BioMed Research International

Volume 2016 (2016), Article ID 6727290, 10 pages

http://dx.doi.org/10.1155/2016/6727290

## Multigrid Nonlocal Gaussian Mixture Model for Segmentation of Brain Tissues in Magnetic Resonance Images

^{1}School of Math and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China^{2}Jiangsu Key Laboratory of Meteorological Observation and Information Processing, Nanjing University of Information Science and Technology, Nanjing 210044, China^{3}School of Information Science and Engineering, Changzhou University, Changzhou 213164, China

Received 13 June 2016; Accepted 22 July 2016

Academic Editor: Yong Xia

Copyright © 2016 Yunjie Chen 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

We propose a novel segmentation method based on regional and nonlocal information to overcome the impact of image intensity inhomogeneities and noise in human brain magnetic resonance images. With the consideration of the spatial distribution of different tissues in brain images, our method does not need preestimation or precorrection procedures for intensity inhomogeneities and noise. A nonlocal information based Gaussian mixture model (NGMM) is proposed to reduce the effect of noise. To reduce the effect of intensity inhomogeneity, the multigrid nonlocal Gaussian mixture model (MNGMM) is proposed to segment brain MR images in each nonoverlapping multigrid generated by using a new multigrid generation method. Therefore the proposed model can simultaneously overcome the impact of noise and intensity inhomogeneity and automatically classify 2D and 3D MR data into tissues of white matter, gray matter, and cerebral spinal fluid. To maintain the statistical reliability and spatial continuity of the segmentation, a fusion strategy is adopted to integrate the clustering results from different grid. The experiments on synthetic and clinical brain MR images demonstrate the superior performance of the proposed model comparing with several state-of-the-art algorithms.

#### 1. Introduction

Magnetic resonance imaging (MRI) is a helpful method for diagnosis of brain diseases, such as Alzheimer’s disease and schizophrenia. Accurate tissues segmentation, including gray matter (GM), white matter (WM), and cerebrospinal fluid (CSF), plays an important role in clinical practice and hence has attracted extensive research attention.

Many methods have been proposed for MR image segmentation in the past several decades. These approaches can be classified in terms of different criteria. For example, edge based methods [1, 2], region based methods [3, 4], and clustering based methods [5–7]. Unfortunately, most segmentation methods are hindered by various imaging artifacts such as noise and intensity inhomogeneities.

Intensity inhomogeneity, also known as bias field, arises from the imperfections of the image acquisition process and changes the absolute intensity for a given tissue class in different locations, which usually makes the intensity distribution within a particular tissue class flatter. Most traditional intensity based methods cannot obtain satisfactory results due to the impact of intensity inhomogeneity.

The observed MRI signal is the product of the true signal generated by the underlying anatomy and a spatially varying field factor and an additive noise :where , , , and are the observed intensity, true intensity, bias field, and noise at the th voxel, respectively. is the total number of pixels in the MR image. Many techniques [6, 8, 9] often ignore the noise and take the logarithmic transform on both sides of (1):

Many methods have been proposed to correct or estimate intensity inhomogeneities. Collewet et al. proposed a method based on measuring the coil sensitivity functions [8]. Based on the observation that the bias field is smooth, another group of methods overcome the impact of intensity inhomogeneity without estimating the bias field [9, 10]. However, most of them may lose edge information [11].

In this paper, we first propose an improved nonlocal Gaussian mixture model by introducing the nonlocal information into GMM model to reduce the effect of the noise. Then, a new multigrid generation method is presented, and we simplify the NGMM into a local version to eliminate the effect of the bias field and the intensity variation of intratissues. Finally, we propose a fusion strategy to integrate the results from different local regions. The experiments on both synthetic and clinical brain MR images show that our method can obtain more accurate results.

#### 2. Nonlocal Gaussian Mixture Model

Gaussian mixture model (GMM) has been widely used in many applications due to its excellent approximation properties. Suppose a MR image has a mixture of components, the mixture density function of pixel can be written as where is the mixture weights and is the parameter vector. is a standard Gaussian distribution of the th component and contains the parameters of the Gaussian distribution. and are the mean and variance, respectively. Then the entire distribution can be written as

The problem is how to find the best parameters :

Equation (5) can be calculated by using the expectation-maximization (EM) method [7]. In the E-Step, the algorithm calculates the expected value of the th weight:

In the M-Step, the parameters of the th Gaussian distribution can be calculated:

Based on the initialization, and are calculated iteratively until the stop criteria are reached. Finally, the pixel can be classified into the th class when . From (3), we can find that the GMM only considers the intensity distribution, which makes the method sensitive to the intensity inhomogeneity and noise.

In order to reduce the effect of the intensity inhomogeneity, Wells et al. [6] proposed a method to simultaneously estimate the bias field and segment the image into different classes. However, the method only addressed the bias field without analyzing the inhomogeneities in inner tissues. In order to ensure the smoothness of the bias field, a low-pass filter is utilized to convolve the bias field, which makes the estimated bias field inaccurate. Furthermore, the method is sensitive to the noise.

In order to reduce the effect of the noise and preserve more detailed information, we improve the Gaussian mixture model by using nonlocal information. The nonlocal information has been widely used for denoising purposes [12, 13]. Following the idea of nonlocal means method [12], we use the nonlocal information to adapt which can be defined aswhere is the weight function based on the similarity between the neighbor patch of each neighbor pixel to that of center pixel and satisfies the conditions and . For pixel and its neighbor pixel , the weight function is defined aswhere and are the neighbor patches around pixels and with width . Broadly speaking, if the neighbor patches of two pixels and are similar, it is more probable that these pixels belong to the same tissue, and the corresponding values of weight function would be higher. Conversely, if these two pixels are quite different, the value of the weight function should be small. acts as a filtering parameter to control the decay of the exponential function. is the Euclidean distance and is the standard deviation of the Gaussian kernel. Due to the fast decay of the exponential kernel, large distances between estimated patches lead to nearly zero weights. Essentially, the weight function aims to take advantage of the redundancy present in natural structures. Therefore, by using nonlocal information, the nonlocal Gaussian mixture model can reduce the effect of noise and preserve details of the edges.

#### 3. Multigrid Nonlocal Gaussian Mixture Method (MGMM)

Without considering the intensity inhomogeneity, the nonlocal Gaussian mixture model can only reduce the effect of noise but cannot obtain satisfactory results for the image containing severe intensity inhomogeneities. Zhu and Jiang [14] proposed a multicontext fuzzy clustering method (MCFC) to reduce the effect of intensity inhomogeneity by using fuzzy clustering method on each nonoverlapping regions and a fusion strategy to integrate the clustering outcomes form different regions. However, this method is sensitive to noise and it only uses traditional multigrid generation method, which makes the method inaccurate. Following the idea of MCFC, we improve the GMM by using the nonoverlapping multigrid. This idea is based on the following four assumptions:(1)The brain image has been skull-stripped. In this paper, we use the cut based method [15].(2)Bias field is smooth and slowly varying.(3)Within each grid, the number of clusters must equal and there are considerable numbers of pixels in each tissue class.(4)Within a grid, all pixels of the same tissue have similar true intensities.

The brain image without skull only has cerebrospinal fluid, gray matter, and white matter. Then, we set with assumption (3). The bias field is smooth and slowly varying, which makes it probable that the bias field values in small local region be regarded as constant. Then the multigrid segmentation method is less sensitive to the bias field. However, each pixel is processed only in its single local grid, which makes the method very sensitive to the size of the grid and unable to preserve the statistical reliability and spatial continuity of segmentation results. This can be illustrated in Figure 1.