Robust kernelized local information fuzzy C-means clustering for brain magnetic resonance image segmentation

Abstract

Brain tissue segmentation from magnetic resonance (MR) images is an importance task for clinical use. The segmentation process becomes more challenging in the presence of noise, grayscale inhomogeneity, and other image artifacts. In this paper, we propose a robust kernelized local information fuzzy C-means clustering algorithm (RKLIFCM). It incorporates local information into the segmentation process (both grayscale and spatial) for more homogeneous segmentation. In addition, the Gaussian radial basis kernel function is adopted as a distance metric to replace the standard Euclidean distance. The main advantages of the new algorithm are: efficient utilization of local grayscale and spatial information, robustness to noise, ability to preserve image details, free from any parameter initialization, and with high speed as it runs on image histogram. We compared the proposed algorithm with 7 soft clustering algorithms that run on both image histogram and image pixels to segment brain MR images. Experimental results demonstrate that the proposed RKLIFCM algorithm is able to overcome the influence of noise and achieve higher segmentation accuracy with low computational complexity.

Keywords

Brain tissue segmentation magnetic resonance images fuzzy C-means grayscale and spatial information Gaussian radial basis kernel image histogram

1 Introduction

Image segmentation is to divide an image into non-overlapping homogeneous objects. Since magnetic resonance imaging (MRI) is a powerful, noninvasive imaging technique to scan human organs, and has good contrast for soft tissues, it becomes one of the main imaging modalities to understand and diagnose human tissues. Basically, brain MR images are classified into white matter (WM), gray matter (GM), and cerebrospinal fluid (CSF). Segmentation of brain MR images is an indispensable step for brain quantification and studying brain anatomical changes over time [1]. In addition, it is a prerequisite step for studying tumor growth of low grade gliomas as the lesion diffusivity differs according to the surrounding tissues [2, 3]. In case of brain MR images, uncertainty factors exist that make the segmentation process challenging. Among such factors are: partial volume effect [4], grayscale non-uniformity [5], bias field [6], and Rician noise [7].

There are a plenty of segmentation algorithms for MR image in literature [1 , 8–10]. These algorithms can be classified into [10]: thresholding, clustering, region growing, edge detection, and model-based. Among these algorithms, clustering algorithms have attracted much attention. Clustering is an unsupervised learning scheme where similar data points are grouped into the same cluster and can be hard or soft. Soft clustering algorithms assign for every data point a membership value to all clusters. The most common soft clustering algorithm used for MR image segmentation is the fuzzy C-means (FCM) [11, 12] which is the main focus of this paper.

Standard FCM algorithm yields good results when noise and other image artifacts do not exist but this is not always guaranteed. To increase its robustness, local neighborhood information should be included in the clustering process. Several attempts have been made to improve the performance by including local spatial and grayscale information [13 –25]. However, how much contextual information should be used efficiently is non-trivial.

Ahmed et al. [13] proposed spatial FCM (FCM_S) algorithm by adding grayscale information from neighborhood to influence the labeling of a pixel from its neighbors. Chen and Zhang [15] improved the computation efficiency of FCM_S by proposing two variants to replace the neighborhood term with mean and median filtered images, respectively. For further speeding up, Szilagyi et al. [14] proposed enhanced FCM (EnFCM) which was applied to histogram of a new linearly-weighted sum image. The EnFCM depends only on grayscales of the image regardless of the image size and is therefore very fast. Cai et al. [16] introduced a fast generalized FCM (FGFCM) by constructing a new image using similarity measure that incorporated both spatial and grayscale information. Yang and Tsai [17] proposed Gaussian kernel-based FCM algorithm to generalize FCM_S by introducing an adaptive parameter to control the local information of every cluster. Recently, Elazab et al. [25] introduced a new parameter to regularize the effect of local information to be adaptive for every pixel.

Krinidis and Chatiz [19] proposed a robust fuzzy local information C-means (FLICM) clustering to overcome the problem of setting parameters in FCM_S and its variants. FLICM uses new fuzzy factor that combines both spatial and grayscale information of neighboring pixels. Gong et al. [22] developed kernel weighted FLICM (KWFLICM) algorithm to be more robust to severe noise using tradeoff weighted fuzzy factor and kernel distance measure. Ji et al. [21] used image patches to replace pixels in the clustering process and constructed a weighting scheme to enable the pixels in each image patch to have anisotropic weights. Recently, the same author in [23] employed robust spatially constrained FCM algorithm by introducing a factor for the spatial direction based on the posterior probabilities and prior probabilities. Li et al. [18] presented an algorithm for brain tissue classification and bias estimation using a coherent local intensity clustering. Later, they introduced multiplicative intrinsic component optimization (MICO) [24] for high level bias field estimation and improved the accuracy of tissue segmentation.

Although the aforementioned algorithms have good results, their performance is subject to specific conditions and may have one or more of the following drawbacks: prior adjusting of crucial parameters [13 –16], limited segmentation accuracy in images corrupted with high levels of noise [11 , 24], lack of robustness to outliers [13 , 21], high computational cost [13 , 21–23], and losing small image details e.g. CSF, [14 , 23]. In this paper, new contextual information is incorporated into the FCM algorithm for better overcoming these problems.

The rest of this paper is organized as follows. Algorithms related to FCM are briefly presented in Section 2. The proposed algorithm is then explained in details in Section 3. Experiments on synthetic and clinical MR images are presented in Section 4. Sections 5 and 6 are devoted to discussion and conclusion, respectively.

2 Related work

Unlike hard clustering, the conventional FCM algorithm [11, 12] assigns a membership value for every element to all clusters in the image space. Let a set of data patterns X = {x₁, x₂, … x_N} ⊂ R^m in m-dimensional space and set of clusters centers v = {v₁, v₂, … v_c}, there is a membership value for every element x_i in jth cluster. The quadratic objective function of the FCM algorithm to be optimized is [12]: $J_{FCM} = \sum_{i = 1}^{N} \sum_{j = 1}^{c} u_{ij}^{m} d^{2} (x_{i}, v_{j})$ (1) where c is positive integer 2 < c ⪡ N, m is a weighting exponent to the degree of fuzziness (m > 1), d² (x_i, v_j) is the distance metric between element x_i and center v_j, and u_ij represents the membership value of every element to each cluster. The membership function u_ij is subject to the following conditions: $\forall i \in [1, N], j \in [1, c] \sum_{j = 1}^{c} u_{ij} = 1, u_{ij} \in [0, 1]$ (2)

Lagrange multiplier is used to minimize the objective function in Equation (1) under constraints given in Equation (2). The membership function and cluster centers are calculated iteratively in an alternating process until the objective function reaches a local minimum using the following formulas: $u_{ij} = \frac{1}{\sum_{k = 1}^{c} {(\frac{d_{ij}}{d_{ik}})}^{\frac{2}{m - 1}}}$ (3) $v_{j} = \frac{\sum_{i = 1}^{N} u_{ij}^{m} x_{i}}{\sum_{i = 1}^{N} u_{ij}^{m}}$ (4)

The FCM algorithm passes through the following steps:

Step (1): Initialize threshold ɛ, m = 2, loop counter t = 0, c and U⁽⁰⁾.

Step (2): Calculate cluster centers $v_{j}^{(t)}$ using U^(t) as in (4).

Step (3): Calculate the membership U^(t+1) using (3).

Step (4): If max |U^(t+1) - U^(t)|| < ɛ or t > 100 then stop, otherwise, update t = t + 1 and go to step 2.

Obviously, the objective function of FCM algorithm does not include any contextual information. Therefore, the immunity to noise and image artifacts is limited. To overcome this problem, Ahmed et al. [13] modified the objective function of FCM by adding a regularization term that included the grayscale information of neighborhood. This term influenced the pixel being classified towards a more homogenous clustering. The new objective function was denoted as FCM_S and defined as follows: $J_{FCM_S} = \sum_{i = 1}^{N} \sum_{j = 1}^{c} u_{ij}^{m} d_{ij}^{2} + \frac{α}{N_{R}} \sum_{i = 1}^{N} \sum_{j = 1}^{c} u_{ij}^{m} (\sum_{x_{r} \in N_{i}} d_{rj}^{2})$ (5) where α is the parameter used to control the effect of the local neighborhoods with 0 < α ≤ 1, N_i is the set of pixels around i and N_R is the cardinality of N_i. Subsequently, the new membership functions and cluster centers are: $u_{ij} = \frac{{(d_{ij}^{2} + \frac{α}{N_{r}} \sum_{x_{r} \in N_{i}} d_{rj}^{2})}^{- \frac{1}{m - 1}}}{\sum_{k = 1}^{c} {(d_{ik}^{2} + \frac{α}{N_{r}} \sum_{x_{r} \in N_{i}} d_{rk}^{2})}^{- \frac{1}{m - 1}}}$ (6) $v_{j} = \frac{\sum_{i = 1}^{N} u_{ij}^{m} (x_{i} + \frac{α}{N_{r}} \sum_{x_{r} \in N_{i}} x_{r})}{(1 + α) \sum_{i = 1}^{N} u_{ij}^{m}} .$ (7)

The third summation in Equation (5) requires additional loop which makes FCM_S very computationally expensive. To tackle this shortcoming and to be more robust, Chen and Zhang [15] replaced $\frac{1}{N_{R}} \sum_{x_{r} \in N_{i}} d_{rj}^{2}$ with ${\bar{d}}_{ij}^{2}$ that is the Euclidean distance between a predefined filtered image $\bar{x}$ that can be calculated once in advance using either median or average filters and the cluster centers with the following objective function: $J_{{FCM}_{S_{1}, 2}} = \sum_{i = 1}^{N} \sum_{j = 1}^{c} u_{ij}^{m} d_{ij}^{2} + α \sum_{i = 1}^{N} \sum_{j = 1}^{c} u_{ij}^{m} {\bar{d}}_{ij}^{2} .$ (8)

The parameter α given in Equations. (5) and (8) controls the effect of local information and has to be set manually which is hard to be initialized without prior knowledge about the noise type and level. Yang and Tsai [17] tried to adjust α adaptively for every cluster using a new parameter η_j that is calculated in every iteration through Gaussian kernel function. Although this parameter improved the result, it is computationally expensive since it has to be updated in each iteration. To handle this drawback, Elazab et al. [25] proposed an adaptively regularized kernel-based FCM framework (ARKFCM) with new parameter to control the local information of every pixel. The ARKFCM framework has three forms for the local average grayscale being replaced by the grayscale of: the average filter (ARKFCM₁), median filter (ARKFCM₂), and devised weighted image (ARKFCM_w).

To overcome the problem of prior parameter adjustment, Krinidis and Chatiz [19] introduced a fuzzy local information C-means (FLICM) that combined both spatial and grayscale information of the neighboring pixels. The fuzzy factor G_ij is added to the standard FCM in Equation (1). Although it is slow, FLICM is robust to noise and artifacts. Later, Gong et al. [22] proposed kernel weighted FLICM (KWFLICM) algorithm with a trade-off weighted fuzzy factor to replace G_ij with $\bar{G}$ _ij that adaptively controls the local neighbor relationship but it significantly increases the computational complexity. Generally, FLICM and KWFLICM are robust to noise and yield good accuracy in presence of severe noise while losing small image details, e.g. CSF. However, Szilagyi [26] pointed out a theoretical mistake in FLICM and KWFLICM that, the iterative optimization nature of FLICM and KWFLICM did not minimize their objective functions, instead, they iterated until the partition matrices converged.

Szilagyi et al. [14] took another direction and introduced EnFCM algorithm where the clustering process is done on the histogram of the grayscales. Since the number of grayscale values are much smaller than the total number of pixels, the EnFCM algorithm is extremely fast. The EnFCM algorithm is guided by a linearly-weighted image ξ that is calculated once in advance from original image to incorporate local information as follows: $ξ_{i} = \frac{1}{1 + α} (x_{i} + \frac{α}{N_{R}} \sum_{x_{r} \in N_{i}} x_{r})$ (9) where ξ_i is the grayscale of pixel i in the image ξ, N_i represents the neighbors of pixel i and N_R is their cardinality. The parameter α is set manually to control the effect of local information as in FCM_S. The objective function of EnFCM algorithm [14] is: $J_{EnFCM} = \sum_{i = 1}^{q} \sum_{j = 1}^{c} γ_{i} u_{ij}^{m} {(ξ_{i} - v_{j})}^{2}$ (10) where q is number of grayscales and γ_i is the number pixels having grayscale equal to i such that $\sum_{i = 1}^{q} γ_{i} = N$ . The membership function and cluster centers that minimize Equation (10) with constraints given in Equation (2) can be calculated using: $u_{ij =} \frac{{(ξ_{i} - v_{j})}^{(\frac{- 2}{m - 1})}}{\sum_{k = 1}^{c} {(ξ_{i} - v_{k})}^{(\frac{- 2}{m - 1})}}$ (11) $v_{j} = \frac{\sum_{i = 1}^{N} γ_{i} u_{ij}^{m} ξ_{i}}{\sum_{i = 1}^{N} u_{ij}^{m}}$ (12)

Although the computational complexity of EnFCM algorithm is very low, the performance is heavily affected by the parameters α that has to be adjusted manually. Moreover, it is still quite sensitive to noise and outliers since local information is not properly encoded.

Cai et al. [16] took advantage of EnFCM and introduced FGFCM with a new factor S_ij that incorporates both local spatial and grayscale information to replace the parameter α: $S_{ij} = {\begin{matrix} - \frac{max (| p_{i} - p_{j} |, | q_{i} - q_{j} |)}{λ_{s}} - \frac{{∥ x_{i} - x_{j} ∥}^{2}}{λ g \times σ_{i}^{2}}, \\ \exp & i \neq i \\ 0, & i \neq j \end{matrix}$ (13) where the pixel i is the center of the local window, pixels j are the set of its neighbors, and (p_i, q_i) is the spatial coordinates of the pixel i. The parameters λ_s and λ_g represent the scale factors of the spread of local spatial information and local grayscale similarity, respectively, while $σ_{i}^{2}$ is defined as [16]: $σ_{i} = \sqrt{\frac{\sum_{j \in N_{i}} {‖ x_{i} - x_{j} ‖}^{2}}{N_{R}}}$ (14)

The FGFCM embeds S_ij into calculation of new weighted image as an alternate to Equation (9) as follows: $ξ_{i} = \frac{\sum_{j \in N_{i}} S_{ij} x_{j}}{\sum_{j \in N_{i}} x_{j}}$ (15)

The subsequent estimation of the membership function and cluster centers are the same as Equation (11) and Equation (12), respectively. Although the resulting segmentation is better than EnFCM algorithm, the overall performance is completely dependent on λ_s and λ_g values. Specifically, the initialization of λ_g which controls the grayscale similarity, can be robust to noise if it is large, and if it is small enough, the segmented image can maintain its sharpness and details [19]. Since knowledge about image noise is not known in advance, initializing λ_s and λ_g will be very difficult and has to be set experimentally via trial-and error.

In this paper, we propose a robust kernelized local information fuzzy C-means clustering algorithm (RKLIFCM) with an effective method to incorporate both spatial and grayscale information. The new formulation comes with no parameter initialization (except number of clusters), and runs on the image histogram. In addition, we employed Gaussian radial basis function (GRBF) to replace the Euclidean distance metric.

3 Proposed algorithm

3.1 Motivation

The weighted image used in EnFCM algorithm uses a parameter α to control the grayscale effect of the pixel in the local window with a fixed value which is not appropriate since noise level differs from one window to another. In addition, EnFCM does not include any local spatial information of the pixel neighbors which means the value of α is independent of the window size. On the other hand, FGFCM algorithm uses spatial and grayscale information but the spatial information is not properly encoded. For example, when using 3 × 3 the spatial information effect will be the same for all pixels within the local window. Moreover, the parameters λ_s and λ_g have a great impact on the calculation of the local information and consequently the segmentation results. In addition, the EnFCM and FGFCM algorithms still use the Euclidean distance that is non-robust to noise and outliers. Motivated by the particular strengths of EnFCM, FGFCM algorithms and the individual strengths of FCM_S algorithm and its variants, we introduce a new weighted image that properly incorporates both grayscale and spatial local information to suppress the effect of noise. Details are shown below.

3.2 The proposed locally weighted image

To be adaptive to noise level and to overcome the drawbacks of EnFCM and FGFCM algorithms, we introduce a new factor w_ij that utilize the local spatial and grayscale information efficiently without any prior adjustment of special parameters. Given an image I ={ x₁, x₂, x₃, …, x_N } with N number of pixels, the local window centered on pixel i is extracted for the subsequent processing. First, we calculate the standard deviation σ_i of the grayscales around the central pixel using: $σ_{i} = {[\frac{\sum_{j \in N_{i}, j \neq i} (x_{j} - {\bar{x}}_{i})^{2}}{(N_{j} - 1)}]}^{1 / 2}$ (16) where ${\bar{x}}_{i}$ is the average grayscale of neighboring pixels j and N_j is their cardinality. Next, we apply the standard deviation values σ_i to the following exponential function to measure the grayscale information denoted as G_ij using: $G_{ij} = exp [- (\frac{σ_{ij} - {\bar{σ}}_{i}}{σ_{ij}})]$ (17) where ${\bar{σ}}_{i}$ is the average standard deviation of the local window centered on pixel i. The G_ij can reflect the damping extent of the local window as it changes automatically with different grayscale values of pixels. In Equation (17), if the standard deviation of the current pixel σ_ij is closer to the average standard deviation ${\bar{σ}}_{ij}$ the smaller the G_ij and vice versa.

To make the effect of the neighboring pixels change flexibly according to the distance from the central pixel, we use the following exponential function to measure the spatial information denoted as S_ij as follows: $S_{ij} = exp [- (d_{ij})]$ (18) where d_ij is the Euclidean distance between the central pixel i and its neighbors j falling in the same window. The value of S_ij is higher for those pixels close to the center of the window and lower otherwise. The weights associated to the pixels in the local window is normalized product of G_ij and S_ij: $η_{ij} = G_{ij} \times S_{ij}$ (19) $w_{ij} = \frac{η_{ij}}{\sum_{j \in N_{i}} η_{ij}}$ (20)

Finally, the new weighted image $\bar{ξ}$ is calculated using: ${\bar{ξ}}_{i} = \sum_{j \in N_{i}} w_{ij} \times x_{ij}$ (21) where ${\bar{ξ}}_{i}$ is the grayscale of the pixel i in the image $\bar{ξ}$ , x_ij represents the grayscales of the local window centered on pixel i, and N_i is the set of its neighbors.

Unlike EnFCM and FGFCM algorithms, our generated weighted image does not depend on predefined crucial parameters that can affect the performance to result in more robust and accurate segmentation. Fig. 1 shows the calculation steps of the weighted image $\bar{ξ}$ and the effect of damping extent using the G_ij and S_ij, respectively.

3.3 Non-Euclidean distance using kernel function

It is known that the kernel functions are able to project data into higher dimensional space where the data could be more easily separated [27]. To employ the kernel function, a so-called kernel trick is employed to transform linear algorithm to nonlinear one by dot product operation [28]. Using the kernel trick, the Euclidean distance term ${‖ {\bar{ξ}}_{i} - v_{j} ‖}^{2}$ can be replaced with ${‖ ϕ ({\bar{ξ}}_{i}) - (v_{j}) ‖}^{2}$ that can be rewritten as: ${‖ ϕ ({\bar{ξ}}_{i}) - (v_{j}) ‖}^{2} = 2 (1 - K ({\bar{ξ}}_{i}, v_{j}))$ (22) where K is a GRBF kernel [20]. $K (x_{i}, v_{j}) = exp (- \frac{{‖ {\bar{ξ}}_{i} - v_{j} ‖}^{2}}{2 σ^{2}})$ (23) where σ is the kernel width. In fact, the choice of the kernel width σ has a great impact on the performance of GBRF as if σ is large, the exponential effect will be almost linear and if it is small, the cluster boundaries will be sensitive to outliers [29]. Similar to [14], we calculate σ based on the distance variances among all pixels of the new weighted image. $σ = {[\frac{\sum_{i = 1}^{N} {(d_{i} - \bar{d})}^{2}}{N - 1}]}^{\frac{1}{2}}$ (24) where $d_{i} = ‖ {\bar{ξ}}_{i} - \bar{ξ} ‖$ is the distance from the grayscale of pixel i to the grayscale average $\bar{ξ}$ of all pixels in the new weighted image and $\bar{d}$ is the average of all distances d_i.

3.4 The proposed RKLIFCM

The RKLIFCM algorithm runs on the image histogram as EnFCM and FGFCM algorithms but with the new weighted image ${\bar{ξ}}_{i}$ and employs the GBRF kernel function. The objective function of the proposed algorithm is: $J_{RKLIFCM} = 2 [\sum_{i = 1}^{l} \sum_{j = 1}^{c} γ_{i} u_{ij}^{m} (1 - K ({\bar{ξ}}_{i}, v_{j}))]$ (25)

Under the conditions specified in Equation (2), the minimization of J_RKLIFCM (u, v) can be calculated through an alternate optimization procedure using Lagrange multiplier with the following membership function and cluster centers: $u_{ij} = \frac{{(1 - K ({\bar{ξ}}_{i}, v_{j}))}^{- \frac{1}{m - 1}}}{\sum_{k = 1}^{c} {(1 - K ({\bar{ξ}}_{i}, v_{j}))}^{- \frac{1}{m - 1}}}$ (26) $v_{j} = \frac{\sum_{i = 1}^{l} {γ_{i} u_{ij}^{m} K ({\bar{ξ}}_{i}, v_{j}) \bar{ξ}}_{i}}{\sum_{i = 1}^{l} γ_{i} u_{ij}^{m} K ({\bar{ξ}}_{i}, v_{j})}$ (27)

The main steps for the RKLIFCM algorithm are summarized below:

Initialize threshold ɛ = 0.01, m = 2, loop counter t = 0, v and u⁽⁰⁾.

Calculate the new weighted image $\bar{ξ}$ using Equations from (16) to (21).

Calculate cluster centers $v_{j}^{(t)}$ using u^(t) as in Equation (27).

Calculate the membership function u^(t+1) with Equation (26).

Check the termination condition: if max ||u^(t+1) - u^(t)|| < ɛ or t > 100 then stop, otherwise, update t = t + 1 and go to step (3).

4 Experiments

Here, we present the experiments on both real and synthetic MR images corrupted with different types of noises. For validation, the proposed RKLIFCM algorithm is compared with 7 different soft clustering algorithms from literature. We include 3 algorithms that run on grayscale histograms namely, the fast version of FCM [12], EnFCM [14], and FGFCM [16]. The other 4 algorithms run on the image grayscales namely, FLICM [19], KWFLICM [22], MICO [24], and ARKFCM₂ as representative of ARKFCM framework that was reported with the best performance [25]. All algorithms are implemented using Matlab software, unless otherwise stated, the experiments are carried out using 3 × 3 window, maximum number of iterations t = 100 and ɛ = 0.01. The segmentation accuracy is evaluated using the Jaccard Similarity (JS) metric. Given S₁ be the segmented volume and S₂ be the corresponding volume of interest, the JS is calculated as follows [30]: $JS (S_{1}, S_{2}) = \frac{| S_{1} \cap S_{2} |}{| S_{1} \cup S_{2} |} .$ (28)

4.1 Experiments on clinical 3T brain MR images

We first test the RKLIFCM algorithm on clinical brain MR images scanned using Philips Medical Systems Intera 3T obtained from brain development project (http://brain-development.org/). This experiment includes coronal slice (number 160) with 256 × 150 pixels. The segmentation results of all algorithms are shown in Fig. 2 from black to white, respectively, background, CSF, GM, and WM.

4.2 Experiments on synthetic brain MR images

We carried out the experiments using Simulated Brain Database (SBD) [31] which includes a set of realistic MR volumes produced by an MR imaging simulator with ground truths for CSF, GM, and WM available.

The first experiment was carried out on a T1-weighted axial slice (number 100) with 217 × 181 pixels, sagittal slice (number 100) with 181 × 217 pixels, and coronal slice (number 100) with 181 × 181 pixels all corrupted with 7% noise and 20% grayscale non-uniformity to be segmented into WM, GM, and CSF. Figures 3, 4 and 5 show the segmentation results of the axial, sagittal and coronal slices, respectively, while Table 1 summarizes the JSs of the WM, GM, CSF, and their averages.

The second experiment was carried out on T1-weighted axial slice (number 91) with 217 × 181 pixels, sagittal slice (number 100) with 181 × 217 pixels, and coronal slice (number 100) with 181 × 181 pixels with different type of noise. We used Gaussian white noise with mean = 0 and variance = 0.01 to severely corrupt those slices to test the performance of all algorithms. The segmentations results together with JSs of all algorithms are shown in Figs. 6, 7 and 8 and summarized in Table 2, respectively.

The third experiment is conducted to check the robustness to Rician noise. We used a different T1-weighted axial slice (number 91) with 217 × 181 pixels where the image details (CSF areas) appear to be more challenging. The segmentation results and the JSs of all algorithms are given in Fig. 9 and Table 3, respectively.

5 Discussion

Brain MR image segmentation is an important and fundamental step for both research and clinics. Due to potential presence of partial volume effect, noise, and other image artifacts, brain MR image segmentation becomes very challenging. FCM clustering is a reasonable choice by considering local information of neighborhood to tackle those problems but the accuracy and computational cost need to be enhanced.

In this paper, we propose RKLIFCM, with new way to incorporate both local grayscale and spatial information of the local neighborhood into a newly formulated image $\bar{ξ}$ . Our algorithm runs on the grayscale histogram of $\bar{ξ}$ rather than the grayscale values, which makes the clustering process very fast regardless of image size. In addition, a GRBF kernel is utilized to replace Euclidean distance for better partitioning and to be less sensitive to outliers. We experimented RKLIFCM algorithm on both clinical and synthetic brain MR images with different noises and compared our results with 7 recent soft clustering algorithms to be discussed below.

5.1 Segmentation accuracy

Visually, the test on the 3T clinical brain MR images (Fig. 2) shows that the proposed algorithm can have good segmentation on clinical images without skull peeling and has better segmentation performance than other algorithms especially in preserving details of CSF.

The segmentation of the axial slice with 7% noise and 20% grayscale non-uniformity, RKLIFCM algorithm shows better performance than the other 7 algorithm in preserving the CSF areas and keeping smooth edges (Fig. 3). The average JS of the RKLIFCM is 0.893 which is higher than the other 7 algorithms with range 0.1–5.8% (Table 1). The accuracy becomes more distinctive in preserving details in the sagittal slice corrupted with the same noise (Fig. 4) with average JS = 0.83 which is higher than the other 7 algorithms with range 0.5–9.4% (Table 1). Similarly in the coronal slice experiment (Fig. 5), RKLIFCM gives JS = 0.859 which is still better than the other 7 algorithms with rang 0.8% –3.5% (Table 1). Interestingly for this experiment, ARKFCM₂ algorithm also performs well and has very competitive results with JSs between 0.1 to 0.8% below the RKLIFCM algorithm. However, the borders resulting from RKLIFCM algorithm are smoother than ARKFCM₂ algorithm (Figs. 3i, 4i, and 5i). Furtherer more, the RKLIFCM algorithm is faster than ARKFCM₂, that will be discussed later.

In the second experiment on the SBD images, the Gaussian white noise is used to severely corrupt the slice which is more challenging with small CSF areas. The average JS of RKLIFCM for the axial slice shown in Fig. 6 is 0.864 which are better than the other 7 algorithms with range 1.2–14.2% (Table 2). For the sagittal slice shown in Fig. 7, the average JS of RKLIFCM is 0.812 which is better than the 7 other algorithms with range 0.3–16.6% (Table 2). For the coronal slice shown in Fig. 8, the average JS of RKLIFCM is 0.86 which is higher than the 7 other algorithms with range 1–13.2% (Table 2). Unlike the first experiment, ARKFCM₂ yields lower performance and suffers from noise as the median filtering can be less effective to Gaussian noise (Figs. 6i, 7i, and 8i). On the other hand, RKLIFCM algorithms shows higher immunity to the Gaussian noise with good results.

For the third experiment of the axial slice corrupted with 10% Rician noise shown in Fig. 9, the average JS of RKLIFCM is 0.886 which is better than the 7 other algorithms with range 0.1–6% (Table 3). The higher JSs of the RKLIFCM algorithm conclude that it can achieve better balance in preserving image details in presence of different noise than the 7 other algorithms. Also in this experiment, the ARKFCM₂ algorithm has high accuracy, 0.1 % less than RKLIFCM algorithm, but with higher running time and less smooth borders (Fig. 9i).

5.2 Computational complexity

The computational cost of the objective function of fast FCM algorithm runs on the grayscale histogram of the image instead of grayscales which means clustering process does not depend on the image size. Basically, grayscales of the MR images can be sufficiently encoded using 8-bit resolution that means, at most, there is 256 different grayscales (l) which are much smaller than the image size (N). Therefore, the time complexity of the FCM based algorithms that run on the grayscale histogram are O (lci) where l is the maximum grayscale in the image, c is the number of clusters and i is the number of iterations the objective function needs to converge. However, not all the fast FCM algorithms have the same running time because number of iterations differs as well as the method used to include of the local information.

The fast FCM algorithm with no contextual information is extremely fast with running time less than 0.01 second. However, the segmentation accuracy is very poor and needs to be enhanced (Figs. 3c, 4c, 5c, 6c, 7c, 8c and 9c and Tables 1, 2 and 3). The EnFCM algorithm includes local grayscale information using a linearly-weighted sum of neighbors in the local window as given in Equation (9). The calculation of the weighted image can be done once in advance which has less influence on the running time (0.04–0.12 seconds). Yet, the segmentation accuracy is still not satisfactory (Figs. 3d, 4d, 5d, 6d, 7d, 8d, and 9d and Tables 1, 2 and 3). For the FGFCM algorithm, although the segmentation accuracy is improved (Figs. 3e, 4e, 5e, 6e, 7e, 8e, and 9e and Tables 1, 2 and 3), the running time increases significantly compared with EnFCM algorithm as the major time is spent in calculating the S_ij measure from the neighbors (0.32–0.46 seconds). In addition, the FGFCM algorithm requires more iterations for the objective function to converge. For the proposed RKLIFCM algorithm, the calculation of w_ij increases the running time slightly while the objective function requires less number of iterations to converge (0.16–0.27 seconds) but with higher segmentation accuracy (Figs. 3j, 4j, 5j, 6j, 7j, 8j, and 9j and Tables 1, 2 and 3).

On the other hand, the other 4 algorithms that run on grayscales of the image (ARKFCM₂, FLICM, KWFLICM and MICO) have higher running time. The ARKFCM₂ considers the local grayscale information only and this can be calculated once in advance. Therefore, the running time can be reduced but it is still image working on the image grayscales which makes ARKFCM₂ slower than the histogram-based versions of FCM algorithm. The FLICM algorithm is even slower as it needs an additional loop on the neighborhood to calculate G_ij in every iteration that increases the running time (2.59–4.57 seconds). As an extension of FLICM, KWFLICM uses $\bar{G}$ _ij that requires two additional loops on the neighborhood at each iteration, so its running time is drastically increased to be the highest (122–166 seconds). For MICO algorithm, the convexity of its energy function has low running in the presence of less noise but tends to increase because of many iterations are required in the presence of high level noise (0.51–0.67 seconds).

To evaluate the computational complexity, we run each algorithm 20 times with randomly initialized membership function and cluster centers then record the running times and number of iterations. In this comparison, we did not include the fast FCM algorithm as it does not include local information, the FLICM and KWFLICM algorithms as they consume so much time. The average running time of these algorithms is summarized in Table 4 and Fig. 10 while average number of iterations is shown in Fig. 11. We can notice that EnFCM algorithm has the lowest running time because of the simple method to compose the local grayscale information but not the best accuracy. On the other hand, the proposed RKLIFCM algorithm consumes low running time, lowest number of iterations and with the best segmentation accuracy.

5.3 Relationship with other algorithms

The RKLIFCM algorithm has relationship with the other algorithms due to the flexible configuration of w_ij. It can be easily noticed that RKLIFCM algorithm goes directly to the fast FCM algorithm when the local weights of w_ij in Equation (20) are all zeroes except the central pixel. Similarly, RKLIFCM algorithm can be a special case of EnFCM algorithm if local weights of w_ij are all equal and multiplied by particular value i.e. w_ij is replaced with $\frac{α}{N_{R}}$ . Also, RKLIFCM algorithm will be typical to the FGFCM algorithm if w_ij = S_ij with central weight of the local window equal to zero. On the other hand, RKLIFCM algorithm can be quite similar to ARKFCM₁ and FCM_S₁, and ARKFCM₂ and FCM_S₂ algorithms if the center is, respectively, replaced with the average and median value of the grayscales.

5.4 Effect of neighborhood size

The local neighborhood window size has a great influence on the performance of the RKLIFCM algorithm in particular, the running time and less influence on the segmentation accuracy. We have tested the RKLIFCM algorithm on window size 5 × 5 pixels and we found that the running time increased about from 1.5 to 2 times with less effect on the segmentation accuracy. Increasing window size to 7 × 7 pixels or higher will significantly increase the running time as much work has to be done for the calculation of w_ij of the local window. Thus, it is recommended to use a local window of size 3 × 3 pixels for calculating the w_ij.

5.5 Limitation

It is found that the RKLIFCM algorithm can be sensitive to severe noise, specifically in case of MR images that have many small details (e.g. the CSF areas in the sagittal and the coronal slices given in Figs. 7i, and 8i). This may be fixed by taking directional information into consideration in our future work.

6 Conclusion

We proposed a new version of the FCM clustering algorithm for MR brain tissue segmentation, RKLIFCM. It adaptively incorporates both grayscale and spatial information to improve the segmentation accuracy. In addition, it runs on the grayscales histogram which speed up the calculations from O (Nci) to $O (lc \hat{ı})$ with reduced number of iterations $\hat{ı}$ . We also adopted the GRBF kernel metric to enhance the robustness against noise. We validated the proposed RKLIFCM algorithm on synthetic brain MR images corrupted with different types of noise and tested it to segment clinical 3T weighted brain MR images. For the synthetic brain MR images, the proposed algorithm achieved higher JSs (Tables 1, 2 and 3) than the other 7 recent soft clustering algorithms, and could preserve small image details (Figs. 3, 4, 5, 6, 7, 8 and 9). On the other hand, the segmentation results of the clinical 3T-weighted brain MR images using the proposed algorithm showed good balance between smooth borders and preserving image details without being affected by the skull (Fig. 2). The proposed algorithm is capable of keeping a balance between high segmentation accuracy and low computational cost (Table 4 and Figs. 10 and 11). It may be a potential tool for brain MR image segmentation.

Footnotes

Acknowledgments

This work has been supported by: National Program on Key Basic Research Project (Nos. 2013CB733800, 2012CB733803), Key Joint Program of National Natural Science Foundation and Guangdong Province (No. U1201257), and Guangdong Innovative Research Team Program (No. 201001D0104648280).

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