Fuzzy Clustering Techniques with Spatial Information Deepthi P Hudedagaddi, B K Tripathy SCOPE, VIT University Vellore-632014 Tamil Nadu, India
[email protected],
[email protected]
Abstract— A conventional FCM algorithm does not fully utilize the spatial information in the image. In this paper, we introduce modifications of fuzzy cmeans (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.Modifications of FCM and intuitionistic FCM with respect to spatial information are introduced in this paper. Keywords—Clustering, Spatial information, FCM,
Intuitionistic I. 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 . However, some of these methods do not exploit the multispectral information of the MRI signal. Fuzzy c-means (FCM) clustering is an unsupervised 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[1]. Medical image segmentation plays an important role in a variety of biomedical-imaging applications, such as the quantification of tissue volumes, diagnosis, localization of pathology, study of anatomical structure, treatment planning, and computer-integrated surgery. However, segmentation of medical images involves three main image-related problems. First, images contain noise that can alter the intensity of a pixel such that its classification becomes uncertain. Second, images exhibit intensity inhomogeneity where the
intensity level of a single tissue class varies gradually over the extent of the image. Third, images have finite pixel size and are subject to partial volume averaging where individual pixel volumes contain a mixture of tissue classes so that the intensity of a pixel in the image may not be consistent with any one class. To overcome these problems, many segmentation techniques have been proposed in the past decades.In addition,several hybrid algorithms of fuzzy and rough set based clustering algorithms have been developed and applied on images[2-6,21,22]. 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 conventional 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 neighbourhood of a pixel to influence its labeling. The modified FCM algorithm improved the results of conventional FCM methods on noisy images.
II. EXISTING METHODS Pham and Prince proposed an adaptive FCM algorithm [10] and its extension to 3D data [11], which incorporated a spatial penalty term into the objective function to enable the estimated membership functions to be spatially smoothed. Ahmed et al. [12] modified the objective function of the standard FCM algorithm to compensate for
intensity inhomogeneity and to allow the labeling of a pixel to be influenced by the labels in its immediate neighborhood. Liew and Yan [13] used a B-spline surface to model the bias field and incorporated the spatial continuity constraints into fuzzy clustering algorithm. Zhang and Chen [14] replaced the original Euclidean distance with a kernel-induced distance and supplemented the objective function with a spatial penalty term, which modeled the spatial continuity compensation. Incorporating spatial information into the membership function, Chuang et al. [1] proposed a modified FCM algorithm which was less sensitive to noise and yielded more homogeneous segmented regions. L. Szilágri et al. [15] proposed an efficient FCM clustering model for compensating intensity inhomogeneity and segmentation of magnetic resonance (MR) images, which drastically reduced the processing time without causing relevant change in terms of accuracy. Recently, local intensity information has been taken into account to deal with intensity inhomogeneity in fuzzy segmentation method. For example, Li et al. [16] proposed a new fuzzy energy minimization method based on coherent local intensity clustering (CLIC) for simultaneous tissue classification and bias field estimation of MR images. CLIC algorithm draws upon intensity information in local regions; therefore, it can be used to segment images with intensity inhomogeneity. However, spatial information is not taken into account in the CLIC algorithm; as a result, the CLIC algorithm is sensitive to noise. Fuzzy sets and Intuitionistic Fuzzy sets[19,20] are extensively used in developing clustering algorithms.
𝑢𝑖𝑗=
−
(𝑘) 𝑢𝑖𝑗 |},
where ϵ is a termination criterion between 0 and 1, whereas k are the iteration steps. This procedure converges to a local minimum or a saddle point of Jm.The algorithm is composed of the following steps: 1.
Assign initial cluster centers or means for c clusters. 2. Calculate the distance dik between data objects xk and centroids vi using Euclidean formula d(x, y) = √(x1 − y1 )2 + (x2 − y2 )2 + ⋯ + (xn − yn )2 (2) 3. Generate the fuzzy partition matrix or membership matrix U: If dij > 0 then μik =
1
(3)
2 m−1
dik ∑C ) j=1( djk
Else 4.
μik = 1 The cluster centroids are calculated using the formula m
Vi =
6.
where m is any real number greater than 1, uij is the degree of membership of xi in the cluster j, xi is the ith of d-dimensional measured data, cj is the ddimension center of the cluster, and ||*|| is any norm expressing the similarity between any measured data and the center. Fuzzy partitioning is carried out through an iterative optimization of the objective function shown above, with the update of membership uij and the cluster centers cj by:
(3)
𝑚 ∑𝑁 𝑗=1 𝑢𝑖𝑗
(𝑘+1)
Fuzzy c-means (FCM) is a method of clustering which allows one piece of data to belong to two or more clusters. This method developed by Dunn in 1973 and improved by Bezdek in 1981[16] is frequently used in pattern recognition. It is based on minimization of the following objective function: (1)
𝑚 ∑𝑁 𝑖=1 𝑢𝑖𝑗 𝑥𝑖
This iteration will stop when maxij{| 𝑢𝑖𝑗
5.
1≤m≤∞
2 ||𝑥𝑖 −𝑐𝑗 || 𝑚−1 ∑𝐶 𝑘=1(||𝑥𝑖 −𝑐𝑘|| )
𝑐𝑗 =
A. Fuzzy C-Means Algorithm (FCM)
𝐶 𝑚 2 𝐽𝑚 = ∑𝑁 𝑖=1 ∑𝑗=1 𝑢𝑖𝑗 ||𝑥𝑖 − 𝑐𝑗 ||
(2)
1
B.
∑N j=1(µij ) xj ∑N j=1(µij )
m
(4)
Calculate new partition matrix by using step 2 and 3 If ‖U (r) − U (r+1) ‖ < ε then stop else repeat from step 4.
Intuitionistic Fuzzy Algorithm (IFCM)
C-Means
The Intuitionistic fuzzy c-means proposed by T. Chaira[17] brings in to account a new parameter that helps in increasing the accuracy of clustering. This parameter is known as the hesitation value and denoted by π. 1. 2.
3.
Assign initial cluster centers or means for c clusters. Calculate the distance dik between data objects xk and centroids vi using Euclidean formula (2) Generate the fuzzy partition matrix or membership matrix U: If dij > 0 then
μik =
4.
1
Else μik = 1 Compute the hesitation matrix π using πA (x) = 1 − μA (x) −
5.
6.
8.
1−μA (x) 1+λμA (x)
| x ∈ X (6)
Compute the modified membership matrix U’ using μ′ik = μik + πik (7) The cluster centroids are calculated using the formula Vi =
7.
(5)
2 dik m−1 ∑C ( ) j=1 d jk
′ m ∑N j=1(µij ) xj ′ m ∑N j=1(µij )
(8)
The sFCM clustering algorithm has two-steps. For every iteration, the conventional FCM algorithm is followed in the first step. Later, the centroid and the membership functions are updated. In the second step, the spatial function hij is calculated and then the new membership function (5) is computed. B.Spatial Intuitionistic Fuzzy C Means(sIFCM) The sIFCM algorithm is given as follows: 1. Provide the initial values for the centroids vi where i =1,…,c 2. Compute the membership function as follows: 1 𝑢𝑖𝑗=
Calculate new partition matrix by using step 2 to 5 If ‖U′((r) − U′(r+1) ‖ < 𝜀 then stop else repeat from step 4.
||𝑥𝑗 −𝑣𝑖 || ∑𝐶 𝑘=1(||𝑥𝑗 −𝑣𝑘|| )
for all i = 1,…,c and j = 1,…,N Compute the hesitation value as: 𝜋ij (𝑥) = 1 − uij (𝑥) − (1 − 𝑢𝑖𝑗 (𝑥))/(1 + 𝜆𝑢𝑖𝑗 (𝑥) for all i = 1,…c , 𝜆 >0 and j = 1,…,N Compute the membership function as: μ′ik = μik + πik for all i = 1,…c and j = 1,…,N Calculate the spatial function as
3. III.
SPATIAL FUZZY C MEANS
A. Spatial FCM(sFCM) 4.
Chuang. et al[1] proposed the following sFCM and proved their results to be better than conventional FCM. The spatial FCM with parameter p and q is denoted sFCMp,q. Note that sFCM1,0 is identical to the conventional 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
hij = ∑𝑘𝜖𝑁𝐵(𝑥𝑗) 𝑢𝑖𝑘
p q
u′ij = ∑c
uij hij
p q k=1 ukj hkj
5.
hij = ∑𝑘𝜖𝑁𝐵(𝑥𝑗) 𝑢𝑖𝑘
(10)
where p and q indicate the relative weightage of the initial membership and the spatial function respectively. In case of a noisy image, the spatial function reduces the number of misclassified pixels by taking the neighbouring pixels into account.
for all i = 1,…c and j = 1,…,N Compute the new membership function which incorporates the spatial function as:
6.
hij = ∑𝑘𝜖𝑁𝐵(𝑥𝑗) 𝑢𝑖𝑘 for all i = 1,…c and j = 1,…,N c Set uij = uij” for all j = 1,…,N and i = 1,…c Calculate the new centroids as follows: 𝑚 ∑𝑁 𝑗=1 𝑢𝑖𝑗 𝑥𝑗 𝑣𝑖 = 𝑚 ∑𝑁 𝑗=1 𝑢𝑖𝑗 for i=1,..,c If |uij(new)-uij(old)|
