Computational and Mathematical Methods in Medicine

Volume 2015 (2015), Article ID 485495, 12 pages

http://dx.doi.org/10.1155/2015/485495

## Segmentation of Brain Tissues from Magnetic Resonance Images Using Adaptively Regularized Kernel-Based Fuzzy -Means Clustering

^{1}Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, 1068 Xueyuan Boulevard, Shenzhen 518055, China^{2}University of Chinese Academy of Sciences, 52 Sanlihe Road, Beijing 100864, China^{3}Faculty of Computers and Information, Mansoura University, Elgomhouria Street, Mansoura 35516, Egypt

Received 29 September 2015; Accepted 23 November 2015

Academic Editor: Jesús Picó

Copyright © 2015 Ahmed Elazab et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An adaptively regularized kernel-based fuzzy -means clustering framework is proposed for segmentation of brain magnetic resonance images. The framework can be in the form of three algorithms for the local average grayscale being replaced by the grayscale of the average filter, median filter, and devised weighted images, respectively. The algorithms employ the heterogeneity of grayscales in the neighborhood and exploit this measure for local contextual information and replace the standard Euclidean distance with Gaussian radial basis kernel functions. The main advantages are adaptiveness to local context, enhanced robustness to preserve image details, independence of clustering parameters, and decreased computational costs. The algorithms have been validated against both synthetic and clinical magnetic resonance images with different types and levels of noises and compared with 6 recent soft clustering algorithms. Experimental results show that the proposed algorithms are superior in preserving image details and segmentation accuracy while maintaining a low computational complexity.

#### 1. Introduction

Image segmentation is to partition an image into meaningful nonoverlapping regions with similar features. Segmentation of brain magnetic resonance (MR) images is necessary to differentiate white matter (WM), gray matter (GM), and cerebrospinal fluid (CSF). Such segmentation is essential for studying anatomical structure changes and brain quantification [1]. It is also a prerequisite for tumor growth modeling as tumors diffuse at different rates according to the surrounding tissues [2]. Due to potential existence of noise, bias field, and partial volume effect, segmentation of brain images remains challenging.

Image segmentation techniques can be roughly categorized into [3] thresholding, region growing, clustering, edge detection, and model-based methods. Clustering is an unsupervised learning strategy that groups similar patterns into clusters and can be hard or soft. Soft clustering is preferred as every pixel can be assigned to all clusters with different membership values [4, 5]. The most popular soft clustering methods applied to MR images are [4] fuzzy -means (FCM) clustering [6, 7], mixture modeling, and hybrid methods of both.

Although the FCM algorithm comes with good accuracy in the absence of noise, it is sensitive to noise and other imaging artifacts. Therefore, enhancements have been tried to improve its performance by including local spatial and grayscale information [8–14] which will be briefly elaborated in Section 2.

A mixture model composed of a finite number of Gaussians has been employed for brain MR image segmentation. The main strategy to incorporate the local information into the mixture model is to use hidden Markov random fields for more accurate segmentation [15]. Nikou et al. [16] proposed a hierarchical and spatially constrained mixture model that takes into account spatial information by imposing distinct smoothness priors on the probabilities of each cluster and pixel neighborhoods. In [17], a nonparametric Bayesian model for tissue classification of brain MR images known as Dirichlet process mixture model was explored. Nguyen and Wu [18] introduced a way to incorporate spatial information between neighboring pixels into the Gaussian mixture model (GMM).

To be more robust to noise and attain fast convergence, FCM and GMM can be combined. Chatzis and Varvarigou [19] embedded the hidden Markov random field model into the FCM objective function to explore the spatial information. Chatzis [20] introduced a methodology for training finite mixture models under fuzzy clustering principle with a dissimilarity function to incorporate the explicit information into the fuzzy clustering procedure. Recently, Ji et al. [21] employed robust spatially constrained FCM (RSCFCM) algorithm for brain MR image segmentation by introducing a factor for the spatial direction based on the posterior probabilities and prior probabilities.

In [22], Li et al. presented an algorithm for brain tissue classification and bias estimation using a coherent local intensity clustering. Later they explored multiplicative intrinsic component optimization (MICO) [23] to improve the robustness and accuracy of tissue segmentation in the presence of high level bias field.

Generally, the current brain MR image segmentation algorithms suffer from one or more of the following shortcomings: lack of robustness to outliers [8, 9, 13], high computational cost [8, 13, 14, 16, 21], prior adjusting of crucial or many parameters [8–11, 21], limited segmentation accuracy in the presence of high level noise [8, 11, 19, 22, 23], and loss of such image details like CSF [9, 13, 14, 21]. In this paper, a new soft clustering framework is to be explored for better handling of the aforementioned segmentation problems.

The rest of this paper is organized as follows. Related work of FCM algorithm is presented in Section 2. The proposed framework is then elaborated in Section 3. Experiments on synthetic and clinical MR images are presented in Section 4. Sections 5 and 6 are devoted to discussion and conclusion, respectively.

#### 2. Related Work

The FCM algorithm in its original form assigns a membership value to each pixel for all clusters in the image space. For an image with set of grayscales at pixel in -dimensional space and cluster centers with being a positive integer (), there is a membership value for each pixel in the th cluster (). The objective function of the FCM algorithm is [7]where is a weighting exponent to the degree of fuzziness, that is, , and is the grayscale Euclidean distance between pixel and center . The membership should be constrained to the following:

The membership function and cluster centers are updated iteratively in an alternating process known as alternate optimization. The membership function and cluster centers are

As the objective function in (1) does not include any local information, the original FCM is very sensitive to noise and the accuracy of clustering in the presence of noise and image artifacts will decrease. To overcome this problem, Ahmed et al. [8] modified the objective function by adding a term for the spatial information of neighboring pixels. This algorithm is denoted as with the following objective function:where is a parameter to control the spatial information of neighbors with , is the set of pixels around pixel , and is the cardinality of .

The algorithm is computationally expensive as the local neighborhood term has to be calculated in each iteration step. To overcome this drawback, Chen and Zhang [10] replaced the term with , where is the grayscale of a filtered image that could be calculated once in advance, and used kernel function to replace the Euclidean distance. The enhancement could be in two forms, that is, by using the average filter and by adopting the median filter. Their objective function is as follows:

Although the accuracy has been improved, it is sensitive to high level noises and different types of noises. In addition, the parameter , which has a great impact on the performance, is set manually with care and requires prior information about noise.

Yang and Tsai [12] proposed a Gaussian kernel-based FCM method with the parameter calculated in every iteration to replace for every cluster. Similar to and , this method has two forms: GKFCM1 and GKFCM2 for average and median filters, respectively. The parameter is estimated using kernel functions:where is the kernel function. The replacement of with could yield better results than and . However, for good estimation of , cluster centers should be well separated which might not be always true; hence the algorithm has to iterate many times to converge. Moreover, the learning scheme requires a large number of patterns and many cluster centers to find the optimal value for .

To tackle the problem of parameter adjustment, Krinidis and Chatzis [13] proposed the FLICM algorithm with a fuzzy factor that combined both spatial and grayscale information of the neighboring pixels. The fuzzy factor was embedded into (1) as follows:where pixel is the center of the local window, pixel is in the neighborhood, and is the spatial Euclidean distance between pixels and .

Although FLICM algorithm enhances robustness to noise and artifacts, it is slow since the fuzzy factor () has to be calculated in each iteration. Moreover, is heavily affected by spatial Euclidean distance from the central pixel to its neighboring pixels to lose small image details due to the smoothing effect.

To enhance the FLICM algorithm, Gong et al. [14] developed KWFLICM algorithm with a trade-off weighted fuzzy factor to control the local neighbor relationship and replaced the Euclidean distance with kernel function. The weighted fuzzy factor of KWFLICM iswhere is the trade-off weighted fuzzy factor of pixel in the local window around the central pixel and is the kernel metric function. The trade-off weighted fuzzy factor combines both the local spatial and grayscale information [14]. Because of the trade-off weighted fuzzy factor, its computational cost increases substantially. In addition, the algorithm is unable to preserve small image details.

In addition to the abovementioned shortcomings, Szilágyi [24] pointed out serious theoretical mistakes in FLICM and KWFLICM. It was shown that the iterative optimization nature of FLICM and KWFLICM did not minimize their objective functions; instead, they iterated until the partition matrices converged. Furthermore, their objective functions intended to employ local contextual information but theoretically failed and were not even suitable for creating a valid partition [24].

To this end, a new way to modify the existing FCM clustering is explored with adaptive regularization for contextual information. The proposed framework utilizes a new parameter to control the effect of pixel neighbors based on the heterogeneity of local grayscale distribution. A weighted image is devised that combines the local contextual information with respect to the heterogeneity of local grayscale distribution and the original grayscale that is calculated once in advance to reduce the computational cost. To improve segmentation accuracy and robustness to outliers, a kernel function is employed to replace the Euclidean distance metric. Validation against both synthetic and clinical MR data has been carried out to compare the proposed algorithms with 6 recent soft clustering algorithms in terms of segmentation accuracy and computational costs.

#### 3. Proposed Algorithms

We introduce a regularizing parameter to enhance segmentation robustness and preserve image details, devise a weighted image, and adopt the Gaussian radial basis function (GRBF) for better accuracy.

##### 3.1. The Introduced Regularization Term

The parameter used in [8–10] is usually set in advance to control the desirable amount of contextual information. Indeed, using a fixed for every pixel is not appropriate since noise level differs from one window to another. In addition, setting such parameter needs prior knowledge about noise which is not always available in reality. Hence, adaptive calculation of is necessary according to the pixel being processed.

To be adaptive to noise level of the pixel being processed, we first calculate the local variation coefficient (LVC) to estimate the discrepancy of grayscales in the local window to be normalized with respect to the local average grayscale. In the presence of noise to have high heterogeneity between the central pixel and its neighbors, LVC will increase. Considerwhere is the grayscale of any pixel falling in the local window around the pixel , is the cardinality of , and is its mean grayscale. Next, is applied to an exponential function to derive the weights within the local window:

The ultimate weight assigned to every pixel is associated with the average grayscale of the local window:

The parameter assigns higher values for those pixels with high LVC (for pixel being brighter than the average grayscale of its neighbors, will be , and will be large when the sum of LVC within its neighborhood is large) and lower values otherwise. When the local average grayscale is equal to the grayscale of the central pixel, will be zero and the algorithm will behave as the standard FCM algorithm. The value 2 in (11) is set through experiments to balance between the convergence rate and the capability to preserve details. The proposed parameter is embedded into (5) to replace . Figure 1 shows the calculation of with different cases of noise.