## 1 Introduction  
Magnetic resonance (MR) image segmentation is a key step after magnetic resonance imaging (MRI), and its results directly affect diagnosis and treatment [1; 2; 3]. Numerous models have been proposed to achieve this, including clustering models [4; 5; 6], level set models [7; 8; 9; 10], active contour models [11; 12], and so on [13; 14; 15]. Accurate segmentation results have important guiding significance for clinical diagnosis. However, MRI quality can be easily destroyed by noise and intensity inhomogeneity because of a series of complex reasons [16]. Intensity inhomogeneity, which is also called bias field, appears as the smooth and slow change of the gray value of a pixel for the same tissue [17]. Hence, these defects in the process of brain MRI segmentation directly lead to segmentation errors [18]. Among all the segmentation methods, the clustering model is widely used, for which hard and soft clustering are the two main methods [19]. K-means clustering is the most representative model of hard clustering. Soft clustering can be further subdivided into mixture models and fuzzy c-means (FCM) [20], where FCM assumes pixels pertain to more than one class and has been widely studied and applied. However, it is worth noting that FCM does not take any spatial information into consideration [21]. Many modified models incorporate spatial information into FCM to improve its robustness. The bias-corrected FCM (BCFCM) model was presented by [PERSON] et al. [22], which introduced the Euclidean distance between pixels in the neighborhood and its cluster center as the constraint term for the objective function of fuzzy clustering. However, the spatial information used in BCFCM is not anisotropic (i.e., pixels in the neighborhood have same the weight for central pixel). Also, the distance between neighborhood and cluster center must be recalculated during each iteration. To decrease time consumption and complexity, [PERSON] et al. [23] first filtered the image before iterative operation, and then proposed two improved models using mean and median filtering, namely, FCM_S1 and FCM_S2, respectively. Both models can reduce the energy minimization time. Nonetheless, both mean and median filtering inevitably blur the edge information. Further, the degree of fuzzy of segmentation results gradually rises with the increasing number of iteration steps. [PERSON] et al. [24] presented the fuzzy local information c-means (FLICM) model by introducing fuzzy parameters to directly manipulate the image and to reduce the number of input parameters. FLICM is highly robust, but it easily obtains incorrect segmentation results at the boundaries of different tissues and does not take the bias field into consideration. [PERSON] et al. [25] incorporated nonlocal and local information into the energy formulation and designed a weighted image patch-based FCM (WIPFCM) model to further improve robustness. They substituted the original image with image patches, which provided anisotropic weighting for the image pixels. Most of the models described above cannot correct the bias field caused by intensity inhomogeneity. Hence, [PERSON] et al. designed a local intensity clustering (LIC) [26] and multiplicative intrinsic component optimization (MICO) model [27], respectively. The two models apply an edge-based level set model and a clustering model, respectively. MICO also takes full advantage of a smoothly and slowly changing bias field. Both LIC as well as MICO can correct the bias field of an inhomogeneous intensity image, but they do not deal well with noise in MRI. Another popular model is the bias correction embedded fuzzy c-means (BCEFCM) [28] proposed by [PERSON] et al., which adds the compute unified device architecture (CUDA) accelerated nonlocal means denoising model to remove noisy pixels. They assumed an original image consisting of a real image and a bias field, and then convolved the bias field by a kernel function in each iteration.