An Adaptive Fuzzy C-Means Algorithm for Improving MRI Segmentation

In this paper, we propose new fuzzy c-means method for improving the magnetic resonance imaging (MRI) segmentation. The proposed method called “possiblistic fuzzy c-means (PFCM)” which hybrids the fuzzy c-means (FCM) and possiblistic c-means (PCM) functions. It is realized by modifying the objective function of the conventional PCM algorithm with Gaussian exponent weights to produce memberships and possibilities simultaneously, along with the usual point prototypes or cluster centers for each cluster. The membership values can be interpreted as degrees of possibility of the points belonging to the classes, i.e., the compatibilities of the points with the class prototypes. For that, the proposed algorithm is capable to avoid various problems of existing fuzzy clustering methods that solve the defect of noise sensitivity and overcomes the coincident clusters problem of PCM. The efficiency of the proposed algorithm is demonstrated by extensive segmentation experiments by applying them to the challenging applications: gray matter/white matter segmentation in magnetic resonance image (MRI) datasets and by comparison with other state of the art algorithms. The experimental results show that the proposed method produces accurate and stable results.


Introduction
Clustering is one of the most popular classification methods and has found many applications in pattern classification and image segmentation [1][2][3][4][5][6].With increasing use of magnetic resonance imaging (MRI) for diagnosis, treatment planning and clinical studies, it has become almost necessary for radiological experts to make clinical diagnosis and treatment planning by using computers.Medical images tend to suffer much more noise than realistic images due to the nature of the acquisition devices.This of course poses great challenges to any image segmentation technique.In order to reduce noise, many devises increase the partial voluming, that is, they average acquisition on a thick slice.This leads to blurring the edges between the objects, which make decisions very hard for automatic tools [1].The different acquisition modalities, the different image manipulations and variability of organs all contribute to a large verity of medical images.It can be safely said that there is no single image segmentation method that suits all possible images.This can pose great problems for any segmentation method.Therefore, several types of image segmentation techni-ques [3-8] were found to achieve accurate segmentations.Among them, the fuzzy clustering methods are of considerable benefits for MRI brain image segmentation [9,10] because the uncertainty of MRI image is widely presented in data.Also the MRIs contain weak boundaries i.e. pixels inside the region and on the boundaries have similar intensity; this causes difficulty working with methods based on edge detection techniques [7,8].The fuzzy c-means clustering algorithms fall into two categories: fuzzy c-means (FCM) [9] and possibilistic c-means (PCM) [10].Many extensions of the FCM algorithm have been proposed to overcome above fuzzy clustering problem and reduce errors in the segmentation process [9][10][11][12][13].There are also other methods for enhancing the FCM performance.For example, to improve the segmentation performance, one can combine the pixel-wise classification with pre-processing (noise cleaning in the original image) [11,13] and post-processing (noise cleaning on the classified data).Xue et al. [13] proposed an algorithm where they firstly denoise images and then classify the pixels using the standard FCM method.These methods can reduce the noise to a certain extent, but still have some drawbacks such as increasing com-putational time, complexity and introducing unwanted smoothing [14,15].Liew et al. [16] proposed a spatial FCM clustering algorithm for clustering and segmenting the images by using both the feature space and spatial information.Another variant of FCM algorithm called the robust fuzzy c-means (RFCM) algorithm was proposed in [17].
Pham and Prince [18] modified the FCM objective function by introducing a spatial penalty for enabling the iterative algorithm to estimate spatially smooth membership functions.Ahmed et al. [8] introduced a neighborhood averaging additive term into the objective function of FCM.They named the algorithm bias corrected FCM (BCFCM).Liew and Yan [19] introduced a spatial constraint to a fuzzy cluster method where the inhomogeneity field was modeled by a B-spline surface.The spatial voxel connectivity was implemented by a dissimilarity index, which enforced the connectivity constraint only in the homogeneous areas.This way preserves significantly the tissue boundaries.Zanaty and Aljahdali [11] introduced a new local similarity measure by combining spatial and gray level distances.They used their method as an alternative pre-filtering to an enhanced fuzzy c-means algorithm (EnFCM) [20].Kang et al. [21] proposed a spatial homogeneity-based FCM (SHFCM).Wang et al. [22] incorporated both the local spatial context and the non-local information into the standard FCM cluster algorithm.They used a novel dissimilarity measure in place of the usual distance metric.In those methods, effect of noise can be overcome by incorporating possibility (typicality) function in addition to membership function.For that, the possibilistic c-means algorithm (PCM) was developed in [23].This technique combines FCM and logic with some modification in its membership function for removal of noise from the MRI brain images.It has been shown to be more robust to outliers than FCM.However, the robustness of PFCM comes at the expense of the stability of the algorithm [24].The PCM-based algorithms suffer from the coincident cluster problem that makes them too sensitive to initialization [24].Many efforts have been presented to improve the stability of possibilistic clustering [25][26][27].
Although the previous MRIs segmentation algorithms had suppressed the impact of noise and intensity inhomogeneity to some extents, these algorithms still produce misclassified small regions.The problems of over-segmentation and sensitivity to noise are still the challenge.FCM and PCM are also very sensitive to initialization and sometimes coincident clusters will occur.Moreover, coincident clusters may occur during fuzziness processes which can affect the final segmentation.
In this paper, we propose a new method called PFCM which combines the characteristics of both FCM and PCM algorithms by new weights for accurate MRIs segmentation.The proposed method avoids various problems of existing fuzzy clustering methods, solves the noise sensitivity defect of FCM and overcomes the coincident clusters problem of PCM.In order to reduce the noise effect during segmentation, the proposed algorithm combines the objective functions of conventional FCM algorithm and PCM algorithm.To overcome the problem of coincident clusters of PCM and also for combination of the objective FCM and PCM, a new weight function is proposed that is based on Gaussian membership.In this method the effect of noise is overcome by incorporating possibility (typicality) function in addition to membership function.Consideration of these constraints can greatly control the noise in the image as shown in our experiments.The efficiency of the proposed algorithm is demonstrated by extensive segmentation experiments using real MRIs and comprising with other state of the art algorithms.
The rest of this paper is organized as follows: The theoretical foundation of fuzzy c-means and possibilistic c-means is described in Section 2. In Section 3, the proposed PFCM algorithm is presented.Experimental and comparisons results are given in Section 4. Finally, Section 5 gives our conclusions.

FCM and PCM Algorithms
The FCM algorithm [10] is an iterative clustering method that produces optimal C partitions by minimizing the weighted within the group sum of squared error objective function FCM J : where is the data set in the p-dimensional vector space, n is the number of points, p is the number of data items, C is the number of clusters with    .

 
is the C centers or prototypes of the clusters, v i is the p-dimension center of the cluster i, and is the degree of membership of j x in the i th cluster; j x is the j th of p-dimensional measured data.The fuzzy partition matrix satisfies: fuzzy membership and determines the amount of fuzzi- The parameter m is a weighting exponent for each .
Although FCM and the modified FCM [18,20,22] are us  (6) where eful clustering methods, their memberships do not always correspond well to the degree of belonging of the data, and may be inaccurate in a noisy environment, because the real data unavoidably involves noise.To alleviate weakness of FCM, and to produce memberships that have a good explanation for the degree of belonging of the data.Wang et al. [22] relaxed the constrained condition (3) of the fuzzy C-partition to obtain a possibilistic type of membership function and propose PCM for unsupervised clustering.The component generated by the PCM corresponds to a dense region in the data set; each cluster is independent of the other clusters in the PCM strategy.The objective function of the PCM can be formulated as follows: is the scale parameter at the i th cluster, and is the possibilistic typicality value of training sample j x belonging to the cluster   , 1, i m  is a weighting factor called the possibilistic Zanaty and Sultan [11] proposed a parameter.method for automatic fuzzy algorithms by considering some spatial constraints on the objective function.The algorithm starts of subdivide the data a set of sing well-known fuzzy method [10] (see Equations ( 1)-( 4)).Assume, the data is divided into Our validity function is proposed to use the intracluster distance measure, which is simply the distance between a centre of cluster A and cluster centre B multiplied by the objective function of fuzzy.We can define the validity function as: where

  min B lusters are the maximum and minimu of c m values
A and B respectively.
 .This al rithm works iteratively the num increases automatically according to the decision of validity function in Equations ( 9) and ( 10), more discussion can be shown in [11].
Typical of other cluster approaches, the PCM also go ber of clusters depends on initialization.In PCM technique [25], the clusters do not have a lot of mobility, since each data point is classified as only one cluster at a time rather than all the clusters simultaneously.Therefore, a suitable initialization is required for the algorithms to converge to nearly global minimum.
with the following constraints:

C
A solution of the objective function can be obtained vi a an iterative process where the degrees of membership, typicality and the cluster centers are updated as follows: Open Access OJMI [26] improved this method by adding a new penalty to the objective function in order noise affects.The objective unction can be written as: (13) to control the with the following constraints: Rajendran and Dhanasekaran [24] define a clustering algorithm that combines the characteristics of both fuzzy and possibilistic c-means.Memberships and typicalities are important for the correct feature of data substructure in clustering problems.Thus, an objective function in this method that depends on both memberships and typicalities can be shown as: with the following constraints:   1 1, 1, , and , 0 A solution of the objective function can be obtained e degrees of membership, typicality and the cluster centers are updated as follows: via an iterative process where th The above equation influenced by all C cluster centers, ij influenced just by the i th cluster center c i .The possibilistic term distributes the t ij with respect to every n data points, but not by means of every C clusters.T membership can be described as relative determines the degree to which a data fit in to cluster in accordance with other clusters and is helpful in correctly la the data ized and the mized [26].(21) s indicate that membership u ij is while possibility t is hus, typicality, it beling a data point.Possibility can be observed as absolute typicality, it determines the degree to which a data point belongs to a cluster correctly, it can decrease the consequence of noise.Joining both membership and possibility can yield to good clustering result [28].

The Proposed PFCM Algorithm
The choice of an appropriate objective function is a key to the success of cluster analysis and to obtain better quality clustering results; hence, clustering optimization is based on the objective function [28].To identify a suitable objective function, one may start from the following set of requissrements: the distance between points assigned to a cluster should be minim distance between clusters should to be maxi To obtain an appropriate objective function, we take into consideration the following: • The distance between clusters and the data points allocated to them must be reduced.The desirability between data and clusters is modeled by the objective function.Hung et al. [29] provides a prototype-driven n exponential separation strength between used for this parameter by ev modified PCM technique which considerably improves the function of FCM because of a learning of parameter.The learning procedure of is dependent on a clusters and is updated at every iteration.
As for the common value ery data for iterations, we propose a new weight function ij w which is based on Gaussian membership of a point p. achieving every point of the data set has a weight in relation to every cluster.The usage of weights produces good classification particularly in the case of noisy data.The weight is calculated as follows: where w is the weight of the point j in relation to class i .This weight is used to modify the fuzzy and typical partition.To classify a data point, the cluster centroid h the data point, it is membership; and for estimating the centroids, the typicality is used for alleviating the undesirable effect of outliers.The objective function is comf two expressions: the first is t and it uses a fuzziness weighting exponent, the second is as to be closest to posed o he fuzzy function possibililstic function and it uses a typical weighting exponent; but the two coefficients in the objective function are only used as exhibitor of membership and typicality.
A new relation, lightly different, enabling a more rapid decrease in the function and increase in the membership and the typicality when they tend toward 1 and decrease this degree when they tend toward 0. This relation is to add weighting Gaussian exponent as exhibitor of distance in the two under objective functions.
To solve the noise sensitivity defect of FCM which has an influence on the estimation of centroids, and to overcome the coincident clusters problem of PCM, the hybridization of possibilistic c-means (PCM) and fuzzy c-means (FCM) is proposed that often avoids various problems.We will use the proposed weight ij w and the term   in the objective function for alleviating th e noise affect and to decrease the distances between clusters and centers and to avoid coincident clusters.Thus the objective function of the proposed method (PFCM) can be formulated as follows: where   and  is a user defined constants and the param is a ghting exponent for each fuzzy memb are tested in [30], hey proved that n between the data shape and m.For i r shape will fit better if m = 3 is use re sion can be found in [10] v can be obtained by using the following theorem [25]: denotes an image with n pixels to be partitioned into C classes (clusters), where i x represents feature data.The algorithm is an iterative optimization that minimizes the objective function defined by Equation ( 23) with the constraints Then ij u ij t and i v must satisfy the following equalities: , Proof: The minimization of constraint problem m J in Equation ( 23) under the given constraints ca solved of using the Lagrange multiplier method.We define a new objective function with the constraint condition of (Equation ( 23)) as follows: n be Taking the partial derivative of L m with respect t an 1 0 1 From Equation ( 27), we get:  and then setting them to equal to zero, we have: By substitution from Equation ( 29) into Equation ( 28), we get The process of finding the best clusters is continued to update the centres and the membership using Equations (30) and respectively.The algorithm for carrying out PFCM for segmentation of MRI brain images can now be stated from the following steps: 1) Select the number of clusters "C" and fuzziness factor "m" 2) Select initial class center prototypes  , randomly and ε ,a very small num 3) Find the value of w ij using Equation ( 22

Experimental Results
The experiments were performed on three different sets T1-weigthed and T2-weighted MRIs: in the first test, we used T1-weigthed and T2-weighted at various noise levels (0%, 3%, 7% que s (0% and 20%) as shown in Figure 1.T1-weigthed corrupted by 129 × 129 pixels [30] (see Figure 2) is used in t. t includes simulated volumetric MRI e third se sting of ten classes as shown in Figure 3.The advantages of using digital phantoms rather than real image data for soft segmentation methods include prior knowlge of the true tissue types and control over image pameters such as modality, slice thickness, noise, and intensity in homogeneities.Through our implementation, we set the following rameters: m = 2, ε = 0.0001 and 2   .The quality of the segmentation algorithm is of tation process.The comm as proposed in [31] is vital importance to the segmen parison score S for each algorith defined as follows: where A represents the set of pixels belonging to a class as found by a particular method and ref A represents the reference cluster pixels.

Noisy T1-Weigthed and T2-Weighted MRI
The proposed technique is applied to T1-weigthed and T2-weighted MRI [32] (as shown in Figures 1(a) and (b)) at various noise levels (0%, 3%, 5% and 9%) and RF levels (0% and 20%).To prove the efficiency of proposed algorithm, several noise levels are added to these data sets, while S (Equation ( 33)) is evaluated for each segment in T1-weigthed and T2-weighted.Figure 4 shows the segmentation output of T1-w and T2-weighted M I at noise lev evels (0% pose we have an image cont ining v segments, the accuracy scores can be computed from: eigthed R els (0%, 3%, 5% and 9%) and RF l and 20%).The average of segmentation accuracy scores (average) for each image is indicated in Table 1.Supa 1 .
Table 1 describes the average of the proposed method when applied to the test images.For example in Figure 4 when noise = 0% and RF = 0%, average is equal to 0.98% and 0.95% for T1-weigthed and T2-weighted MRI respectively.The obtained results show that the proposed algorithms are very robust to noise and i ensity homogeneities jdenbos lent gorithm has desired performh nt and inhomogeneities.According to Zi rage > 0.7 indicates excel [32] statement that ave agreement; the proposed al ance in cortical segmentation.
The best average is achieved for low noise and RF levels, for which values of average are higher than 0.94.According to Table 1, the proposed technique is stable at 88% at noise level 9% and RF 20%, this result is satisfactory for segmenting the weak boundary tissues.

T1-Weighted MRI Phantom
We used a high-resolution T1-weighted MRI (with slice thickness of 1mm, 6% noise and RF 20%) obtained from the simulated brain database of McGill University [32] (see Figure 2).In this test, beside evaluating the proposed method and the most recent fuzzy c-means such as: Pal et al. [26], Wang et al. [22], Rajendran and D ana-   Obviously, the proposed method acquires the best segmentation performance.The proposed method appears to be stable and achieve better performance than Pal et al. [26], Wang et al. [22], Rajendran and Dhanasekaran [24] and Zanaty and Aljahdali [11] by factor 12%, 8%, 7%, and 3% respectively.

Simulated MR Data
In this section, we experiment the proposed method and some existing methods such as: Pal et al. [26], Wang et al. [22], Rajendran and Dhanasekaran [24] and Zanaty and Aljahdali [11] when applied to the nine classes (slices# 51-59) from 3D simulated data (see Figure 3).Ta- ble 3 shows the corresponding accuracy scores (%) of the proposed and the existing methods for the nine classes.Obviously, the FCM gives the worst segmentation accuracy for all classes, while other methods give satisfactory results.On the other hand, the method of Pal et al. [26], Wang et al. [22], Rajendran and Dhanasekaran [24] and Zanaty and Aljahdali [11] acquire the good segmentation performance in case of classes 9, 3, 4, and 8 respectively.Overall, the proposed method is more stable and achieves much better performance than the others in all different classes even with misleading of true tissue of validity indexes.

Conclusion
This paper has presented a new approach called possiblsistic fuzzy c-means (PFCM) which combines FCM and PCM to overcome the weakness of both methods.The proposed algorithm is formulated by modifying the object e ed by other pixels and to suppress the iv function of PCM algorithm to allow the labeling of a pixel to be influenc vels (0%, 3%, 5%, 7%, and 9%) and RF Next test, we have experimented the pr ).We have noted that the proposed gment noisy MRI images (at using T1-weigthed MRI with 6% noi accuracy of each segment is evaluated.The al. [22] by factor 7%, Rajendran and by factor 8% and Zanaty and Aljahdali [11] by factor 3%. In addition, fo ulated 3D data (brain volume consists of ten slices), the average accuracy o he propos algorithm ha uated and comp Pal [26], Wang et al. [22], Rajendran and Dhanasekaran [24] and Zanaty [11].We hav d tha aver et ve an provemen , 6%, and 7% the proposed algorithm is demonstrated by comparing its performance against the existing Pal et al. [26], Wang et al. [22], Rajendran and Dhanasekaran [24] and Zanaty and Aljahdali [11].The proposed method achieves better performance than Pal et al. [26]

•
Coincident clusters may occur and must to be controlled.•Selecting the initialization sensitive parameters for decreasing noises affect.