Computational and Mathematical Methods in Medicine

Volume 2019, Article ID 4762490, 13 pages

https://doi.org/10.1155/2019/4762490

## Brain Tissue Segmentation and Bias Field Correction of MR Image Based on Spatially Coherent FCM with Nonlocal Constraints

^{1}College of Physics and Information Engineering, Minnan Normal University, Zhangzhou 363000, China^{2}Electronic Engineering College, Heilongjiang University, Harbin 150080, China

Correspondence should be addressed to Jianhua Song; moc.361@hjs_gd89

Received 27 September 2018; Accepted 11 February 2019; Published 3 March 2019

Guest Editor: Tomas Krilavičius

Copyright © 2019 Jianhua Song and Zhe Zhang. 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

Influenced by poor radio frequency field uniformity and gradient-driven eddy currents, intensity inhomogeneity (or bias field) and noise appear in brain magnetic resonance (MR) image. However, some traditional fuzzy c-means clustering algorithms with local spatial constraints often cannot obtain satisfactory segmentation performance. Therefore, an objective function based on spatial coherence for brain MR image segmentation and intensity inhomogeneity correction simultaneously is constructed in this paper. First, a novel similarity measure including local neighboring information is designed to improve the separability of MR data in Gaussian kernel mapping space without image smoothing, and the similarity measure incorporates the spatial distance and grayscale difference between cluster centroid and its neighborhood pixels. Second, the objective function with an adaptive nonlocal spatial regularization term is drawn upon to compensate the drawback of the local spatial information. Meanwhile, bias field information is also embedded into the similarity measure of clustering algorithm. From the comparison between the proposed algorithm and the state-of-the-art methods, our model is more robust to noise in the brain magnetic resonance image, and the bias field is also effectively estimated.

#### 1. Introduction

Magnetic resonance image has been widely used in diagnostic imaging for general check-up in clinical application, especially the detection and diagnosis of brain diseases. The volume change for brain tissues often indicates various diseases [1], such as brain tumor, leukoencephalopathy, olivopontocerebellar atrophy (OPCA), etc. Therefore, brain tissue segmentation of MR image has become a very important medical treatment step. However, brain MR image has some lacks such as noise, intensity inhomogeneity, low contrast, the partial volume effect, and so on, which brings serious obstacles to segment the brain MR images. To this end, the multitudinous brain MR image segmentation methods have been put forward by using the theory of fuzzy set, random field, and level set.

Currently, there are two popular methods-based models for medical image segmentation: the random field theory [2–4] and the fuzzy c-means (FCM) algorithm. Random field is density-based unsupervised method where it finds the maximum likelihood estimate of the parameters from a given dataset. However random field algorithm has the disadvantages in high complexity and slow convergence and will drop into local optimization. FCM clustering is another efficient method used in image segmentation because it has robust characteristics for ambiguity and can retain much more information than random field algorithm [5]. Therefore, FCM has been widely applied in different types of image segmentation [6–8]. The neighboring pixels in 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. However, the conventional FCM algorithm does not fully utilize this spatial information. Pedrycz and Waletzky [9] took advantage of the available classified information and actively applied it as part of their optimization procedures. Szilagyi et al. [10] proposed the enhanced FCM (EnFCM) algorithm to accelerate the image segmentation process in which the pixels of an image are replaced the gray-level histogram and the statistical number and calculation are much smaller than FCM. In order to further reduce the computation time and improve the parameter inflexibility, Cai et al. [11] presented a fast generalized FCM (FGFCM) method, and FGFCM introduced a flexible locality factor *S*_{ij} incorporating simultaneously both the gray-level difference and spatial distance in a local window. Ji et al. [12] proposed a robust spatially constrained fuzzy c-means (RSCFCM) algorithm for brain MR image segmentation. First, a spatial factor is constructed based on the posterior probabilities and prior probabilities and takes the spatial direction into account. Second, the negative log-posterior is utilized as dissimilarity function by taking the prior probabilities into account.

FCM with spatial constraint and its variants greatly improved the antinoise performance compared with FCM, but when the noise is very serious in the image, the performance of the algorithm may be worse. Therefore, the nonlocal spatial information was often used and incorporated into the distance metric of FCM in recent years [13–16]. Zhao [14] brought in a nonlocal adaptive regularization term in its energy function, and the control factor is adaptive determined to adjust the balance of the objective function. Feng et al. [15] proposed a FCM method with specific nonlocal information for the segmentation of synthetic aperture radar (SAR) image. Ma et al. [16] proposed a modified FGFCM approach by introducing nonlocal constraint term, and local distance metric and nonlocal distance metric are used, respectively, in its objective function. By introducing nonlocal constraint term, the features of image can be used more comprehensively and effectively, and the robustness to noise of FCM-based algorithm is significantly improved. However, there generally exists intensity inhomogeneity in brain MR images. Therefore, it is necessary to further design relevant algorithms to correct the intensity inhomogeneity. Sled et al. designed a set of software package for the estimation of bias field [17], and the characteristic of the method is nonparametric nonuniform intensity normalization or N3 for short; the distribution of the true tissue intensities can be achieved by an iterative method. Tustison et al. [18] improved the N3 algorithm based on modified B-spline approximation and hierarchical optimization algorithm (called N4ITK); N4ITK can also automatically perform without the priori knowledge. Liew and Hong Yan [19] introduced a spatial constraint to a fuzzy cluster method where the inhomogeneity field was modeled by a B-spline surface. The spatial pixel connectivity was implemented by a dissimilarity index, which enforced the connectivity constraint only in the homogeneous areas. Ji et al. proposed the modified possibilistic FCM (MPFCM) algorithm for bias field [20], generalized rough fuzzy c-means (GRFCM) algorithm, [21] and fuzzy local Gaussian mixture model (FLGMM) for brain MR image segmentation [22], respectively. Those methods can estimate bias field and segment the MR images simultaneously.

In this paper, a brain tissue classification and intensity inhomogeneity correction model of MR image based on spatially coherent FCM with nonlocal constraints is proposed. In this model, firstly, both the local constraint term and nonlocal regularization term about brain MR image are incorporated into the objective function, and an adaptive control factor is used to maintain the balance between them. Secondly, the similarity measure is designed in Gaussian kernel mapping space without image filtering, and the detail information and the edge of the image can be preserved well. Meanwhile, bias field model is also embedded into the objective function of clustering algorithm. Therefore, after the intensity inhomogeneity of the MR image is corrected, the segmentation accuracy is improved significantly. Experiments demonstrate that this algorithm can not only effectively estimate the bias field of MR image but also has stronger antinoise ability.

#### 2. Preliminary Theory

##### 2.1. Fuzzy Clustering Algorithms

Let denote an image with *N* pixels, where is gray value of the *k*th pixel in the image. FCM clustering aims at partitioning into *c* clusters by minimizing the following objective function:where denotes the *i*th cluster prototype, denotes the membership degree of belonging to *i*th cluster and follows , denotes the number of centers, denotes the Euclidean norm, and the parameter is a weight exponent on each fuzzy membership that determines the amount of fuzziness of the resulting partition.

Ahmed et al. proposed a modification to FCM objective function by introducing a term that allows the labeling of a pixel to be influenced by the labels in its immediate neighborhood [23]. This effect acts as a regularizer and biases the solution toward piecewise homogeneous labeling. It proved useful in segmenting images corrupted by noise. The modified objective function is given bywhere represents the neighbor voxels of and and stand for the number of voxels in the neighborhood of the *k*th voxel. The parameter controls the intensity of the neighboring effect. One disadvantage of BCFCM is that the neighborhood labeling is computed in each iteration step, something that is very time consuming.

##### 2.2. Spatially Coherent Fuzzy c-Means Clustering (SCFCM)

In view of some drawbacks of standard FCM algorithm, a modified scheme is proposed by Despotović et al. [24]. The similarity measure is replaced by introducing a weight factor , and the objective function iswhere is a weight factor including both local spatial information and grayscale difference and is designed as follows:where denotes a local neighboring window around , denotes the membership degree of neighborhood pixels belonging to the *i*th cluster, is the absolute intensity difference between the study pixel and its neighbor, is the Manhattan distance between the coordinates of pixel and , and and denote the coordinates and in the image, respectively. By minimizing equation (3) by Lagrangian multiplier approach, and can be derived as shown in the following equation:

Compared with the FCM, this algorithm has two advantages: firstly, it enhances the robustness to all kinds of noise. The constraint item of neighborhood information is included into the similarity measure, so as to effectively utilize the local information of the image. Secondly, it considers anisotropic neighborhood information, and more details and edges information can be preserved. However, the influence of bias field for MR images in segmentation algorithm is not mentioned.

##### 2.3. Coherent Local Intensity Clustering Model

Bias field of the MR image usually embodies slowly and smoothly varying for the pixel grayscale of the local region across an image. Meanwhile, in a neighboring local window of the image, bias field can be approximately considered as a constant. Therefore, the most popular model can be described in equation (6) [25]; let denote the observed image, denote the unknown bias field, denote the true image to be restored, and *n* denote the additive noise.

In the observed image, noise *n* is often assumed to obey Gaussian distribution with zero mean and variance , and the gray level value of the true image approximately takes a constant in a local window. Therefore, the gray level of the observed image can be approximated to obey Gaussian distribution with mean and variance . In coherent local intensity clustering (CLIC) model [26], a novel metric introducing spatially coherent local intensity convergence criterion for bias field estimation and image segmentation simultaneously is proposed. A Gaussian kernel weight parameter is introduced into the similarity measure of each pixel gray level and its neighbor pixels , and the objective function of CLIC iswhere is the clustering prototype with bias field in the selective region , is the weight of a truncated Gaussian kernel allocated for the intensity , and the weight parameter can be defined aswhere denotes the standard deviation, denotes a normalization factor to standardize Gaussian kernel, and denotes a radius to measure the size of the local region.

In the CLIC model, intensity inhomogeneity of the MR image can be effectively corrected and can reduce the misclassification rate, but there are some drawbacks in CLIC. When computing the distance metric between the central pixel and its surrounding pixels in a local region, the model only used the local neighborhood information of the pixel without considering the global structure information of the entire image. As a result, the antinoise performance of this algorithm is unsatisfactory.

#### 3. Proposed Method

The standard FCM algorithm has the shortcoming of being sensitive to noise. Though, the modified FCM algorithms are improved by adding the spatial information, it is difficult to get a satisfied segmentation result for noise robustness. Therefore, an improved FCM approach based on CLIC and SCFCM is proposed; its objective function is constructed according to the local constraint term and global regularization term; the similarity measure including local neighboring information is designed in Gaussian kernel mapping space, and brain tissue classification and intensity inhomogeneity correction can be realized simultaneously.

##### 3.1. Nonlocal Weighted Constraint

In a discrete noisy image , is the *k*th pixel and *y*_{k} is its nonlocal weighted average value. The derivation method of the nonlocal weighted average can be acquired in [27], and the mathematical expression of *y*_{k} iswhere indicates a search region of radius around , denotes nonlocal weight coefficient depending on similarity measure between and its neighboring pixels in window , and satisfies the constraint conditions and . The weight coefficient is computed as follows:where and denote the grayscale vectors in the square neighborhood and of radius around and , respectively, and is the weighted Euclidean distance between and ; its expression is defined in equation (11). is the same as in equation (8), and denotes a control factor to adjust the variation of the similarity measure .where is the *p*th pixel in the grayscale vectors and is defined as follows:where and , denote the coordinates of the *p*th component in a preselected region and .

##### 3.2. Objective Function

In order to correct bias field and classify the brain tissues simultaneously, the modified objective function-incorporated local constraint term and nonlocal regularization term is as follows:where is the membership degree of belonging to the *i*th cluster, denotes a local square region of the radius around the center , is the *i*th cluster center in , denotes the nonlocal weighted average value of , denotes the weight factor of local neighborhood information in equation (4), and the definitions of and are the same as equation (7). is a trade-off weight factor to adjust the balance of local neighborhood information and nonlocal constraints information, and the definition of parameter iswhere denotes the gray level mean of all pixels in the local region , and denotes the variance of pixel gray values in the same window. The larger the value is, the smaller the influence of the noise is. The factor can be obtained adaptively with the change of the local window without being given in advance.

Theorem. *Assume , and . On the basis of Lagrange multiplier approach, equation (13) is minimized with respect to , and can be derived as shown in the following equation:*

*Proof. *According the method of Lagrange multiplier, equation (13) can be converted to unconstrained optimization problem:where is the Lagrange multiplier of the constraint condition , by computing the partial derivatives of polynomial in regard to and , respectively, and set , , as shown in the following equation:The following equation can be obtained by mathematical derivation of equation (17):Substituting equation (19) into equation (17), we obtain the following equation:And then substituting equation (20) into equation (19), the following equation can be obtained:Similarly, set , that isThe following equation can be obtained from equation (22) by mathematical derivation:We adopt the same mathematical derivation process to estimate bias field , for fixed and , and computing the partial derivative of with respect to , set , that is can be obtained from equation (24).The theorem proves to be completed.

Finally, the framework of the proposed algorithm can be summarized in Table 1.