

ORIGINAL ARTICLE 

Year : 2019  Volume
: 5
 Issue : 2  Page : 6875 

Local Gauss multiplicative components method for brain magnetic resonance image segmentation
Jie Cheng, Haiqing Yin, Lingling Jiang, Junyu Zheng, Su Wei
College of Science, China University of Petroleum, Qingdao, China
Date of Web Publication  23Sep2019 
Correspondence Address: Haiqing Yin College of Science, China University of Petroleum, Qingdao 266580 China
Source of Support: None, Conflict of Interest: None  Check 
DOI: 10.4103/digm.digm_7_19
Background and Objectives: In magnetic resonance (MR) images' quantitative analysis, there are often considerable difficulties due to factors, such as intensity inhomogeneities and low contrast. Here, we construct a new image segmentation method to solve the MR image segmentation problem caused by internal and external factors. Materials and Methods: We downloaded a series of MR images as research objects through the BrainWeb (http://www.bic.mni.mcgill.ca/brainweb/). There is low contrast information between different components in these images. In addition, we randomly added a certain degree of bias field information to the images. We proposed a model that can simultaneously perform bias field estimation and image segmentation. Our idea is to make use of the property that observed image can be decomposed into multiplicative components. First, the bias field representation is given by a series of smooth basic functions; the required true image is represented as the function of observed image and bias field. Then, the segmentation model of Gaussian probability distribution with different means and variances is constructed by local information. Results: Qualitative experiments (intensity inhomogeneity images) show that our model achieves satisfactory segmentation results with very few (<10) iterations for severe intensity inhomogeneities image segmentation, while quantitative experiments (20 brain MR images) show that the proposed model can achieve higher accuracy in segmentation. Conclusions: Different from the existing model, our model is constructed based on the local information of the true image, and the influence of abovementioned factors is better avoided and obtain satisfactory results.
Keywords: Bias field correction, image segmentation, magnetic resonance image, probability distribution
How to cite this article: Cheng J, Yin H, Jiang L, Zheng J, Wei S. Local Gauss multiplicative components method for brain magnetic resonance image segmentation. Digit Med 2019;5:6875 
How to cite this URL: Cheng J, Yin H, Jiang L, Zheng J, Wei S. Local Gauss multiplicative components method for brain magnetic resonance image segmentation. Digit Med [serial online] 2019 [cited 2020 Oct 20];5:6875. Available from: http://www.digitmedicine.com/text.asp?2019/5/2/68/267611 
Introduction   
Magnetic resonance (MR) image segmentation is a crucial step in the field of medical diagnosis and clinical application. However, due to the characteristics of MR devices, they are often affected by intensity inhomogeneities, bias field, low contrast, and additive noise. It is the influence of these factors that lead to overlapping areas of intensity distribution among different organizations and results in a wrong classification. To solve these problems, a large number of related segmentation models have been proposed in recent years.
To segment such images, the bias correction is needed first. The bias field correction is to estimate the bias field by the observed image and then to eliminate the influence on the true image. In general, the observed image can be decomposed into two intrinsic multiplicative components: bias field and true image and one additive zeromean noise part. The intensity inhomogeneity can be obtained by approximate calculation of the smooth bias field component. The existing bias correction methods can be classified into two types: prospective methods and retrospective methods. In the prospective methods, some measures are used to avoid the intensity inhomogeneity in the image acquisition process, and the clinical application is limited because the methods cannot remove the effect on the image. In contrast, the retrospective methods consider using image information to eliminate any intensity inhomogeneities of the MR devices. There are several methods of bias field correction that can do the latter: the polynomial or splinebased,^{[1]} templatebased,^{[2]} and histogrambased^{[3],[4],[5]} methods are among them. In the study by Sled et al.,^{[3]} the bias field is estimated by maximizing the highfrequency information of the tissue intensity distribution, whereas in the study by Likar et al.,^{[5]} it can be estimated by minimizing the image entropy. In addition, it also includes some classical lowpass filtering bias field correction methods,^{[6],[7],[8],[9]} which have the advantages of simplicity and high efficiency. For this type of methods, the intensity inhomogeneity is regarded as a lowfrequency spatial signal, which can be suppressed by highpass filtering.
Usually, bias field correction is a preprocessing step of image segmentation. The bias field correction and segmentation can be regraded as two intertwining processes. Based on this, the segmentationbased bias field correction methods are proposed to combine the two processes so that the both promote each other in the process. In previous studies^{[10],[11]} adaptive fuzzy Cmeans (FCM) methods are proposed. In the method, local gray information is introduced by modifying the objective function to compensate the bias field. A spatial FCM is proposed in the study by Chuang et al.,^{[12]} where the local spatial information is introduced into the objective function. In the study by Qiu et al.,^{[13]} an improved fuzzy algorithm is proposed by introducing fuzzifiers and spatial constraint in the membership function. The recently proposed improved FCM algorithms for MR image segmentation were given in the previous studies.^{[14],[15],[16]} Ashburne et al.^{[17]} proposed a probabilistic framework that simultaneously performs registration, classification, and bias fidel correction. Based on the expectationmaximization algorithms,^{[18],[19],[20],[21]} we developed the approaches that can simultaneously perform segmentation and bias field correction. Li et al.^{[22]} proposed a robust parametric method named multiplicative intrinsic component optimization (MICO). In the model, the bias field is represented as a linear combination of a set of basis functions.
Here, we propose a new model for simultaneous bias field and image segmentation. The model combines the advantages of smooth basis functions' representation of bias field in the study by Li et al.^{[22]} and the traditional local Gaussian distribution (LGD) model.^{[23]} According to the formation of MR image principle, the true image is expressed as a function of the bias field, and then, the true image is used to construct the Gaussian distribution energy framework based on local information. By alternating minimization of the energy functional, the segmentation images with intensity inhomogeneous and low contrast are achieved.
Materials and Methods   
Local Gaussian distribution method
Wang et al.^{[23]} proposed a regionbased active contour model based on local intensity distribution. The method efficiently utilizes the local image intensities, which are described by Gaussian distribution with different means and variances. The means and variances are regarded as spatially varying functions and introduced as two variables of the proposed energy functional in the model. The LGD energy is defined as follows:
Where is a posteriori probability of the subregions :
Where u_{i}(x) and σ_{i}(x) are local intensity means and standard deviations, respectively.
Multiplicative intrinsic component optimization method
Li et al.^{[22]} proposed an energy minimization method called MICO. It takes the bias field b and the true image J as two multiplicative components of the image and integrates them into an energy minimal framework. In this method, image bias field and tissue segmentation are implemented simultaneously. The energy functional is given as:
Where is a column vector valued function and each component u_{i}(x) represents the membership function of pixel x for the region Ω_{i}. And is denoted by:
and
is a column vector where each component C_{i} corresponds to the gray value of the i^{th} tissue. The estimation of the bias field is performed by finding the optimal vector coefficient in the linear combination ) where (·) T is the transpose operator . q is a fuzzifier and can be valued as q ≥ 1. By minimizing the above energy functional, b and U can be obtained simultaneously so that the segmentation of MR images with intensity inhomogeneity can be achieved.
Proposed model
Local Gaussdi multiplicative components method
Based on the above analysis, to achieve the segmentation more accurately, we will propose a new model based on local distribution. According to the process of image formation, the observed signal I(x) can be expressed as the following formula:
Where b(x) is the bias field and is slowly varying in the image domain and J(x) is the true image that we require and need to be segmented. For different tissue regions , J(x) takes N different constants c_{i}. n(x) is additive noise with zeromean. However, due to the effect of intensity and inhomogeneity in the same tissue, the assumption that J(x) is a constant in the same tissue is usually difficult to achieve.
To effectively utilize the properties of the bias field b and the true image J, we use the description of bias field and true image in the study by Li et al.^{[22]} They can be described as follows:
Where W,C, U(X) and G(x) are defined in MICO method section). In this paper, we set a number of basis functions of bias field b to M = 15. As mentioned by Li et al.,^{[22]} J(x) in Equation (6) represents a soft segmentation. To make full use of local intensity information, we will use LGD model to describe the distribution of local intensities. Considering the influence of the bias field b(x), we try to construct a local distribution model of the true image J(x) based on the observed signal I(x) through the relationship in Equation (5). Similar to the study done by Li et al,^{[22]} we consider the bias field b(x) and the true image J(x) as the multiplicative intrinsic components of an observed image. Therefore, given the observed signal I(x), through mathematical transformation, we get . Given a pixel x in the image domain Ω Ω, the neighborhood O_{x} can be segmented by the maximum a posteriori probability (MAP) framework. Let be the a posteriori probability of the subregions given the neighborhood gray valueJ(W,y). According to the Bayes rule,
Where denoted by P_{i,x}(J(W,y)) is the probability density in the region . is a priori probability of the partition among all possible partitions of O_{x} and given all partitions are a priori equally possible.^{[24]} The a priori probability p(J(W,y)) is independent of the choice of the region. Assuming that pixels in each regions are independent, MAP is achieved when is maximized In this paper, is the type of Gaussian distribution defined by Equation (2). Using logarithmic operation and introducing a local weight kernel function, the problem at pixel x can be converted to minimize the following objective functional:
Where u_{i}(x) and are local intensity means and standard deviations, respectively. and is a hardsegmented binary membership function, with . Moreover, ω(x–y) is the nonnegative weight kernel function.
Minimizing Equation (8) for all pixels x in the image domain Ω achieves the final segmentation. In addition, to achieve a soft segmentation result, we introduce a fuzzifier q ≥ 1 in Equation (8) to define the energy functional. Therefore, our ultimate task is to minimize the following doubleintegral energy functional:
Where and
.
Energy minimization
We can solve the minimization of the energy functional by alternately minimizing the Equation (9) with respect to each variable to be solved given the others fixed. The detailed process is as follows.
Optimization of
For fixed and W, we find an optimal and that minimizes E. By calculation, they can be given as:
Optimization of W
We fix to minimize the energy functional Equation (9) with respect to W. We obtain the coefficient function of the bias field as follows:
From the obtained vector W, we can estimate the true image J of the current iteration: .
Optimization of
The values of are divided into two kinds: the hard segmentation (for q = 1) and the soft segmentation (for q > 1). Then we fix other variables and minimize the energy functional E(x) with respect to U(y). When q = 1, according to the previous analysis, we can see that the minimum form is given by:
Where and
For q > 1, we combine the constraints in Equation (4) to minimize E(x), which can be given by:
The overall framework of the algorithm is summarized as follows [Algorithm 1]:
Remark
The proposed model is essentially different from the two involved models.^{[22],[23]} Compared with the previous study^{[23]}(as mentioned in Part “MATERIALS AND METHODS Local Gaussian distribution method “): (1) the construction of our model as Equation (9) is based on the true image J(x) in Equation (6); (2) the bias field b(x) in Equation (6) can be estimated during the implementation process; (3) the study^{[23]} is achieved through the level set φ(x), while the proposed model uses the membership function u=(u_{1},...., u_{N})^{T} in Equation (4) and fuzzifier q. Compared with the study by Li et al.,^{[22]} the proposed model introduces a local weight function, which changes the single integral into the double integral; the variational means and variances are introduced based on the local probability of the true image.
Results   
In this section, the proposed method has been implemented on synthetic and real MR images. We first use the proposed model to perform segmentation tests on images with intensity inhomogeneities, and then compare the model with the MICO model through a series of segmentation to verify the validity of our model. For all the experiments in this section, we set the number of basis functions of bias field to M = 15 and σ = 3 (i.e., the standard deviation of local weight kernel function ω).
Experimental segmentation results on the images with different degree of intensity inhomogeneity images and the corresponding energy functional curves of E are presented in [Figure 1] and [Figure 2]. We denote the images in the upper and lower row as Img1 and Img2, respectively. The original images, the bias field images, the bias field corrected images, and the segmentation results are shown in the left, second, third, and the last columns, respectively. We can see that even if there are severe intensity inhomogeneities in the images, our model can still produce satisfactory segmentation results. In addition, as shown in [Figure 2], since the initialization of the bias field b and the membership function before our iteration, the energy functional reaches a maximum at the second iteration. Moreover, the number of iterations before the stability of the function is related to the degree of intensity inhomogeneity. Usually, through our experiments, it is found that the stability value is achieved before only several iterations. Therefore, we can find the acceptable convergence speed of the model through the figure.  Figure 1: Results of our method on the images with different degree of intensity inhomogeneity. The original images, the bias field images, the bias field corrected images, and the segmentation results are shown in the left, second, third, and the right columns, respectively
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 Figure 2: The energy functional curves of Img1 and Img2 in Figure 1 correspond to each iteration. Img1: Upper row, Img2: Lower row
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To quantitatively evaluate our approach, we give a comparison experiment between the proposed model and the MICO model. In this experiment, the 20 experimental images with ground truth from BrainWeb which can be download from http://www.bic.mni.mcgill.ca/brainweb/. We have added different degrees of intensity inhomogeneities to each of these images. Two of the 20 experimental images are selected and presented in this paper, as shown in [Figure 3]. The original images (Img3 and Img4) and the segmentation results of Img3 and Img4 by the proposed and MICO model are shown in the second, third, fourth, and fifth rows, respectively. The red circles in the images mark the obvious difference between the results given by the two models, where there is lowcontrast information between the different tissues. The proposed model can better segment these regions using local information.  Figure 3: The segmentation results of the proposed (LGMC) and MICO methods for two of the 20 experimental images. The segmentation results, the segmented WM, GM, and CSF are shown in the first, second, third, and fourth columns, respectively. WM: White matter, GM: Gray matter, CSF: Cerebrospinal fluid, LGMC: Local Gauss multiplicative components, MICO: multiplicative intrinsic component optimization
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To more theoretically demonstrate the difference between the two models, we evaluate the accuracy by the Dice ratios on the 20 experimental images, as shown in [Figure 4]. The Dice ratio for white and gray matter is shown in the left and right subgraphs of [Figure 4], respectively. As we can see, our method achieves relatively shorter box plot and higher Dice ratio values, which clearly demonstrates the advantages of our model in terms of accuracy.  Figure 4: Quantitative evaluation for the segmentation results of the proposed (local Gauss multiplicative components) and multiplicative intrinsic component optimization model for 20 images using Dice ratio. WM: White matter, GM: Gray matter, LGMC: Local Gauss multiplicative components, MICO: Multiplicative intrinsic component optimization
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Conclusions   
In this paper, we propose a novel LGD MR image segmentation (local Gauss multiplicative components) model based on multiplicative intrinsic components. The proposed model is a segmentation method that combines probability theory with image decomposability. The design of the proposed model considers the influence of bias field in the previous model design process, and takes the decomposed true image as the target object of the model design, which not only avoids the influence of inhomogeneities in the image but also realizes the estimation and segmentation of image offset field at the same time. In the experimental part, we performed selfverification experiment and comparative experiment. The images in the selfvalidation experiment with intensity inhomogeneity, since the new model only applies the corrected real images, and the experiment obtains an accurate segmentation at a satisfactory convergence speed. In contrast experiments, the proposed model achieves high precision segmentation in the form of local probability estimation for regions with lowcontrast information between different organizations in the experimental images. In conclusion, compared with the previous models, the proposed model has the advantages of efficiency and accuracy for brain MR image segmentation and provides new ideas for segmentation of such images.
Financial support and sponsorship
Nil.
Conflicts of interest
There are no conflicts of interest.
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[Figure 1], [Figure 2], [Figure 3], [Figure 4]
