methods Method FCM IF-FCM Proposed Method
Jaccard Index 0.73 0.77 0.84
Dice Index 0.72 0.76 0.83
(a)
(c)
(b)
(d)
Fig. 1: Input image and corresponding segmented image obtained using proposed method with different α values (a) α = 0.2, (b) α = 0.4, (c) α = 0.6, (d) α = 0.8
V.
C ONCLUSION
In this paper, we have segmented MRI brain images using Picture Fuzzy Clustering (PFC). PFC is generalization version of traditional FCM algorithm and Intuitionistic Fuzzy Clustering. The proposed method is applied to MRI brain image segmentation. To evaluate the performance, two quantitative measure i.e Jaccard and Dice index are used. The segmentation result of the proposed method depends on the exponent value α. From experiments, we found that the proposed method gives better segmentation result at α = 0.8. We compared the results with traditional FCM and Intuitionistic Fuzzy C-Means (IFFCM) algorithm. It is clear from our experimental results that the proposed algorithm outperforms the other methods. R EFERENCES [1]
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