Fuzzy Local Gaussian Mixture Model for Brain MR ... - IEEE Xplore

IEEE TRANSACTIONS ON INFORMATION TECHNOLOGY IN BIOMEDICINE, VOL. 16, NO. 3, MAY 2012

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Fuzzy Local Gaussian Mixture Model for Brain MR Image Segmentation Zexuan Ji, Yong Xia, Member, IEEE, Quansen Sun, Qiang Chen, Member, IEEE, Deshen Xia, and David Dagan Feng, Fellow, IEEE

Abstract—Accurate brain tissue segmentation from magnetic resonance (MR) images is an essential step in quantitative brain image analysis. However, due to the existence of noise and intensity inhomogeneity in brain MR images, many segmentation algorithms suffer from limited accuracy. In this paper, we assume that the local image data within each voxel’s neighborhood satisfy the Gaussian mixture model (GMM), and thus propose the fuzzy local GMM (FLGMM) algorithm for automated brain MR image segmentation. This algorithm estimates the segmentation result that maximizes the posterior probability by minimizing an objective energy function, in which a truncated Gaussian kernel function is used to impose the spatial constraint and fuzzy memberships are employed to balance the contribution of each GMM. We compared our algorithm to state-of-the-art segmentation approaches in both synthetic and clinical data. Our results show that the proposed algorithm can largely overcome the difficulties raised by noise, low contrast, and bias field, and substantially improve the accuracy of brain MR image segmentation. Index Terms—Bias field correction, fuzzy C-means (FCMs), Gaussian mixture model (GMM), image segmentation, MRI.

Manuscript received May 1, 2011; revised September 7, 2011 and November 20, 2011; accepted January 16, 2012. Date of publication January 24, 2012; date of current version May 4, 2012. This work was supported in part by the Australian Research Council grants, in part by the National Natural Science Foundation of China under Grant 60805003, in part by the National Science Foundation of Jiangsu Province under Grant BK2008411, and in part by the Doctoral Fund of Ministry of Education of China (Research Fund for the Doctoral Program) under Grant 200802880017. Z. Ji is with the School of Computer Science and Technology, Nanjing University of Science and Technology, Nanjing 210094, China, and also with the School of Information Technologies, University of Sydney, Sydney, N.S.W. 2006, Australia (e-mail: [email protected]). Y. Xia is with the School of Information Technologies, University of Sydney, Sydney, N.S.W. 2006, Australia, and also with the Department of PET and Nuclear Medicine, Royal Prince Alfred Hospital, Sydney, N.S.W. 2050, Australia (e-mail: [email protected]). Q. Sun and D. Xia are with the School of Computer Science and Technology, Nanjing University of Science and Technology, Nanjing 210094, China (e-mail: [email protected]; [email protected]). Q. Chen is with the Department of Radiology, Stanford University, Stanford, CA 94305 USA, and also with the School of Computer Science and Technology, Nanjing University of Science and Technology, Nanjing 210094, China (e-mail: [email protected]). D. D. Feng is with the School of Information Technologies, University of Sydney, Sydney, N.S.W. 2006, Australia, and also with the Med-X Research Institute, Shanghai JiaoTong University, Shanghai 200025, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TITB.2012.2185852

I. INTRODUCTION EGMENTATION of major brain tissues, including gray matter (GM), white matter (WM), and cerebrospinal fluid, from magnetic resonance (MR) images plays an important role in both clinical practice and neuroscience research. However, due to the nonuniform magnetic field or susceptibility effects, brain MR images may contain a smoothly varying bias field, which is also referred to as the intensity inhomogeneity or intensity nonuniformity [1]. As a result, the intensities of the same tissue vary across voxel locations and may lead to segmentation errors. Therefore, bias field correction and segmentation should be interleaved in an iterative process so that they can benefit from each other and yield better results. Many brain MR image segmentation approaches with bias field correction have been proposed in the literature [2]–[12]. Among them, those based on the expectation-maximization (EM) algorithm [2]–[4] and fuzzy C-mean (FCM) clustering [5]–[12] are the most popular ones. Pham and Prince. [8] proposed an adaptive FCM (AFCM) algorithm, which incorporates a spatial penalty term into the objective function to enable the estimated membership functions to be spatially smoothed. Ahmed et al. [9] added a neighborhood averaging term to the objective function, and thus developed the bias-corrected FCM (BCFCM) algorithm. Liew and Yan. [10] used a B-spline surface to model the bias field and incorporated the spatial continuity constraints into fuzzy clustering algorithms. Li et al. [6] proposed an energy-minimization approach to the coherent local intensity clustering (CLIC), with the aim of achieving tissue classification and bias field correction simultaneously. In our previous work [11], [12], we incorporated the global information into the CLIC model to enhance its robustness to the involved control parameters, then we proposed a newly modified possibilistic FCMs clustering algorithm (MPFCM) for bias field estimation and segmentation of brain MR image. Although such modification improves the segmentation accuracy, it also dramatically increases the computational complexity. Various kernel techniques have been used to improve the performance of clustering approaches. Chen and Zhang [13] replaced the original Euclidean distance with a kernel-induced distance and supplemented the objective function with a spatial penalty term, which models the spatial continuity compensation. Yang and Tsai [14] proposed an adaptive Gaussian-kernel-based FCM (GKFCM) algorithm with the spatial bias correction. Liao et al. [15] developed a spatially constrained fast kernel FCM (SFKFCM) clustering algorithm to improve the computational efficiency. However, the clustering performed in a kernel space is generally very time consuming.

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Alternatively, brain MR images can be segmented by using the Gaussian mixture model (GMM) [16], where the voxel intensities in each target region are modeled by a Gaussian distribution [17]. The GMM parameters are usually estimated by maximizing the likelihood of the observed image via the EM algorithm [2]. A major drawback of the GMM-EM framework is its lack of taking the spatial information and uncertainty of data into consideration. As a result, it may produce less accurate segmentation results. To remedy this drawback, Greenspan et al. [18] and Blekas et al. [19] incorporated the spatial constraints into the GMM. Tran et al. proposed the fuzzy GMM (FGMM) model to address the uncertainty of data and improve parameter estimation [20]. Zeng et al. [21] developed the type-2 FGMM (T2-FGMM) model for density modeling and classification. However, to our knowledge, so far those fuzzy extensions of the GMM-based segmentation algorithm have not been able to overcome the difficulties caused by the intensity inhomogeneity. The conventional GMM implies the stochastic assumption that throughout the image, intensities in the same region are sampled independently from an identical Gaussian distribution. This assumption, however, is invalid for brain MR images due to the existence of the bias field. Based on the fact that the bias field varies very slowly and can be ignored within a small window, in this paper, we assume that the local image data within the neighborhood of each voxel follow the GMM, in which the mean of each Gaussian component is approximated as a tissuedependent constant multiplied by the bias field estimated at this voxel. Thus, we propose the fuzzy local GMM (FLGMM) algorithm for brain MR image segmentation. The objective function of our algorithm is defined as the integration of the weighted GMM energy functions over the entire image. In the objective function, a truncated Gaussian kernel function is used to impose the spatial constraint, and fuzzy memberships are employed to balance the contribution of each GMM to the segmentation process. The proposed algorithm has been compared to other state-of-the-art segmentation algorithms in both simulated and clinical brain MR images.

B. Fuzzy C-Mean Let I = {I(k) ∈ Rd ; 1 ≤ k ≤ n} be a set of d-dimensional image features. The FCM [22] partitions this feature set into c clusters based on minimizing the sum of distances from each feature to every cluster centroid weighted by its corresponding membership. Let the membership function be U = {ui (k)} ∈ Rc×n , where ui (k) ∈ [0, 1] is the degree offeature I(k) belonging to cluster i and follows the constraint ci=1 ui (k) = 1. The quadratic objective function to be minimized is c JFCM = (2) ui (k)m |I(k) − vi |2 dk i=1

where vi is the centroid of cluster i, and m ∈ (1, ∞) is the fuzzy coefficient. C. Gaussian Mixture Model The GMM is a weighted sum of c Gaussian density distributions. With the GMM, the likelihood of the observed data I(k) is as follows: c pi N (I(k)|μi , Σi ) (3) P (I(k)|Θi ) = i=1

where Θi = {pi , μi , Σi } is the assembly of parameters, and pi is the mixing coefficient of ith Gaussian component N (I(k)|μi , Σi ) and follows the constraint ci= pi = 1. The parameters involved in the GMM are denoted by Θ = {Θi , i = 1, . . . , c}, and are usually estimated through maximizing the likelihood of observed data via the EM algorithm, as shown in the following: c n ∗ pi N (I(k)|μi , Σi ) . (4) Θ = arg max Θ

k =1

i=1

III. FLGMM ALGORITHM A. Modeling

II. RELATED WORK A. Bias Field Formulation The bias field in a brain MR image can be modeled as a multiplicative component of an observed image, as shown in the following: I = bJ + n

each tissue type, implies that the true signal J is approximately a piecewise constant map.

(1)

where I is the observed image, J is the true image to be restored, b is an unknown bias field, and n is the additive zero-mean Gaussian noise. The goal of bias field correction is to estimate and eliminate the bias field b from the observed image I. Without loss of generality, the bias field b is assumed to be slowly varying in the entire image domain. Ideally, the intensity J in each tissue should take a specific value vi , reflecting the physical property being measured, e.g., the proton density in MR images. This property, in conjunction with the spatially coherent nature of

Image segmentation aims to partition the image domain Ω into c disjoint regions, such that Ω = {Ωi }ci=1 . For each voxel x, if its neighborhood region is denoted by Ox , {Ωi ∩ Ox }ci=1 forms a partition of Ox . We assume the local image data within Ox satisfy the GMM. Let p (y ∈ Ωi ∩ Ox |I(y)) be the posterior probability of voxel y belonging to the subregion Ωi ∩ Ox on condition that it has the intensity value I(y). According to the Bayes rule, we have p(I(y)|y ∈ Ωi ∩ Ox )p(y ∈ Ωi ∩ Ox ) p(I(y)) (5) where p(I(y)|y ∈ Ωi ∩ Ox ) is the probability distribution within the subregion Ωi ∩ Ox , p (y ∈ Ωi ∩ Ox ) is the prior probability of voxel y belonging to Ωi ∩ Ox , and p(I(y)) is the probability of observing the intensity I(y). Since p(I(y)) does not vary with the class label of voxel y, we can simplify p(y ∈ Ωi ∩ Ox |I(y)) =

JI et al.: FUZZY LOCAL GAUSSIAN MIXTURE MODEL FOR BRAIN MR IMAGE SEGMENTATION

the posterior probability as follows: p(y ∈ Ωi ∩ Ox |I(y)) ∝ p(I(y)|y ∈ Ωi ∩ Ox )p(y ∈ Ωi ∩ Ox ). (6) Let the prior probability p (y ∈ Ωi ∩ Ox ) be denoted by pi (x). Based on the GMM assumption, p(I(y)|y ∈ Ωi ∩ Ox ) follows a Gaussian distribution N (I(y)|μi (x), Σi (x)) with mean μi (x) and covariance matrix Σi (x). Applying natural logarithm to both sides of this equation, we can estimate the maximum a posteriori probability (MAP) solution to this segmentation problem by minimizing the following energy function [23]: ExLGM M

=

Ω i ∩O x

− ln [pi (x)N (I(y)|μi (x), Σi (x))] dy

(7) (x) is the mixing coefficient, and satisfies the constraint where p i c i= pi (x) = 1. To incorporate bias field correction into the segmentation process, we assume the true intensity J in each tissue type i to be a constant vi . Thus, the mean of observed data in each Gaussian component can be approximated as follows: μi (x) = b(x)vi , i = 1, . . . , c

(8)

where b(x) is the bias field at each voxel x. Consequently, the local GMM energy can be expressed as follows: c

weighting function, the FLGMM energy is c FLGM M = −ui (y)m K(x − y) Ex i=1

Ω

ln[pi (x)N (I(y)|b(x)vi , Σi (x))]dy.

(12)

ExFLGM M

over every voxel x in To minimize the GMM energy the image domain Ω, we define the following objective function for our FLGMM: c FLGM M = −ui (y)m K(x − y) J i=1

Ω

Ω

ln[pi (x)N (I(y)|b(x)vi , Σi (x))]dydx.

c i=1

341

(13)

B. Minimization By changing the order of integration, the above objective function can be written as follows: c FLGM M J = ui (y)m di (I(y))dy (14) i=1

where di (I(y)) = K(x − y) × − ln

pi (x) (2π)d/2 |Σi (x)|1/2

1 exp − (I(y) − b(x)vi )T Σi (x)−1 (I(y) − b(x)vi ) dx 2

− ln [pi (x)N (I(y)|b(x)vi , Σi (x))] dy.

(15)

(9) Next, we introduce the fuzzy membership function ui (y) into the calculation of the local GMM energy defined in the neighborhood Ox . Since ui (y) becomes extremely small for y∈ / Ωi ∩ Ox , we have the following FLGMM energy:

can be viewed as a dissimilarity measure between the observed voxel intensity I(y) and the cluster prototype characterized by {b(x)vi , Σi (x)}. The iterative minimization of the objective function J FLGM M subject to the constraints c c ui (x) = 1, and pi (x) = 1 (16)

ExLGM M =

i=1

Ω i ∩O x

ExFLGM M c = −ui (y)m ln[pi (x)N (I(y)|b(x)vi , Σi (x))]dy i=1

i=1

(10)

Ox

where m ∈ (1, ∞) is the fuzzy coefficient. Generally, voxels far away from the neighborhood center x are expected to have less influence than voxels close to x. Therefore, we incorporate a nonnegative and monotonically decreasing weighting function K(d) into the energy function to constrain the influence of different voxels [17]. In this study, we choose K(d) to be the following truncated Gaussian kernel with a localization property [6]: ⎧ ⎨ 1 e−|d|2 /2τ 2 , for |d| ≤ ρ (11) K(d) = a ⎩ 0, else where τ is the standard deviation of the Gaussian kernel, ρ is the radius of the neighborhood, and a is a constant to normalize the Gaussian kernel such that O x K (u)du = 1. With this

i=1

can be derived analogously to the traditional FCM algorithm using the Lagrange multiplier method. This iterative algorithm is summarized as follows. Step 1: Initialization. Initialize the number of clusters, standard deviation, and neighborhood radius of the truncated Gaussian kernel, cluster centroids, and bias field at each voxel. Step 2: Updating parameters. Step 2.1: Updating membership function ⎞ ⎛

1/m −1 −1 c d (I(y)) i ⎠ . (17) ui (y) = ⎝ d (I(y)) j j =1 Step 2.2: Updating covariance matrix Σi (x) ui (y)m K(x − y)((I(y) − b(x)vi )(I(y) − b(x)vi )T )dy = . ui (y)m K(x − y)dy (18)

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TABLE I SUMMARY OF TESTING IMAGES USED IN COMPARATIVE EXPERIMENTS

Step 2.3: Updating bias field c K(x − y)ui (y)m I(y)T Σi (x)−1 vi dy i=1 . b(x) = c K(x − y)ui (y)m viT Σi (x)−1 vi dy i=1 (19) Step 2.4: Updating mixture weight K ∗ um i pi = c m. j =1 K ∗ uj

(20)

Step 2.5: Updating centroids

−1 ui (y)m K(x − y)b(x)2 (Σi (x)−1 )dxdy vi =

m −1 × ui (y) K(x − y)b(x)(Σi (x) I(y))dxdy . (21)

Fig. 1. Illustration of (first column) three synthetic image and their (second to forth columns) intermediate and (fifth column) final segmentation results.

Step 3: Checking the termination condition. If the distance between the newly obtained cluster centers and old ones is less than a user-specified small threshold ε, stop the iteration; otherwise, go to step 2. IV. EXPERIMENTAL RESULTS We compared the proposed FLGMM algorithm to state-ofthe-art segmentation algorithms in both synthetic and clinical brain MR images. We empirically set the parameters used in our algorithm as follows: the fuzzy factor m = 2, standard deviation of the kernel function τ = 4, and neighborhood radius of the kernel function ρ = 10. For all the experiments, we initialize the centroids with k-means clustering. The testing images used in our experiments are summarized in Table I. A. Segmentation of Synthetic Images The first experiment was performed in three synthetic images, which were displayed in the first column of Fig. 1. In the first image, the intensities of the star-shaped object and background have the same mean but different variances. The images in the middle and bottom rows were corrupted by intensity inhomogeneity. The intermediatesegmentation results obtained by running the proposed algorithm for different numbers of iterations were shown in the second to fourth columns, and the final results obtained after the convergence of our algorithm were shown in the fifth column. It is revealed from Fig. 1 that the result gradually improves during the iterative segmentation process. In the final segmentation results, objects and background

Fig. 2. Illustration of (top row) three 3T-weighted brain MR images, (middle row) the estimated bias field, and (bottom row) segmentation results.

can be clearly differentiated despite of the impact of low contrast and intensity inhomogeneity. B. Segmentation of Brain MR Images The second experiment was carried out in 3T- and 7Tweighted clinical brain MR images. Fig. 2 shows three 3Tweighted clinical brain MR images that were used in [5],


343

Fig. 3. Illustration of (left column) two 7T-weighted brain MR images with skull, (middle column) the estimated bias field, and (right column) segmentation results.

together with the estimated bias fields and segmentation results. It is clear from this figure that in spite of the quite obvious bias field and noise in these images, the proposed algorithm can estimate the bias field and achieves satisfactory segmentation results. In Fig. 3, the left column gives two 7T-weighted clinical brain MR images that were used in [6] and [24], respectively, the middle column displays the estimated bias fields, and the right column shows the segmentation results. Although both testing images contain obvious bias field, our algorithm still achieves good segmentation results. It should be noted that the testing image in the top row was not preprocessed by skull stripping. It reveals that the proposed algorithm is immune to the impact of nonbrain regions.

Fig. 4. Illustration of (a) three slices extracted from a simulated T1-weighted MR study, (b) their ground truth, and segmentation results obtained by using (c) the proposed, (d) AFCM, (e) BCFCM, (f) GKFCM, (g) SFKFCM, (h) Wells’, (i) Lemmput’s, (j) CLIC, and (k) MPFCM algorithms.

C. Quantitative Comparison In the third experiment, we quantitatively compared the proposed FLGMM algorithm to eight existing segmentation approaches, including two FCM-based algorithms (AFCM [8] and BCFCM [9]), two kernel FCM-based algorithms (GKFCM [14] and SFKFCM [15]), two EM-based algorithms proposed by Wells et al. [2] and Leemput et al. [3], and two fuzzy membership and local-information-based algorithms (CLIC [6] and MPFCM [12]). To make a fair comparison, all algorithms were initialized by using the k-means clustering. The segmentation accuracy was measured by the Jaccard similarity (JS), which is the ratio between intersection and union of the segmented volume S1 and ground truth volume S2 JS (S1 , S2 ) =

|S1 ∩ S2 | . |S1 ∪ S2 |

(22)

The value of JS ranges from 0 to 1, and a higher JS represents more accurate segmentation. First, we evaluated the aforementioned nine segmentation algorithms in T1-weighted simulated brain MR images selected from the brain web simulated brain database (BrainWeb) [25]. The segmentation results obtained by applying these algorithms to a sagittal, a coronal, and a transverse slice extracted from

Fig. 5. Average JS of (left) GM segmentation and (right) WM segmentation obtained by applying nine segmentation algorithms to simulated brain MR images with increasing levels of intensity inhomogeneity.

a MR image with 1 mm cubic voxels, 100% image intensity inhomogeneity and 3% noise, were displayed in Fig. 4. These algorithms were also applied to the segmentation of 20 MR images, in which the intensity inhomogeneity ranges from 20% to 100%. The segmentation accuracy was measured in terms of the average JS of GM and WM delineation, and was compared in Fig. 5. Both visual and quantitative comparisons show that the proposed FLGMM algorithm is more robust to the intensity inhomogeneity and can produce more accurate segmentation results. We also compared our algorithm to other algorithms in the brain MR image used in [26], which has both obvious intensity inhomogeneity and low contrast. The image and corresponding segmentation results were shown in Fig. 6. The average segmentation accuracy counted in these results was listed in Table II. This experiment demonstrates that the proposed FLGMM method can simultaneously overcome the impact of

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TABLE II COMPARISON OF THE JS VALUES OF THE SEGMENTATION RESULTS SHOWN IN FIG. 6

Fig. 6. Illustration of (a) brain MR images with obvious intensity inhomogeneity and low contrast, (b) its region of interest, (c) ground truth, and segmentation results obtained by using (d) proposed FLGMM algorithm, (e) AFCM algorithm, (f) BCFCM algorithm, (g) GKFCM algorithm, (h) SFKFCM algorithm, (i) Wells’ algorithm, (j) Leemput’s algorithm, (k) CLIC algorithm, and (l) MPFCM algorithm.

intensity inhomogeneity and low contrast, and outperforms other eight approaches in terms of segmentation accuracy. Next, we compared the proposed FLGMM algorithmand other eight segmentation algorithms in T1-weighted clinical adult brain MR images provided by BrainMaps [27]. Six images with noise of around 3% and intensity inhomogeneity ranging from 60% to 80%, together with their segmentation results, were shown in Fig. 7. The accuracy of all these segmentation results was compared in Fig. 8. It demonstrates that the proposed algorithm is more accurate than other eight segmentation algorithms in clinical MR images. D. Segmentation of 3-D MR Images The proposed segmentation algorithm can be easily extended to segment 3-D MR images with intensity inhomogeneity, noise, and low contrast. In this experiment, we applied our algorithm to 20 normal T1-weighted 3-D brain MR images selected from the Internet brain segmentation repository (IBSR) [28]. Fig. 9 shows the segmentation results of our algorithm in four studies and corresponding ground truth. We compared the accuracy of our algorithm to that of six segmentation algorithms provided by the IBSR, including the MAP, adaptive MAP, (BMAP), FCM, tree-structure k-mean, and maximum-likelihood classifier algorithms. The segmentation accuracy of those algorithms was

Fig. 7. Six clinical T1-weighted brain MR images (1st row), their segmentation results obtained by using AFCM (2nd row), BCFCM (3rd row), GKFCM (4th row), SFKFCM (5th row), Wells’ algorithm (6th row), Leemput’s algorithm (7th row), CLIC (8th row), MPFCM (9th row), the proposed algorithm (10th row), and the ground truth (11th row).


345

Fig. 8. JS values of (left) GM segmentation and (right) WM segmentation shown in Fig. 7.

Fig. 10. Tanimoto performance metric of (top) GM segmentation and (bottom) WM segmentation obtained by applying seven segmentation algorithms to 20 3-D IBSR brain MR images.

Fig. 9. Comparison between the (left column) ground truth and (right column) corresponding segmentation results obtained by applying the proposed FLGMM algorithm to 3-D IBSR brain MR image (first row) 7_8, (second row) 100_23, (third row) 8_4, and (forth row) 112_2.

measured by the Tanimoto performance metric and is depicted in Fig. 10. It demonstrates again that our algorithm outperforms other six methods in terms of segmentation accuracy. V. DISCUSSION The FLGMM algorithm provides an effective way to estimate the intensity inhomogeneity in brain MR images. The smoothness of the derived estimation of the bias field is intrinsically ensured by the convolutions with the truncated Gaussian kernel.

As a result, this model can estimate the bias field during iterations and no extra effort is needed for smoothing the bias field. With the spatial information, this model can reduce the impact of noise. Meanwhile, the membership function in our model depends on both the mean and variance of local Gaussian component, which are spatially varying. In low-contrast areas, various types of tissues may have similar means, but different variances. Therefore, our model can distinguish the tissues in low-contrast areas. The FLGMM algorithm is essentially different from the FGMM [18], [19] whose parameters and objective function are defined globally without considering the local spatial information. In our algorithm, we use GMM to describe local data and employ a truncated Gaussian kernel to ensure the smoothness of the estimated bias field and to constrain the voxels in the neighborhood. The FLGMM algorithm is also different from the GKFCM algorithms [13]–[15], which transform all cluster centroids and data samples into a high-dimensional feature space with an implicit mapping so that a class of robust non-Euclidean distance measures is introduced. Moreover, the GKFCM algorithms use the constant means and variances in a globally defined energy. Our algorithm improves the traditional FCM algorithm by replacing the original Euclidean distance with the probability density of each voxel, which is based on a newly defined local GMM energy. In our algorithm, the means and standard deviation of Gaussian kernel are updated iteratively and varying spatially. Comparing with our previous work MPFCM [12], both the proposed algorithm (FLGMM) and MPFCM utilized the basic idea from CLIC algorithm [6] to estimate the bias field. The

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TABLE III MEAN ± STANDARD DEVIATION OF THE TIME COST OF OUR ALGORITHM

Fig. 11. JS values of the segmentation results obtained by using different settings of Gaussian kernel parameters.

major differences between FLGMM and MPFCM include the definition of the distance measure and the construction of local information. First, the dissimilarity measure used in this paper (FLGMM) was defined based on local GMM probability, which means that we used the local GMM probability to measure the distance between the data points and clustering centroids. MPFCM just utilized the Euclidean distance as the distance measure. Consequently, the proposed algorithm in this paper used a second-order statistics (including the mean and variance) to measure the distance, whereas MPFCM just used a firstorder statistics. Second, FLGMM assumed that the local image data within each voxel’s neighborhood satisfied the GMM, and thus introduced the FLGMM to construct the local information. MPFCM introduced an adaptive method to compute the weights for the neighborhood of each pixel in the image. The FLGMM algorithm suffers from manually setting of two parameters: the standard deviation of the kernel function τ and neighborhood radius of the kernel function ρ. The Gaussian kernel influences the bias field estimation and the smoothness of segmentations. Fig. 11 shows the JS results of the segmentation with different parameter selections. The original image is obtained from BrainMaps [27]. The left figure shows the influence of the standard deviation, while the right figure shows the influence of the radius. The accuracy of segmentations increases with the increasing of τ and ρ. When τ > 5 or ρ > 10, the JS values of GM and WM increase slightly, while the time consumption would have a significant increase. It is very hard to find the best parameter setting for the segmentation accuracy. But the larger the radius, the smooth the segmentations. Considering the segmentation accuracy and the time consumption of the algorithm, we suggest that 4 ≤ τ ≤ 7 and 8 ≤ ρ ≤ 15. In our experiments, we set τ = 4 and ρ = 10 for general MR images. It should be noted that our algorithm is a bit time consuming, especially when applying to large 3-D image volumes. The high complexity of our algorithm is largely caused by its iterative nature and repeated bias field estimation. To evaluate its computational complexity, the proposed algorithm was tested in 40 2-D and 40 3-D images selected from the BrainWeb dataset [25] and IBSR [28]. The mean and standard deviation of the time cost of our algorithm in this experiment (Intel Dual-Core 2.93 GHz Processor, 2G memory, and MATLAB Version 7.1) were listed in Table III.

VI. CONCLUSION In this paper, we assume that the local image within the neighborhood of each voxel follows the GMM, and thus propose the FLGMM algorithm for brain MR image segmentation. This algorithm uses a truncated Gaussian kernel function to incorporate spatial constraints into local GMMs, and employs the fuzzy membership function to balance the contribution of each GMM to the segmentation process. Our results in both synthetic and clinical images show that the proposed algorithm can largely overcome the difficulties raised by noise, low contrast, and bias fields, and is capable of producing more accurate segmentation results than several state-of-the-art algorithms. However, similar to other FCM- and EM-based segmentation approaches, the FLGMM algorithm has a nonconvex objective function, and hence may be trapped by local optima. Our future work will focus on reducing its computational complexity and improving its robustness to initialization. ACKNOWLEDGMENT The authors would like to thanks the anonymous reviewers whose comments greatly improved the paper. REFERENCES [1] U. Vovk, F. Pernus, and B. Likar, “A review of methods for correction of intensity inhomogeneity in MRI,” IEEE Trans. Med. Imag., vol. 26, no. 3, pp. 405–421, Mar. 2007. [2] W. Wells, E. Grimson, R. Kikinis, and F. Jolesz, “Adaptive segmentation of MRI data,” IEEE Trans. Med. Imag., vol. 15, no. 4, pp. 429–442, Apr. 1996. [3] V. Leemput, K. Maes, D. Vandermeulen, and P. Suetens, “Automated model-based bias field correction of MR images of the brain,” IEEE Trans. Med. Imag., vol. 18, no. 10, pp. 885–896, Oct. 1999. [4] Y. Zhang, M. Brady, and S. Smith, “Segmentation of brain MR images through a hidden Markov random field model and the expectationmaximization algorithm,” IEEE Trans. Med. Imag., vol. 20, no. 1, pp. 45– 57, Jan. 2001. [5] C. Li, C. Gatenby, L. Wang, and J. Gore, “A robust parametric method for bias field estimation and segmentation of MR images,” in Proc. IEEE Conf. Comput. Vision Pattern Recog., 2009, pp. 218–223. [6] C. Li, C. Xu, A. Anderson, and J. Gore, “MRI tissue classification and bias field estimation based on coherent local intensity clustering: A unified energy minimization framework,” in Proc. 21st Int. Conf. Inf. Process. Med. Imag., Lecture Notes in Computer Science, 2009, vol. 5636, pp. 288– 299. [7] K. Sikka, N. Sinha, P. K. Singh, and A. K. Mishra, “A fully automated algorithm under modified FCM framework for improved brain MR image segmentation,” Magn. Reson. Imag., vol. 27, pp. 994–1004, Jul. 2009. [8] D. Pham and J. Prince, “Adaptive fuzzy segmentation of magnetic resonance images,” IEEE Trans. Med. Imag., vol. 18, no. 9, pp. 737–752, Sep. 1999.


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Zexuan Ji received the B.E. degree in computer science and technology from the Nanjing University of Science and Technology, Nanjing, China, in 2007, where he is currently working toward the Ph.D. degree at the School of Computer Science and Technology. His research interests include medical imaging, image processing, and pattern recognition.

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Yong Xia (S’05–M’08) photograph and biography not available at the time of publication.

Quansen Sun received the Ph.D. degree in pattern recognition and intelligence system from the Nanjing University of Science and Technology (NUST), Nanjing, China, in 2006. He is currently a Professor in the Department of Computer Science, NUST. He visited the Department of Computer Science and Engineering, The Chinese University of Hong Kong, in 2004 and 2005. His current research interests include pattern recognition, image processing, computer vision, and data fusion.

Qiang Chen (M’10) received the B.Sc. degree in computer science and Ph.D. degree in pattern recognition and intelligence system from the Nanjing University of Science and Technology, Nanjing, China, in 2002 and 2007, respectively. He is currently an Associate Professor with the School of Computer Science and Technology, Nanjing University of Science and Technology. He visited the Department of Radiology, Stanford University, Stanford, CA from 2011 to 2012. His main research interests include image segmentation, object tracking, image denoising, and image restoration.

Deshen Xia received the B.Sc. degree in radiology from the Nanjing Institute of Technology, Nanjing, China, in 1963, and the Ph.D. degree from the Faculty of Science, Rouen University, Rouen, France, in 1988. He is currently a Professor in the Faculty of Computer Science and Technology, Nanjing University of Science and Technology (NUST), Nanjing, China, where he is also the Director of Image Recognition Lab, an Honorary Professor in ESIC/ELEC, Rouen, and a Research member in Computer Graphics Lab, National Center of Scientific Research (CNRS), Paris, France. NUSTHe is the direction member of National Remote Sensing Federation, China, and the direction member of Micro-Computer Application Association of Province Jiangsu, China. His research interests include image processing, analysis, and recognition, remote sensing, medical imaging, and mathematics in imaging.

David Dagan Feng (S’88–M’88–SM’94–F’03) photograph and biography not available at the time of publication.