Strong fuzzy c-means in medical image data analysis

This paper presents a robust fuzzy c-means (FCM) for an automatic effective segmentation of breast and brain magnetic resonance images (MRI). This paper obtains novel objective functions for proposed robust fuzzy c-means by replacing original Euclidean distance with properties of kernel function on feature space and using Tsallis entropy. By minimizing the proposed effective objective functions, this paper gets membership partition matrices and equations for successive prototypes. In order to reduce the computational complexity and running time, center initialization algorithm is introduced for initializing the initial cluster center. The initial experimental works have done on synthetic image and benchmark dataset to investigate the effectiveness of proposed, and then the proposed method has been implemented to differentiate the different region of real breast and brain magnetic resonance images. In order to identify the validity of proposed fuzzy c-means methods, segmentation accuracy is computed by using silhouette method. The experimental results show that the proposed method is more capable in segmentation of medical images than existed methods.     

Effective fuzzy c-means clustering algorithms for data clustering problems

Clustering is a well known technique in identifying intrinsic structures and find out useful information from large amount of data. One of the most extensively used clustering techniques is the fuzzy c-means algorithm. However, computational task becomes a problem in standard objective function of fuzzy c-means due to large amount of data, measurement uncertainty in data objects. Further, the fuzzy c-means suffer to set the optimal parameters for the clustering method. Hence the goal of this paper is to produce an alternative generalization of FCM clustering techniques in order to deal with the more complicated data; called quadratic entropy based fuzzy c-means. This paper is dealing with the effective quadratic entropy fuzzy c-means using the combination of regularization function, quadratic terms, mean distance functions, and kernel distance functions. It gives a complete framework of quadratic entropy approaching for constructing effective quadratic entropy based fuzzy clustering algorithms. This paper establishes an effective way of estimating memberships and updating centers by minimizing the proposed objective functions. In order to reduce the number iterations of proposed techniques this article proposes a new algorithm to initialize the cluster centers.In order to obtain the cluster validity and choosing the number of clusters in using proposed techniques, we use silhouette method. First time, this paper segments the synthetic control chart time series directly using our proposed methods for examining the performance of methods and it shows that the proposed clustering techniques have advantages over the existing standard FCM and very recent ClusterM-k-NN in segmenting synthetic control chart time series.     

Fuzziness parameter selection in fuzzy c-means: The perspective of cluster validation

Fuzzy c-means （FCM） algorithm is an important clustering method in pattern recognition, while the fuzziness parameter, m, in FCM algorithm is a key parameter that can significantly affect the result of clustering. Cluster validity index （CVI） is a kind of criterion function to validate the clustering results, thereby determining the optimal cluster number of a data set. From the perspective of cluster validation, we propose a novel method to select the optimal value of m in FCM, and four well-known CVIs, namely XB, VK, VT, and SC, for fuzzy clustering are used. In this method, the optimal value of m is determined when CVIs reach their minimum values. Experimental results on four synthetic data sets and four real data sets have demonstrated that the range of m is [2, 3.5] and the optimal interval is [2.5, 3].     

基于遗传FCM算法的文本聚类

本文提出基于遗传FCM算法的文本聚类方法,首先采用LSI方法对文本特征进行降维,然后通过聚类有效性分析得到文本的类别数,最后再采用遗传FCM算法对文本进行聚类,这种方法较好的克服了FCM算法收敛于局部最优的缺陷,很好的解决了FCM算法对初值敏感的问题。实验表明提出的方法具有较好的聚类性能。     

Image Segmentation Based on Adaptive Cluster Prototype Estimation

An image segmentation algorithm based on adaptive fuzzy c-means (FCM) clustering is presented in this paper. In the conventional FCM clustering algorithm, cluster assignment is based solely on the distribution of pixel attributes in the feature space, and does not take into consideration the spatial distribution of pixels in an image. By introducing a novel dissimilarity index in the modified FCM objective function, the new adaptive fuzzy clustering algorithm is capable of utilizing local contextual information to impose local spatial continuity, thus exploiting the high inter-pixel correlation inherent in most real-world images. The incorporation of local spatial continuity allows the suppression of noise and helps to resolve classification ambiguity. To account for smooth intensity variation within each homogenous region in an image, a multiplicative field is introduced to each of the fixed FCM cluster prototype. The multiplicative field effectively makes the fixed cluster prototype adaptive to slow smooth within-cluster intensity variation, and allows homogenous regions with slow smooth intensity variation to be segmented as a whole. Experimental results with synthetic and real color images have shown the effectiveness of the proposed algorithm.     

A Possibilistic Fuzzy c-Means Clustering Algorithm

In 1997, we proposed the fuzzy-possibilistic c-means (FPCM) model and algorithm that generated both membership and typicality values when clustering unlabeled data. FPCM constrains the typicality values so that the sum over all data points of typicalities to a cluster is one. The row sum constraint produces unrealistic typicality values for large data sets. In this paper, we propose a new model called possibilistic-fuzzy c-means (PFCM) model. PFCM produces memberships and possibilities simultaneously, along with the usual point prototypes or cluster centers for each cluster. PFCM is a hybridization of possibilistic c-means (PCM) and fuzzy c-means (FCM) that often avoids various problems of PCM, FCM and FPCM. PFCM solves the noise sensitivity defect of FCM, overcomes the coincident clusters problem of PCM and eliminates the row sum constraints of FPCM. We derive the first-order necessary conditions for extrema of the PFCM objective function, and use them as the basis for a standard alternating optimization approach to finding local minima of the PFCM objective functional. Several numerical examples are given that compare FCM and PCM to PFCM. Our examples show that PFCM compares favorably to both of the previous models. Since PFCM prototypes are less sensitive to outliers and can avoid coincident clusters, PFCM is a strong candidate for fuzzy rule-based system identification.     

一种基于决策粗糙集的模糊C均值聚类数的确定方法

Fuzzy C-Means(FCM)是模糊聚类中聚类效果较好且应用较为广泛的聚类算法,但是其对初始聚类数的敏感性导致如何选择一个较好的C值 变得十分重要。因此,确定FCM的聚类数是使用FCM进行聚类分析时的一个至关重要的步骤。通过扩展决策粗糙集模型进行聚类的有效性分析,并进一步确定FCM的聚类数,从而避免了使用FCM时不好的初始化所带来的影响。文中提出了一种基于扩展粗糙集模型的模糊C均值聚类数的确定方法,并通过图像分割实验来验证聚类的效果。实验通过比对不同聚类数下分类结果的代价获得了一个较好的分割结果,并将结果与Z.Yu等人于2015年提出的蚁群模糊C均值混合算法(AFHA)以及提高的AFHA算法(IAFHA)进行对比,结果表明所提方法的聚类结果较好,图像分割效果较明显,Bezdek分割系数比AFHA和IAFHA算法的更高,且在Xie-Beni系数上也有较大优势。     

Alpha-cut implemented fuzzy clustering algorithms and switching regressions.

In the fuzzy c-means (FCM) clustering algorithm, almost none of the data points have a membership value of 1. Moreover, noise and outliers may cause difficulties in obtaining appropriate clustering results from the FCM algorithm. The embedding of FCM into switching regressions, called the fuzzy c-regressions (FCRs), still has the same drawbacks as FCM. In this paper, we propose the alpha-cut implemented fuzzy clustering algorithms, referred to as FCMalpha, which allow the data points being able to completely belong to one cluster. The proposed FCMalpha algorithms can form a cluster core for each cluster, where data points inside a cluster core will have a membership value of 1 so that it can resolve the drawbacks of FCM. On the other hand, the fuzziness index m plays different roles for FCM and FCMalpha. We find that the clustering results obtained by FCMalpha are more robust to noise and outliers than FCM when a larger m is used. Moreover, the cluster cores generated by FCMalpha are workable for various data shape clusters, so that FCMalpha is very suitable for embedding into switching regressions. The embedding of FCMalpha into switching regressions is called FCRalpha. The proposed FCRalpha provides better results than FCR for environments with noise or outliers. Numerical examples show the robustness and the superiority of our proposed methods.     

Novel segmentation algorithm in segmenting medical images

The aim of this paper is to develop an effective fuzzy c-means (FCM) technique for segmentation of Magnetic Resonance Images (MRI) which is seriously affected by intensity inhomogeneities that are created by radio-frequency coils. The weighted bias field information is employed in this work to deal the intensity inhomogeneities during the segmentation of MRI. In order to segment the general shaped MRI dataset which is corrupted by intensity inhomogeneities and other artifacts, the effective objective function of fuzzy c-means is constructed by replacing the Euclidean distance with kernel-induced distance. In this paper, the initial cluster centers are assigned using the proposed center initialization algorithm for executing the effective FCM iteratively. To assess the performance of proposed method in comparison with other existed methods, experiments are performed on synthetic image, real breast and brain MRIs. The clustering results are validated using Silhouette accuracy index. The experimental results demonstrate that our proposed method is a promising technique for effective segmentation of medical images.     

Fuzzy curve-tracing algorithm

This paper presents a fuzzy clustering algorithm for the extraction of a smooth curve from unordered noisy data. In this method, the input data are first clustered into different regions using the fuzzy c-means algorithm and each region is represented by its cluster center. Neighboring cluster centers are linked to produce a graph according to the average class membership values. Loops in the graph are removed to form a curve according to spatial relations of the cluster centers. The input samples are then reclustered using the fuzzy c-means (FCM) algorithm, with the constraint that the curve must be smooth. The method has been tested with both open and closed curves with good results.     

新的模糊聚类有效性指标

在经典的模糊C均值（FCM）算法中,聚类数需要预先给出,否则算法无法工作,这在一定程度上限制了FCM算法的应用范围。针对FCM算法中聚类数需要预先设定问题,提出了一种新的模糊聚类有效性指标。首先,通过运行FCM算法得到隶属度矩阵;然后,通过隶属度矩阵计算类内紧密性和类间重叠性;最后,利用类内的紧密性和类间的重叠性定义了一个新的聚类有效性指标。该指标克服了FCM算法中类数需要预先设定的缺点,利用该指标可以发现最符合数据自然分布的类的数目。通过对人工数据集和实际数据集的测试表明,对于模糊因子取1.8,2.0和2.2三个不同的常用值,均能发现最优聚类数。     

A fuzzy clustering algorithm based on evolutionary programming

In this paper, a fuzzy clustering method based on evolutionary programming (EPFCM) is proposed. The algorithm benefits from the global search strategy of evolutionary programming, to improve fuzzy c-means algorithm (FCM). The cluster validity can be measured by some cluster validity indices. To increase the convergence speed of the algorithm, we exploit the modified algorithm to change the number of cluster centers dynamically. Experiments demonstrate EPFCM can find the proper number of clusters, and the result of clustering does not depend critically on the choice of the initial cluster centers. The probability of trapping into the local optima will be very lower than FCM.     

A Novel Similarity-Based Fuzzy Clustering Algorithm by Integrating PCM and Mountain Method

The fuzzy c-means (FCM) and possibilistic c-means (PCM) algorithms have been utilized in a wide variety of fields and applications. Although many methods are derived from the FCM and PCM for clustering various types of spatial data, relational clustering has received much less attention. Most fuzzy clustering methods can only process the spatial data (e.g., in Euclidean space) instead of the nonspatial data (e.g., where the Pearson's correlation coefficient is used as similarity measure). In this paper, we propose a novel clustering method, similarity-based PCM (SPCM), which is fitted for clustering nonspatial data without requesting users to specify the cluster number. The main idea behind the SPCM is to extend the PCM for similarity-based clustering applications by integration with the mountain method. The SPCM has the merit that it can automatically generate clustering results without requesting users to specify the cluster number. Through performance evaluation on real and synthetic data sets, the SPCM method is shown to perform excellently for similarity-based clustering in clustering quality, even in a noisy environment with outliers. This complements the deficiency of other fuzzy clustering methods when applied to similarity-based clustering applications.     

Simulated annealing using a reversible jump Markov chain Monte Carlo algorithm for fuzzy clustering

In this paper, an approach for automatically clustering a data set into a number of fuzzy partitions with a simulated annealing using a reversible jump Markov chain Monte Carlo algorithm is proposed. This is in contrast to the widely used fuzzy clustering scheme, the fuzzy c-means (FCM) algorithm, which requires the a priori knowledge of the number of clusters. The said approach performs the clustering by optimizing a cluster validity index, the Xie-Beni index. It makes use of the homogeneous reversible jump Markov chain Monte Carlo (RJMCMC) kernel as the proposal so that the algorithm is able to jump between different dimensions, i.e., number of clusters, until the correct value is obtained. Different moves, like birth, death, split, merge, and update, are used for sampling a candidate state given the current state. The effectiveness of the proposed technique in optimizing the Xie-Beni index and thereby determining the appropriate clustering is demonstrated for both artificial and real-life data sets. In a part of the investigation, the utility of the fuzzy clustering scheme for classifying pixels in an IRS satellite image of Kolkata is studied. A technique for reducing the computation efforts in the case of satellite image data is incorporated.     

基于重叠度增量的模糊聚类有效性函数

提出用重叠度来刻画模糊类间的距离,在此基础上针对模糊划分总重叠度有随类数增加而单调递增的趋势,提出基于重叠度增量的聚类有效性函数。该算法由重叠度增量最大值来确定最佳聚类数,不但克服了传统有效性函数的单调问题,而且计算简单。基于模糊C-均值聚类算法（FCM）,应用多组测试数据对其进行性能分析,并与当前广泛应用且具代表性的有效性函数进行深入比较。仿真结果表明,该函数的有效性和优越性。     

Double indices-induced FCM clustering and its integration with fuzzy subspace clustering

As one of the most popular algorithms for cluster analysis, fuzzy c-means (FCM) and its variants have been widely studied. In this paper, a novel generalized version called double indices-induced FCM (DI-FCM) is developed from another perspective. DI-FCM introduces a power exponent r into the constraints of the objective function such that the fuzziness index m is generalized and a new criterion of selecting an appropriate fuzziness index m is defined. Furthermore, it can be explained from the viewpoint of entropy concept that the power exponent r facilitates the introduction of entropy-based constraints into fuzzy clustering algorithms. As an attractive and judicious application, DI-FCM is integrated with a fuzzy subspace clustering (FSC) algorithm so that a new fuzzy subspace clustering algorithm called double indices-induced fuzzy subspace clustering (DI-FSC) algorithm is proposed for high-dimensional data. DI-FSC replaces the commonly used Euclidean distance with the feature-weighted distance, resulting in having two fuzzy matrices in the objective function. A convergence proof of DI-FSC is also established by applying Zangwill’s convergence theorem. Several experiments on both artificial data and real data were conducted and the experimental results show the effectiveness of the proposed algorithm.     

Hadoop based uncertain possibilistic kernelized c-means algorithms for image segmentation and a comparative analysis

Over the years data clustering algorithms have been used for image segmentation. Due to the presence of uncertainty in real life datasets, several uncertainty based data clustering algorithms have been developed. The c-means clustering algorithms form one such family of algorithms. Starting with the fuzzy c-means (FCM) a subfamily of this family comprises of rough c-means (RCM), intuitionistic fuzzy c-means (IFCM) and their hybrids like rough fuzzy c-means (RFCM) and rough intuitionistic fuzzy c-means (RIFCM). In the basic subfamily of this family of algorithms, the Euclidean distance was being used to measure the similarity of data. However, the sub family of algorithms obtained replacing the Euclidean distance by kernel based similarities produced better results. Especially, these algorithms were useful in handling viably cluster data points which are linearly inseparable in original input space. During this period it was inferred by Krishnapuram and Keller that the membership constraints in some rudimentary uncertainty based clustering techniques like fuzzy c-means imparts them a probabilistic nature, hence they suggested its possibilistic version. In fact all the other member algorithms from basic subfamily have been extended to incorporate this new notion. Currently, the use of image data is growing vigorously and constantly, accounting to huge figures leading to big data. Moreover, since image segmentation happens to be one of the most time consuming processes, industries are in the need of algorithms which can solve this problem at a rapid pace and with high accuracy. In this paper, we propose to combine the notions of kernel and possibilistic approach together in a distributed environment provided by Apache™ Hadoop. We integrate this combined notion with map-reduce paradigm of Hadoop and put forth three novel algorithms; Hadoop based possibilistic kernelized rough c-means (HPKRCM), Hadoop based possibilistic kernelized rough fuzzy c-means (HPKRFCM) and Hadoop based possibilistic kernelized rough intuitionistic fuzzy c-means (HPKRIFCM) and study their efficiency in image segmentation. We compare their running times and analyze their efficiencies with the corresponding algorithms from the other three sub families on four different types of images, three different kernels and six different efficiency measures; the Davis Bouldin index (DB), Dunn index (D), alpha index (α), rho index (ρ), alpha star index (α*) and gamma index (γ). Our analysis shows that the hyper-tangent kernel with Hadoop based possibilistic kernelized rough intuitionistic fuzzy c-means is the best one for image segmentation among all these clustering algorithms. Also, the times taken to render segmented images by the proposed algorithms are drastically low in comparison to the other algorithms. The implementations of the algorithms have been carried out in Java and for the proposed algorithms we have used Hadoop framework installed on CentOS. For statistical plotting we have used matplotlib (python library).     

An interval number distance- and ranking-based method for remotely sensed image fuzzy clustering

ABSTRACT

Fuzzy c-means clustering is an important non-supervised classification method for remote-sensing images and is based on type-1 fuzzy set theory. Type-1 fuzzy sets use singleton values to express the membership grade; therefore, such sets cannot describe the uncertainty of the membership grade. Interval type-2 fuzzy c-means (IT2FCM) clustering and relevant methods are based on interval type-2 fuzzy sets. Real vectors are used to describe the clustering centres, and the average values of the upper and lower membership grades are used to determine the classification of each pixel. Thus, the width information for interval clustering centres and interval membership grades are ignored. The main contribution of this article is to propose an improved IT2FCM* algorithm by adopting interval number distance (IND) and ranking methods, which use the width information of interval clustering centres and interval membership grades, thus distinguishing this method from existing fuzzy clustering methods. Three different IND definitions are tested, and the distance definition proposed by Li shows the best performance. The second contribution of this work is that two fuzzy cluster validity indices, FS- and XB-, are improved using the IND. Three types of multi/hyperspectral remote-sensing data sets are used to test this algorithm, and the experimental results show that the IT2FCM* algorithm based on the IND proposed by Li performs better than the IT2FCM algorithm using four cluster validity indices, the confusion matrix, and the kappa coefficient (κ). Additionally, the improved FS- index has more indicative ability than the original FS- index.     

Optimality test for generalized FCM and its application to parameter selection

In cluster analysis, the fuzzy c-means (FCM) clustering algorithm is the best known and most widely used method. It was proven to converge to either a local minimum or saddle points by Bezdek et al. Wei and Mendel produced efficient optimality tests for FCM fixed points. Recently, a weighting exponent selection for FCM was proposed by Yu et al. Inspired by these results, we unify several alternative FCM algorithms into one model, called the generalized fuzzy c-means (GFCM). This GFCM model presents a wide variation of FCM algorithms and can easily lead to new and interesting clustering algorithms. Moreover, we construct a general optimality test for GFCM fixed points. This is applied to theoretically choose the parameters in the GFCM model. The experimental results demonstrate the precision of the theoretical analysis.     

A fast and robust image segmentation using FCM with spatial information

Automated segmentation of images has been considered an important intermediate processing task to extract semantic meaning from pixels. In general, the fuzzy c-means approach (FCM) is highly effective for image segmentation. But for the conventional FCM image segmentation algorithm, cluster assignment is based solely on the distribution of pixel attributes in the feature space, and the spatial distribution of pixels in an image is not taken into consideration. In this paper, we present a novel FCM image segmentation scheme by utilizing local contextual information and the high inter-pixel correlation inherent. Firstly, a local spatial similarity measure model is established, and the initial clustering center and initial membership are determined adaptively based on local spatial similarity measure model. Secondly, the fuzzy membership function is modified according to the high inter-pixel correlation inherent. Finally, the image is segmented by using the modified FCM algorithm. Experimental results showed the proposed method achieves competitive segmentation results compared to other FCM-based methods, and is in general faster.