An improved intuitionistic fuzzy c-means clustering algorithm incorporating local information for brain image segmentation
Graphical abstract
Original and segmented simulated brain image by different algorithms: (a) axial view of original simulated T1-weighted brain image with INU = 0 and 1% noise, (b) skull stripping simulated brain image, (c) manual segmented CSF, GM and WM images, (d) IIFCM algorithm, (e) IFCM algorithm, (f) FLICM algorithm, (g) EnFCM algorithm, (h) FGFCM algorithm, (i) FCM_S1 algorithm, (j) FCM_S2 algorithm, (k) ImFCM algorithm.
Introduction
The segmentation of human brain image from magnetic resonance imaging (MRI) into three brain tissues: cerebrospinal fluid (CSF), gray matter (GM) and white matter (WM) [1] is one of the important components in computer-aided diagnosis and neuroscience research. It helps to detect different diseases such as tumors, edema [2], Alzheimer's Disease (AD) [3] and Schizophrenia [4]. Due to complicated structure of human brain and absence of well-defined boundary between different tissues, segmentation of brain image is a very difficult task. In literature, various segmentation methods such as thresholding based methods [5], atlas-guided methods [6], artificial neural network (ANN) [7], Markov random field model [8], level set methods [9], and clustering methods [10], [11] etc. have been suggested for MRI. Thresholding based method is used when the histograms of different brain tissues namely, GM, WM and CSF are quite distinguishable. Due to complex distribution of intensities of different brain tissues, the determination of appropriate threshold for brain image segmentation is difficult. Atlas-guided method [6] finds a transformation to map a pre-segmented atlas image to the target image that requires to be segmented. However, atlas-based method has a limitation due to the anatomical variability exhibited in the available atlases which does not explicitly model the intensity of the given new image. Artificial neural network (ANN) is a machine learning model in which the learning is achieved by updating the weight assigned to the connection between the different nodes. Due to many interconnections between the different nodes, the ANN model incorporates the spatial information during the segmentation. The Markov random field (MRF) model is itself not a segmentation method but a statistical model which can be used within other segmentation methods. Various segmentation algorithms have used the MRF model to incorporate the spatial information that occurs in the neighboring pixels. The level set method [9] is a non-parametric approach to track shapes and interfaces. It determines contours/surfaces with well-established mathematical theories such as calculus of variations and partial differential equations. Clustering methods are considered to be efficient for MRI brain image segmentation. The basic idea of clustering is to group similar pixels. In literature, various clustering methods such as Expectation-maximization [10], k-means [11] and fuzzy clustering methods [12] have been suggested. The k-means algorithm performs segmentation of the image which involves iterative computation of mean intensity for each cluster and assigning each pixel to the cluster whose mean is closest to the particular pixel. In the EM algorithm, it is assumed that the data follows a Gaussian mixture model. It iterates to determine the posterior probabilities, maximum likelihood estimates of the means, covariances, and mixing parameters of the mixture model to perform clustering. However, it is found to be sensitive to the choice of initial parameters. Among these clustering approaches, the standard fuzzy c-means (FCM) algorithm [12] and its variants are widely used by image and pattern recognition research community. The FCM works well for noiseless images. But, one major disadvantage of the standard FCM algorithm during segmentation is that it does not consider any spatial information [13], [14], [15], which makes it sensitive to noise.
In literature, many researchers have attempted to incorporate the local spatial information in the standard FCM algorithm. Pham [14] introduced the robust fuzzy c-means (RFCM) algorithm with modification of the standard FCM objective function, which includes the local spatial penalty term. This penalty term allows computation of the smooth membership function. It improves the segmentation performance and is insensitive of noise to some extent. Another approach, the fuzzy clustering with spatial constraints (FCM_S) proposed by Ahmed et al. [16], overcomes the problem of the intensity inhomogeneity. This is achieved by adding a term in the objective function of the standard FCM which allows smoothing of pixels by its neighborhood pixels. The FCM_S algorithm is insensitive to noise to some extent but it takes more execution time as it involves computation of neighborhood term in all iterations during clustering process. To reduce the execution time of the FCM_S algorithm, Chen and Zhang [17] suggested two variants of FCM_S algorithm, named the FCM_S1 and FCM_S2 algorithms. The mean filtered image is computed in the FCM_S1 whereas median filtered image is computed in FCM_S2 algorithm in advance, to replace the neighborhood term of the FCM_S algorithm. Shen et al. [2] pointed that RFCM, FCM_S, FCM_S1 and FCM_S2 algorithms loses the continuity from the standard FCM, as these algorithms are formulated with modification of standard FCM objective function. Further, Shen et al. [2] suggested a new algorithm, named improved fuzzy c-means (ImFCM) by introducing the neighborhood attraction term in its distance measure. This neighborhood attraction depends on two factors; one is pixel intensities and other is spatial position of the surrounding pixels. The ImFCM uses two parameters β and ψ whose optimal values are learned using artificial neural network model to adjust the degree of two factors in the neighborhood attraction [2]. However it requires additional execution time to learn parameters using artificial neural network model. Szilagyi et al. [18] suggested the enhanced fuzzy c-means (EnFCM) which is faster. In the EnFCM algorithm, the clustering is carried out on the gray-level histogram of newly generated image, which is obtained from the original image and its local neighbor average image. Generally, in an image, total number of gray-levels is less in comparison to the total number of pixels and this reduces significantly the execution time of EnFCM algorithm. Another similar research work is suggested by Cai et al. [19], the fast generalized fuzzy c-means (FGFCM), which incorporates both local spatial and gray-level information by introducing a similarity measure factor. To incorporate the local spatial information in the standard FCM, Krinidis and Chatzis [20] suggested fuzzy local information c-means (FLICM) algorithm by including a new fuzzy local neighborhood factor in the objective function of standard FCM algorithm. The new fuzzy local neighborhood factor determines gray-level and spatial relationship and does not require any parameter setting.
In addition to sensitivity to noise, another major challenge in segmentation of brain image is to handle the uncertainty that arises on boundary between different tissues. To handle this type of uncertainty, Atanassov [21] suggested the generalization of fuzzy sets theory, known as intuitionistic fuzzy sets (IFS) theory. To exploit the advantage of IFS theory, Pelekis et al. [22] proposed the fuzzy clustering of intuitionistic fuzzy data, where the qualitative information, which is obtained via intuitionistic membership μp and non-membership degree νp for each data point xp is used. The n-dimensional data point xp represented in IFS theory is a vector of triplet i.e. instead of vector in fuzzy set theory. Xu and Wu [23] suggested the intuitionistic fuzzy c-means (IFCM) algorithm which utilizes the intuitionistic fuzzy distance instead of fuzzy distance. The IFCM successfully handles the uncertainty but one main shortcoming of the IFCM algorithm is that it does not include any local spatial information, which makes its susceptible to noise.
To make IFCM insensitive to noise, we present a new algorithm, named an improved intuitionistic fuzzy c-means (IIFCM), which incorporates the local spatial information in IFCM by introducing a new intuitionistic fuzzy factor. This factor determines both local gray-level and spatial information and is free of any parameter tuning. The IIFCM has following characteristics: (i) the intuitionistic fuzzy factor in the IIFCM incorporates simultaneous both local gray-level and local spatial information; (ii) the balance between image details and noise are automatically maintained by intuitionistic fuzzy local constraints; (iii) no need of pre-processing step to apply the IIFCM; (iv) it utilizes qualitative information which is obtained via intuitionistic fuzzy membership μp, non-membership νp for the given data point xp. The proposed IIFCM algorithm is tested on a synthetic square image and brain images of two publically available brain databases. To check the efficacy of the proposed approach (IIFCM), the performance is evaluated in the terms of similarity index, false negative ratio and false positive ratio, and compared quantitatively with seven existing segmentation methods. A nonparametric statistical test based on multiple comparisons with a control method is also carried out to compare the performance of segmentation algorithms.
The rest part of the paper is structured as follows. Section 2 describes IFS theory, similarity measure and intuitionistic fuzzification of images. Section 3, briefly describes FCM_S, FCM_S1, FCM_S2, EnFCM, FGFCM, FLICM and IFCM algorithms. The proposed IIFCM algorithm is described in Section 4. Experimental results and comparison with seven exiting algorithms are included in Section 5. Finally, Section 6 includes the conclusions.
Section snippets
Intuitionistic fuzzy sets, similarity measure and intuitionistic fuzzification of images
In this section, we describe briefly intuitionistic fuzzy sets, similarity measures between elements and intuitionistic fuzzification of images.
Fuzzy clustering with constraints (FCM_S) and its variants
The fuzzy clustering with constraints (FCM_S) algorithm proposed by Ahmed et al. [16] includes a penalty term in the standard FCM objective function. This penalty term allows the smoothing of a pixel within its specified neighborhood. The optimization problem of the FCM_S algorithm [16] is given as:where X = {x1, x2, …, xN} are N pixels, m (1 < m < ∞) is the fuzzifier constant, c (1 < c < N) is the number of
Proposed algorithm
The IFCM algorithm has two drawbacks: (i) the objective function (Eq. (26)) of the IFCM algorithm does not incorporates any local spatial information as it deals each pixel as a separate point. Generally, noise in the image occurs during image acquisition process, which may change the intensity value of pixel. So, the noisy pixels are always wrongly classified in an image because it shows the abnormal behavior in that neighborhood. (ii) The membership degree of the IFCM (Eq. (28)) is a function
Experimental results and discussion
To demonstrate the efficacy of the proposed IIFCM algorithm, experiment is carried out on synthetic square image and two publically available MRI brain images. The segmentation performance of the IIFCM method is compared with seven existing methods: IFCM, FLICM, EnFCM, ImFCM, FGFCM, FCM_S1 and FCM_S2. These algorithms are implemented in Matlab.
The segmentation results are compared quantitatively in the term of similarity index (ρ), false negative ratio (rfn) and false positive ratio (rfp).
Conclusions
In this paper, we have proposed an improved intuitionistic fuzzy c-means (IIFCM) algorithm that overcomes the disadvantage of the IFCM algorithm. The IIFCM algorithm incorporates both local gray-level and spatial information using an intuitionistic fuzzy factor to handle noise and uncertainty during segmentation. It is free of requirement of any parameter tuning. We have performed experiments on synthetic square image, real and simulated MRI brain images, and compared quantitatively their
Acknowledgments
The authors are thankful to the anonymous reviewers for their constructive suggestions to improve the overall quality of this paper. Moreover, the first author would like to thank the Council of Scientific & Industrial Research (CSIR), New Delhi, India for financial support.
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