Segmentation of brain MR image using fuzzy local Gaussian mixture model with bias field correction

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1 IOSR Journal of VLSI and Signal Proessing (IOSR-JVSP) Volume 2, Issue 2 (Mar. Apr. 2013), PP e-issn: , p-issn No. : Segmentation of brain MR image using fuzzy loal Gaussian mixture model with bias field orretion *Pavan G S, **Nagendra Kumar M,*** Dr.S.Bhargavi IV Sem M.Teh (LSP), Assoiate Professor, Professor Dept. of ECE, SJCIT Chikballapur Abstrat: Aurate brain tissue segmentation from magneti resonane (MR) images is an essential step in quantitative brain image analysis. However, due to the existene of noise and intensity in-homogeneity in brain MR images, many segmentation algorithms suffer from limited auray. Here, we assume that the loal image data within eah voxel s neighborhood satisfy the Gaussian mixture model (GMM), and thus propose the fuzzy loal GMM (FLGMM) algorithm for automated brain MR image segmentation with bias field orretion. This algorithm estimates the segmentation result that maximizes the posterior probability by minimizing an objetive energy funtion, in whih a trunated Gaussian kernel funtion is used to impose the spatial onstraint and fuzzy memberships are employed to balane the ontribution of eah GMM. Our results show that the proposed algorithm an largely overome the diffiulties raised by noise, low ontrast, and bias field, and substantially improve the auray of brain MR image segmentation. I. Introdution SEGMENTATION of major brain tissues, inluding gray matter (GM), white matter (WM), and erebrospinal fluid, from magneti resonane (MR) images plays an important role in both linial pratie and neurosiene researh. However, due to the non-uniform magneti field or suseptibility effets, brain MR images may ontain a smoothly varying bias field, whih is also referred to as the intensity in-homogeneity or intensity non-uniformity [1]. As a result, the intensities of the same tissue vary aross voxel loations and may lead to segmentation errors. Therefore, bias field orretion and segmentation should be interleaved in an iterative proess so that they an benefit from eah other and yield better results. Many brain MR image segmentation approahes with bias field orretion have been proposed in the literature [2] [12]. Among them, those based on the expetation-maximization (EM) algorithm [2] [4] and fuzzy C-mean (FCM) lustering [5] [12] are the most popular ones. Pham and Prine. [8] proposed an adaptive FCM (AFCM) algorithm, whih inorporates a spatial penalty term into the objetive funtion to enable the estimated membership funtions to be spatially smoothed. Ahmed et al. [9] added a neighborhood averaging term to the objetive funtion, and thus developed the biasorreted FCM (BCFCM) algorithm. Liew and Yan. [10] used a B-spline surfae to model the bias field and inorporated the spatial ontinuity onstraints into fuzzy lustering algorithms. Li et al. [6] proposed an energyminimization approah to the oherent loal intensity lustering (CLIC), with the aim of ahieving tissue lassifiation and bias field orretion simultaneously. In our previous work [11], [12], we inorporated the global information into the CLIC model to enhane its robustness to the involved ontrol parameters, then we proposed a newly modified possibilisti FCMs lustering algorithm (MPFCM) for bias field estimation and segmentation of brain MR image. Although suh modifiation improves the segmentation auray, it also dramatially inreases the omputational omplexity. Various kernel tehniques have been used to improve the performane of lustering approahes. Chen and Zhang [13] replaed the original Eulidean distane with a kernel-indued distane and supplemented the objetive funtion with a spatial penalty term, whih models the spatial ontinuity ompensation. Yang and Tsai [14] proposed an adaptive Gaussian-kernel-based FCM (GKFCM) algorithm with the spatial bias orretion. Liao et al. [15] developed a spatially onstrained fast kernel FCM (SFKFCM) lustering algorithm to improve the omputational effiieny. However, the lustering performed in a kernel spae is generally very time onsuming. Alternatively, brain MR images an be segmented by using the Gaussian mixture model (GMM) [16], where the voxel intensities in eah 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 drawbak of the GMM-EM framework is its lak of taking the spatial information and unertainty of data into onsideration. As a result, it may produe less aurate segmentation results. To remedy this drawbak, Greenspan et al. [18] and Blekas et al. [19] inorporated the spatial onstraints into the GMM. Tran et al. proposed the fuzzy GMM (FGMM) model to address the unertainty of data and improve parameter estimation [20]. Zeng et al. [21] developed the type-2 FGMM (T2-FGMM) model for density modeling and lassifiation. 35 Page

2 However, to our knowledge, so far those fuzzy extensions of the GMM-based segmentation algorithm have not been able to overome the diffiulties aused by the intensity in-homogeneity. The onventional GMM implies the stohasti assumption that throughout the image, intensities in the same region are sampled independently from an idential Gaussian distribution. This assumption, however, is invalid for brain MR images due to the existene of the bias field. Based on the fat that the bias field varies very slowly and an be ignored within a small window, in this paper, we assume that the loal image data within the neighborhood of eah voxel follow the GMM, in whih the mean of eah Gaussian omponent is approximated as a tissue dependent onstant multiplied by the bias field estimated at this voxel. Thus, we propose the fuzzy loal GMM (FLGMM) algorithm for brain MR image segmentation. The objetive funtion of our algorithm is defined as the integration of the weighted GMM energy funtions over the entire image. In the objetive funtion, a trunated Gaussian kernel funtion is used to impose the spatial onstraint, and fuzzy memberships are employed to balane the ontribution of eah GMM to the segmentation proess. The proposed algorithm has been ompared to other state-of-the-art segmentation algorithms in both simulated and linial brain MR images. II. Related work A. Bias Field Formulation The bias field in a brain MR image an be modeled as a multipliative omponent of an observed image, as shown in the following: I = bj + n 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 orretion is to estimate and eliminate the bias field b from the observed image I. B. Fuzzy C-Mean Let I = {I(k) R d ; 1 k n} be a set of d-dimensional image features. The FCM [22] partitions this feature set into lusters based on minimizing the sum of distanes from eah feature to every luster anroids weighted by its orresponding membership. Let the membership funtion be U = {u i (k)} R n, where u i (k) [0, 1] is the degree of feature I(k) belonging to luster i and follows the onstraint i=1 ui k = 1. The quadrati objetive funtion to be minimized is J FCM = i=1 ui(k) m I k vi 2 dk.. 2 Where v i is the anroids of luster i, and m (1, ) is the fuzzy oeffiient. C. Gaussian Mixture Model The GMM is a weighted sum of Gaussian density distributions. With the GMM, the likelihood of the observed data I(k) is as follows: P(I(k) Θi)= i=1 p i N(I(k) μ i i).3 Where Θi = {p i, μ i,σ i } is the assembly of parameters, and p i is the mixing oeffiient of i th Gaussian omponent N (I(k) μ i,σi) and follows the onstraint i=1 p i = 1. The parameters involved in the GMM are denoted by Θ = {Θ i, i = 1... }, and are usually estimated through maximizing the likelihood of observed data via the EM algorithm, as shown in the following: n Θ = arg max Θ k=1 [ i=1 p i N I k μ i i ]...4 III. Arhitetural design for proposed system Aurate brain tissue segmentation of MR images has been one of the most important researh areas for several years. It is important to have an aurate segmentation of different brain tissue types for various appliations suh as radiotherapy planning, image-guided interventions, surgial planning by using EEG or fmri information and brain disease studies suh as Alzheimer and MS. For instane a study on Alzheimer disease arried out on department of radiology in university of California at San Franiso medial enter to ompare volume of brain tissues in patients with Alzheimer disease and ontrol subjets. They extrated white matter (WM), Grey matter (GM) and Cerebrospinal fluid (CSF) volumes using quantitative tissue segmentation tehniques. Their results showed signifiant derease in GM and an inrease in the ventriular and sulal CSF for Alzheimer patients whih shows the importane of having an aurate brain tissue segmentation for this study. 36 Page

3 As shown in the figure 1 first we input the MR images and then estimate the bias field using BCFCM and orret the bias field. Seond, use the bias orreted MR image as input to the FLGMM algorithm and obtain the segmented output use full for aforementioned brain disease studies suh as Alzheimer and MS. Figure 1: Arhitetural design for proposed system IV. Algorithms / Tehniques used In this setion we used two tehniques for Aurate brain tissue segmentation by interleaving bias field orretion with FLGMM algorithm. One is BCFCM for bias field estimation and orretion. Seond is FLGMM for segmentation of bias field orreted MR brain images. A. BCFCM Algorithm The BCFCM algorithm for orreting the bias field and segmenting the image into different lusters an be summarized in the following steps. N Step 1.Selet initial lass prototypes {v i } i=1. Set {β k } k=1 to equal and very small values. Step 2.Update the partition matrix using u ik = j =1 1 D ik + α N R γ i p D jk + α N R γ j Step 3.The prototypes of the lusters are obtained in the form of weighted averages of the patterns using v i = Step 4.Estimate the bias term using N p k=1 u ik y k β k + α N R N p 1 + α k=1 u ik β k = y k i=1 p i=1 u ik u ik p v i 1 p 1 y r N k y r β r Repeat Steps 2 4 till termination. The termination riterion is as follows: V new V old < ε Where. is the Eulidean norm, V is a vetor of luster enters, and ε is a small number that an be set by the user. B. FLGMM Algorithm FLGMM for segmentation of bias field orreted MR brain images. Step 1: Initialization. Initialize the number of lusters, standard deviation, and neighborhood radius of the trunated Gaussian kernel, luster entroids, and bias field at eah voxel. Step 2: Updating parameters. Step 2.1: Updating membership funtion 37 Page

4 u i y = i=1 d i (I(y)) d j (I(y)) 1 m 1 1. Step 2.2: Updating ovariane matrix Step 2.3: Updating bias field. = u i y m k x y ( I y b x v i (I y b(x)v i ) T )dy i u i y m k x y dy b(x)= i=1 k x y u i y m (I(y) T i(x) 1 v i )dy i=1 k x y u i y m (v T i i(x) 1 v i )dy Step 2.4: Updating mixture weight p i = k u i m i=1 k u j m Step 2.5: Updating entroids v i = u i y m k x y b x 2 ( i (x) 1 )dxdy 1 u i y m k x y b x. ( i x 1 I(y))dxdy Step 3: Cheking the termination ondition. If the distane between the newly obtained luster enters and old ones is less than a user-speified small threshold ε, stop the iteration; otherwise, go to step 2. V. Experimental results We empirially set the parameters used in our algorithm as follows: the fuzzy fator m= 2, standard deviation of the kernel funtion τ = 4 and neighborhood radius of the kernel funtion ρ = 10. For all the experiments, we initialize the entroids with k-means lustering. A. Segmentation of Syntheti Images The first experiment was performed in three syntheti images, whih were displayed in the first olumn of Figure 2. In the first image, the intensities of the star-shaped objet and bakground have the same mean but different varianes. The images in the middle and bottom rows were orrupted by intensity in-homogeneity. The intermediate segmentation results obtained by running the proposed algorithm for different numbers of iterations were shown in the seond to fourth olumns, and the final results obtained after the onvergene of our algorithm were shown in the fifth olumn. It is revealed from Figure 2 that the result gradually improves during the iterative segmentation proess. Figure 2. Illustration of (first olumn) three syntheti image and their (seond to forth olumns) intermediate and (fifth olumn) final segmentation results. B. Segmentation of Brain MR Images The seond experiment was arried out in 3T- and 7Tweighted linial brain MR images. Figure 3 shows three 3Tweighted linial brain MR images that were used in [5], together with the estimated bias fields and segmentation results. 38 Page

5 It is lear from this figure that in spite of the quite obvious bias field and noise in these images, the proposed algorithm an estimate the bias field and ahieves satisfatory segmentation results. Figure 3. Illustration of (top left olumn) 3T-weighted brain MR images (Axial View), (bottom left olumn) the partition matrix, (middle olumn) estimated bias field, and (bottom right olumn) bias field orreted image, (top right olumn) segmented MR image.. Figure 4. Illustration of (top left olumn) 7T-weighted brain MR images (sagittal view), (bottom left olumn) the partition matrix, (middle olumn) estimated bias field, and (bottom right olumn) bias field orreted image, (top right olumn) segmented MR image. Figure 5. Illustration of (top left olumn) 3T-weighted brain MR images (Coronal View), (bottom left olumn) the partition matrix, (middle olumn) estimated bias field, and (bottom right olumn) bias field orreted image, (top right olumn) segmented MR image. 39 Page

6 C. Quantitative Comparison In the third experiment, we quantitatively ompared the proposed FLGMM algorithm to eight existing segmentation approahes, inluding two FCM-based algorithms (AFCM [8] and BCFCM [9]), two kernel FCMbased 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 loal-information-based algorithms (CLIC [6] and MPFCM [12]). To make a fair omparison, all algorithms were initialized by using the k-means lustering. The segmentation auray was measured by the Jaard similarity (JS), whih is the ratio between intersetion and union of the segmented volume S1 and ground truth volume S2 JS (S 1, S 2 ) = s 1 s 2 s 1 Us 2 The value of JS ranges from 0 to 1, and a higher JS represents more aurate segmentation. Figure 4. Average JS of (left) GM segmentation and (right) WM segmentation obtained by applying nine segmentation algorithms to simulated brain MR images with inreasing levels of intensity in-homogeneity. VI. Conlusion In this paper, we assume that the loal image within the neighborhood of eah voxel follows the GMM, and thus propose the FLGMM algorithm for brain MR image segmentation. This algorithm uses a trunated Gaussian kernel funtion to inorporate spatial onstraints into loal GMMs, and employs the fuzzy membership funtion to balane the ontribution of eah GMM to the segmentation proess. Our results in both syntheti and linial images show that the proposed algorithm an largely overome the diffiulties raised by noise, low ontrast, and bias fields, and is apable of produing more aurate segmentation results than several state-ofthe-art algorithms. VII. Future work Our future work will fous on reduing its omputational omplexity and improving its robustness to initialization. Referenes [1] U. Vovk, F. Pernus, and B. Likar, A review of methods for orretion of intensity inhomogeneity in MRI, IEEE Trans.Med. Imag., vol. 26, no. 3, pp , Mar [2] W. Wells, E. Grimson, R. Kikinis, and F. Jolesz, Adaptive segmentation of MRI data, IEEE Trans. Med. Imag., vol. 15, no. 4, pp , Apr [3] V. Leemput, K. Maes, D. Vandermeulen, and P. Suetens, Automated model-based bias field orretion of MR images of the brain, IEEE Trans. Med. Imag., vol. 18, no. 10, pp , Ot [4] Y. Zhang, M. Brady, and S. Smith, Segmentation of brain MR images through a hidden Markov random field model and the expetationmaximization algorithm, IEEE Trans.Med. Imag., vol. 20, no. 1, pp , Jan [5] C. Li, C. Gatenby, L. Wang, and J. Gore, A robust parametri method for bias field estimation and segmentation of MR images, in Pro. IEEE Conf. Comput. Vision Pattern Reog., 2009, pp [6] C. Li, C. Xu, A. Anderson, and J. Gore, MRI tissue lassifiation and bias field estimation based on oherent loal intensity lustering: A unified energy minimization framework, in Pro. 21st Int. Conf. Inf. Proess. Med. Imag., Leture Notes in Computer Siene, 2009, vol. 5636, pp [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 , Jul [8] D. Pham and J. Prine, Adaptive fuzzy segmentation of magneti resonane images, IEEE Trans.Med. Imag., vol. 18, no. 9, pp , Sep [9] M. Ahmed, S. Yamany, N. Mohamed, A. Farag, and T.Moriarty, A modified fuzzy -means algorithm for bias field estimation and segmentation of MRI data, IEEE Trans. Med. Imag., vol. 21, no. 3, pp , Mar Page

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