Robust fuzzy clustering using nonsymmetric student׳s t finite mixture model for MR image segmentation

Abstract

Accurate tissue segmentation from magnetic resonance (MR) images is an essential step in clinical practice. In this paper, we introduce a robust fuzzy clustering scheme for finite mixture model fitting, which exploits the merits of the mixture of the nonsymmetric Student׳s t-distribution and mean template to reduce the sensitivity of the segmentation results with respect to noise. This approach utilizes a fuzzy objective function regularized by the Kullback–Leibler (KL) divergence term and sets the dissimilarity function as the negative log-likelihood of the nonsymmetric Student׳s t-distribution and mean template. The advantage of this fuzzy clustering scheme is that the spatial relationships among neighbouring pixels are taken into account with the help of the mean template so that the proposed method is more robust to noise than several other existing fuzzy c-means (FCM)-based algorithms. Another advantage is that the application of the nonsymmetric Student׳s t-distribution mixture model allows the proposed model to fit different shapes of observed data. Experiments using synthetic and real MR images show that the proposed model has considerably better segmentation accuracy and robustness against noise compared with several well-known finite mixture models.

Introduction

Accurate segmentation from magnetic resonance (MR) images according to relevant anatomical structure plays a vital role in computer-aided diagnosis [1], [2], [3], [4]. However, segmentation may be considerably difficult because MR images usually suffer from many artefacts, including noise, bias field, and partial volume effect. Therefore, searching for efficient MR image segmentation methods is still a popular research field.

In recent years, statistical approaches, especially fuzzy c-means (FCM)-type fuzzy clustering [5], [6] and neural network method [7] have been successful in addressing MR image segmentation because an FCM-type algorithm has additional flexibility, which allows the pixel to belong to multiple classes with varying degrees of membership. Further, they can preserve more information from the original image than the hard c-means clustering technique. Applying FCM had good segmentation results on images without noise. However, its accuracy in noisy images is not enough [8] mainly because each pixel is addressed as a separate unit; therefore, it did not consider the spatial information in the image space, which makes FCM very sensitive to noise and imaging artefacts. Because MR images are mostly noisy and have low contrast and inhomogeneity, to make FCM more robust to noise and outliers, several modified FCM have been investigated. Such as Liao and Lin [9] developed a spatially constrained fast FCM clustering algorithm to improve the computational efficiency. There are also methods that modified the objective function, which allows a pixel to be labelled by the influence of its neighbourhood labels [10]. This is a very effective way of improving the robustness against noise. There are some other methods, such as that of Zhang et al. [11], that extended the FCM by using Student׳s t-distribution as the distance function rather than the traditional Euclidean distance. Yu and Yang [12] proposed a generalized fuzzy clustering regularization (GFCR) model and then studied its theoretical properties. GFCR presents most variations of FCM with a constraint on membership function. Nefti et al. in [13] presented a new merging method based on clustering of fuzzy sets in the parameter space of membership function. Recently, fuzzy clustering with multiview data is becoming a hot topic in data mining and machine learning. In [14], Jiang et al. proposed a multiple weighted view fuzzy clustering called WV-Co-FCM algorithm which can automatically identify the importance of each view, weight each view and carry out a weighted approach to multiview fuzzy clustering.

Another widely used statistical method is the finite mixture model (FMM), which provides a mathematical-based approach to the statistical modelling of a wide variety of random phenomena. In these model-based techniques, the Gaussian mixture model (GMM) has been selected most widely as a particular case of FMM [15], [16]. However, the main drawback of GMM is its sensitivity to outliers. To solve this problem, Student׳s t mixture model (SMM) [17], [18] is proposed, where the probability distribution function of Student׳s t has longer tails and one more parameter compared to Gaussian distribution. However, the relationship among neighbouring pixels is not taken into account so that GMM and SMM are sensitive to changes in the pixel intensity. To solve this problem, the Markov random field (MRF) [19], [20], [21] and mean template [11], [22] techniques have been applied to impose spatial smoothness constraints on the image pixel. Yang et al. in [1] embed an MRF to the conventional level set energy function to segment glioma in brain MR images. Han et al. [23] integrated an adaptive MRF model and the coupled level-set information into the prior term, and presented a clinical data-driven approach for segmenting the bladder wall. In [24], Feng and Chen described a modified FCM where a prior spatial constraint (defined as refusable level in their paper) was introduced into FCM through MRF theory, so that the spatial information was encoded through mutual influences of neighbouring sites. Chatzis and Varvarigou in [25] proposed a novel fuzzy clustering type treatment of the hidden MRF model. Their method utilized a fuzzy objective function regularized by Kullback–Leibler (KL) divergence information, offered a significant enhancement for the image segmentation. However, among the algorithms based on the MRF-based technique, the log-likelihood function is too complex to use the EM algorithm directly, which causes the cost of computational complexity to rise sharply. Another phenomenon with the FMM approach has generally been identified, and the Gaussian distribution and Student׳s t-distribution are all symmetric shape and have a single peak. However, in real applications, it is noted that the intensity distributions of each label type of the data set do not exhibit a regular shape exactly. To overcome this problem, Vadaparthi et al. [5] presented a skew symmetric mixture model to improve the final segmentation accuracy. Wang et al. [26] defined a local Gaussian distribution fitting energy function to make it possible to distinguish regions by using similar intensity means but different variances. Nguyen et al. [27], [28] also proposed some nonsymmetric mixture models based on the nonsymmetric Gaussian or Student׳s t-distributions. This nonsymmetric mixture model helps to fit different shapes of observed data.

Recently, another family of finite mixture models based on FCM has been successfully applied to image segmentation [29], [30], [31], [32] such as Chatzis and Varvarigou [29], which extended the FCM by providing a fuzzy clustering-type treatment of the finite Student׳s t mixture models. In [30], the authors provided a general way of combining Gaussian distribution with partial membership and declared that FCM could be employed to provide a fuzzy clustering-type methodology for training any type of FMM. As an example, they presented the so-called KL-FCM methods where GMM are trained under the fuzzy clustering principle [31], [32]. In their method, KL divergence term was introduced into the objective function. It has proven to be quite effective for image segmentation. However, one of the main disadvantages of KL-FCM is that it does not consider the spatial relationship of the neighbouring pixels and lacks sufficient robustness to noise. On the other hand, among these methods, the probability distribution function of each label type of the data set is symmetric.

Considering the above reasons, we present a clustering approach that incorporates a nonsymmetric Student׳s t-distribution mixture model and mean template into FCM (FNSK) to segment the MR images. The novel framework utilizes a variant of the FCM scheme, regularized by KL information by considering the spatial information between neighbouring pixels simultaneously. The distance function is measured by multivariate nonsymmetric Student׳s t instead of the Euclidean distance in the traditional FCM. The advantage of the proposed framework is that it allows one to effectively incorporate the information regarding the considered real objects׳ nonsymmetric distributions into the fuzzy clustering training. Our method is different from [31]. With our Student׳s t-distribution has heavily tailed and more robust to outliers while the method in [31] is sensitive to outliers and may lead to excessive sensitivity to small numbers of data points [27]. In addition, in many real applications, the intensity distribution of each label type of the data set is not symmetric. Therefore, the results of the model, which are based on the symmetric Gaussian distribution [31] is slightly poor in these nonsymmetric situations. Although the proposed method follows the idea using the FCM and KL divergence term suggested in [32], the mean template used in this study makes the proposed model more robust to noise. Unlike approach in [32], the nonsymmetric Student׳s t-distribution used in our method might lead to the proposed FNSK that has the flexibility to fit different shapes of observed data. All of these factors make the new model an accurate technique for MR image segmentation. The proposed approach has been applied for segmenting simulation and clinical MR images, and the results are compared to other models based on FCM and FMM schemes.

The remainder of this paper is organized as follows. Section 2 briefly reviews the related works, which are closely related to this paper. In section 3, a full explanation of the proposed model is given. Section 4 will present the parameter estimation. In Section 5, the experimental results are provided to verify the accuracy of the proposed method. Finally, in the concluding section, we summarize this paper.