Rough set based generalized fuzzy c-means algorithm and quantitative indices.

A generalized hybrid unsupervised learning algorithm, which is termed as rough-fuzzy possibilistic c-means (RFPCM), is proposed in this paper. It comprises a judicious integration of the principles of rough and fuzzy sets. While the concept of lower and upper approximations of rough sets deals with uncertainty, vagueness, and incompleteness in class definition, the membership function of fuzzy sets enables efficient handling of overlapping partitions. It incorporates both probabilistic and possibilistic memberships simultaneously to avoid the problems of noise sensitivity of fuzzy c-means and the coincident clusters of PCM. The concept of crisp lower bound and fuzzy boundary of a class, which is introduced in the RFPCM, enables efficient selection of cluster prototypes. The algorithm is generalized in the sense that all existing variants of c-means algorithms can be derived from the proposed algorithm as a special case. Several quantitative indices are introduced based on rough sets for the evaluation of performance of the proposed c-means algorithm. The effectiveness of the algorithm, along with a comparison with other algorithms, has been demonstrated both qualitatively and quantitatively on a set of real-life data sets.     

A multiobjective spatial fuzzy clustering algorithm for image segmentation

This article describes a multiobjective spatial fuzzy clustering algorithm for image segmentation. To obtain satisfactory segmentation performance for noisy images, the proposed method introduces the non-local spatial information derived from the image into fitness functions which respectively consider the global fuzzy compactness and fuzzy separation among the clusters. After producing the set of non-dominated solutions, the final clustering solution is chosen by a cluster validity index utilizing the non-local spatial information. Moreover, to automatically evolve the number of clusters in the proposed method, a real-coded variable string length technique is used to encode the cluster centers in the chromosomes. The proposed method is applied to synthetic and real images contaminated by noise and compared with k-means, fuzzy c-means, two fuzzy c-means clustering algorithms with spatial information and a multiobjective variable string length genetic fuzzy clustering algorithm. The experimental results show that the proposed method behaves well in evolving the number of clusters and obtaining satisfactory performance on noisy image segmentation.     

A new fuzzy clustering algorithm for the segmentation of brain tumor

This paper introduces a new method of clustering algorithm based on interval-valued intuitionistic fuzzy sets (IVIFSs) generated from intuitionistic fuzzy sets to analyze tumor in magnetic resonance (MR) images by reducing time complexity and errors. Based on fuzzy clustering, during the segmentation process one can consider numerous cases of uncertainty involving in membership function, distance measure, fuzzifier, and so on. Due to poor illumination of medical images, uncertainty emerges in their gray levels. This paper concentrates on uncertainty in the allotment of values to the membership function of the uncertain pixels. Proposed method initially pre-processes the brain MR images to remove noise, standardize intensity, and extract brain region. Subsequently IVIFSs are constructed to utilize in the clustering algorithm. Results are compared with the segmented images obtained using histogram thresholding, k-means, fuzzy c-means, intuitionistic fuzzy c-means, and interval type-2 fuzzy c-means algorithms and it has been proven that the proposed method is more effective.     

Generalized rough fuzzy c-means algorithm for brain MR image segmentation

Fuzzy sets and rough sets have been widely used in many clustering algorithms for medical image segmentation, and have recently been combined together to better deal with the uncertainty implied in observed image data. Despite of their wide spread applications, traditional hybrid approaches are sensitive to the empirical weighting parameters and random initialization, and hence may produce less accurate results. In this paper, a novel hybrid clustering approach, namely the generalized rough fuzzy c-means (GRFCM) algorithm is proposed for brain MR image segmentation. In this algorithm, each cluster is characterized by three automatically determined rough-fuzzy regions, and accordingly the membership of each pixel is estimated with respect to the region it locates. The importance of each region is balanced by a weighting parameter, and the bias field in MR images is modeled by a linear combination of orthogonal polynomials. The weighting parameter estimation and bias field correction have been incorporated into the iterative clustering process. Our algorithm has been compared to the existing rough c-means and hybrid clustering algorithms in both synthetic and clinical brain MR images. Experimental results demonstrate that the proposed algorithm is more robust to the initialization, noise, and bias field, and can produce more accurate and reliable segmentations.     

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).     

几何距优化质心结合隶属度约束RFCM的脑MRI图像分割算法

针对现有图像分割算法聚类复杂以及分割精度不够高的问题,提出了基于几何距优化质心和粗糙模糊C-均值（RFCM）相结合的医学图像聚类分割算法。首先建立软集表示的像素集,并计算每个像素与质心之间的距离,然后基于像素和质心之间的最小距离,将像素分组到聚类中。为了将软集应用到粗糙模糊C-均值中,定义了一个模糊软集,进一步将输入图像转换为二值图像,通过计算连通区域的几何距选择适当的质心。最后利用这些新的质心计算更新像素的隶属度值,从而完成模糊聚类划分。在Allen Brain Atlas等三个医学数据库上评估了所提出混合算法的性能,获得的Jaccards系数和分割精度（SA）都优于几种对比算法。实验证明,提出的聚类分割算法具有良好的性能。     

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。     

基于改进的直觉模糊核聚类的图像分割方法

针对噪声图像模糊性的本质，提出了基于改进的直觉模糊核聚类的图像分割方法。采用直觉模糊集描述噪声图像包含的不确定性信息，将图像的灰度信息转换到直觉模糊域进行处理；将模糊核聚类拓展为直觉模糊核聚类，在图像的直觉模糊域进行聚类；通过高斯核函数和欧氏距离分别对像素8-邻域的灰度和空间信息进行建模，综合平衡灰度和空间信息对聚类的作用，并将其作为惩罚项加入到直觉模糊核聚类的目标函数中；通过梯度下降法，推导了迭代求解算法；通过典型的合成图像和自然图像分割实例，验证了所提算法的有效性和鲁棒性。     

Soft sets and soft rough sets

In this study, we establish an interesting connection between two mathematical approaches to vagueness: rough sets and soft sets. Soft set theory is utilized, for the first time, to generalize Pawlak’s rough set model. Based on the novel granulation structures called soft approximation spaces, soft rough approximations and soft rough sets are introduced. Basic properties of soft rough approximations are presented and supported by some illustrative examples. We also define new types of soft sets such as full soft sets, intersection complete soft sets and partition soft sets. The notion of soft rough equal relations is proposed and related properties are examined. We also show that Pawlak’s rough set model can be viewed as a special case of the soft rough sets, and these two notions will coincide provided that the underlying soft set in the soft approximation space is a partition soft set. Moreover, an example containing a comparative analysis between rough sets and soft rough sets is given.     

基于二维直方图的图像模糊聚类分割新方法

基于二维直方图的模糊聚类分割算法可以有效地抑制噪声的干扰。但是,FCM算法用于图像数据聚类时的最大缺陷是运算的开销太大,这就限制了这种方法在图像分割中的应用。该文根据FCM算法和灰度图像的特点,提出了一种适用于灰度图像分割的抑制式模糊C-均值聚类算法(S-FCM)。通过调节抑制因子α来提高分割速度和分类的正确率。实验结果表明,新算法对小目标灰度图像的分割效果优于FCM算法。     

基于优化分块颜色直方图及模糊C聚类的彩色图像检索方法

将数据挖掘的聚类算法应用到基于内容的图像检索中可以有效提高检索的速度和效果。模糊聚类算法更符合图像检索本身所具有的模糊性,但这种方法存在聚类分析时间过久影响检索性能的问题,因此本文提出了一种基于优化分块颜色直方图及模糊C聚类的彩色图像检索方法。首先对图像库中的每幅图像进行分块,并提取出每一块的优化颜色特征信息;然后采用模糊C均值聚类算法对得到的颜色特征向量进行聚类,得到每个图像类的聚类中心;最后计算查询示例图像和对应图像类的图像之间的相似度,按照相似度的大小返回检索结果。实验表明,本文提出的方法不仅具有较高的查全率和查准率,而且提取的特征维数较少,聚类时间短,检索速度快。     

An adaptive spatial information-theoretic fuzzy clustering algorithm for image segmentation

This paper presents an adaptive spatial information-theoretic fuzzy clustering algorithm to improve the robustness of the conventional fuzzy c-means (FCM) clustering algorithms for image segmentation. This is achieved through the incorporation of information-theoretic framework into the FCM-type algorithms. By combining these two concepts and modifying the objective function of the FCM algorithm, we are able to solve the problems of sensitivity to noisy data and the lack of spatial information, and improve the image segmentation results. The experimental results have shown that this robust clustering algorithm is useful for MRI brain image segmentation and it yields better segmentation results when compared to the conventional FCM approach.     

Improved spatial fuzzy c-means clustering for image segmentation using PSO initialization,Mahalanobis distance and post-segmentation correction

In this paper, we propose an improvement method for image segmentation using the fuzzy c-means clustering algorithm (FCM). This algorithm is widely experimented in the field of image segmentation with very successful results. In this work, we suggest further improving these results by acting at three different levels. The first is related to the fuzzy c-means algorithm itself by improving the initialization step using a metaheuristic optimization. The second level is concerned with the integration of the spatial gray-level information of the image in the clustering segmentation process and the use of Mahalanobis distance to reduce the influence of the geometrical shape of the different classes. The final level corresponds to refining the segmentation results by correcting the errors of clustering by reallocating the potentially misclassified pixels. The proposed method, named improved spatial fuzzy c-means IFCMS, was evaluated on several test images including both synthetic images and simulated brain MRI images from the McConnell Brain Imaging Center (BrainWeb) database. This method is compared to the most used FCM-based algorithms of the literature. The results demonstrate the efficiency of the ideas presented.     

On biological validity indices for soft clustering algorithms for gene expression data

Unsupervised clustering methods such as K-means, hierarchical clustering and fuzzy c-means have been widely applied to the analysis of gene expression data to identify biologically relevant groups of genes. Recent studies have suggested that the incorporation of biological information into validation methods to assess the quality of clustering results might be useful in facilitating biological and biomedical knowledge discoveries. In this study, we generalize two bio-validity indices, the biological homogeneity index and the biological stability index, to quantify the abilities of soft clustering algorithms such as fuzzy c-means and model-based clustering. The results of an evaluation of several existing soft clustering algorithms using simulated and real data sets indicate that the soft versions of the indices provide both better precision and better accuracy than the classical ones. The significance of the proposed indices is also discussed.     

基于阴影集的粗糙模糊可能性C均值聚类算法

相对于硬聚类算法,软聚类算法可以更好地表示具有不精确边界的类簇。粗糙集和模糊集均是用于描述不确定数据的有效的数学工具,二者互为补充。研究人员已经将粗糙集和模糊集的概念相结合,并应用到聚类算法中,提出了粗糙模糊可能性C均值聚类算法。而文中通过引入阴影集,有效地解决了粗糙模糊可能性C均值聚类算法中的阈值选择问题。     

HSI颜色模型下基于空间信息的FCM算法

针对提花毛皮样片的花型识别技术,在HSI颜色模型下提出了一种基于空间信息的FCM图像分割算法。算法在HSI颜色模型下获得FCM算法的初始聚类中心,并采用了基于空间信息的模糊C均值聚类方法对图像进行分割。经C++编程验证,算法能有效去除花型图像中的噪声,获得较理想的花型识别结果。     

DCT子空间的邻域加权模糊C均值聚类算法

模糊C均值聚类是一种有效的图像分割方法, 但存在因忽略空间上下文信息和结构信息而易为噪声所干扰的现象. 为此提出了DCT子空间的邻域加权模糊C均值聚类方法. 该方法首先结合分块的思想, 对图像块进行离散余弦变换(discrete cosine transform,DCT), 建立了一个基于图像块局部信息的相似性度量模型; 然后定义目标函数中的欧式距离为邻域加权距离; 最后将该方法应用于加噪的人工合成图像、自然图像和MR图像. 实验结果表明, 该方法能够获得较好的分割效果, 同时具有较强的抗噪性.     

Optimal-selection-based suppressed fuzzy c-means clustering algorithm with self-tuning non local spatial information for image segmentation

Suppressed fuzzy c-means clustering algorithm (S-FCM) is one of the most effective fuzzy clustering algorithms. Even if S-FCM has some advantages, some problems exist. First, it is unreasonable to compulsively modify the membership degree values for all the data points in each iteration step of S-FCM. Furthermore, duo to only utilizing the spatial information derived from the pixel’s neighborhood window to guide the process of image segmentation, S-FCM cannot obtain satisfactory segmentation results on images heavily corrupted by noise. This paper proposes an optimal-selection-based suppressed fuzzy c-means clustering algorithm with self-tuning non local spatial information for image segmentation to solve the above drawbacks of S-FCM. Firstly, an optimal-selection-based suppressed strategy is presented to modify the membership degree values for data points. In detail, during each iteration step, all the data points are ranked based on their biggest membership degree values, and then the membership degree values of the top r ranked data points are modified while the membership degree values of the other data points are not changed. In this paper, the parameter r is determined by the golden section method. Secondly, a novel gray level histogram is constructed by using the self-tuning non local spatial information for each pixel, and then fuzzy c-means clustering algorithm with the optimal-selection-based suppressed strategy is executed on this histogram. The self-tuning non local spatial information of a pixel is derived from the pixels with a similar neighborhood configuration to the given pixel and can preserve more information of the image than the spatial information derived from the pixel’s neighborhood window. This method is applied to Berkeley and other real images heavily contaminated by noise. The image segmentation experiments demonstrate the superiority of the proposed method over other fuzzy algorithms.     

A new ECG beat clustering method based on kernelized fuzzy c-means and hybrid ant colony optimization for continuous domains

The kernelized fuzzy c-means algorithm uses kernel methods to improve the clustering performance of the well known fuzzy c-means algorithm by mapping a given dataset into a higher dimensional space non-linearly. Thus, the newly obtained dataset is more likely to be linearly seprable. However, to further improve the clustering performance, an optimization method is required to overcome the drawbacks of the traditional algorithms such as, sensitivity to initialization, trapping into local minima and lack of prior knowledge for optimum paramaters of the kernel functions. In this paper, to overcome these drawbacks, a new clustering method based on kernelized fuzzy c-means algorithm and a recently proposed ant based optimization algorithm, hybrid ant colony optimization for continuous domains, is proposed. The proposed method is applied to a dataset which is obtained from MIT–BIH arrhythmia database. The dataset consists of six types of ECG beats including, Normal Beat (N), Premature Ventricular Contraction (PVC), Fusion of Ventricular and Normal Beat (F), Artrial Premature Beat (A), Right Bundle Branch Block Beat (R) and Fusion of Paced and Normal Beat (f). Four time domain features are extracted for each beat type and training and test sets are formed. After several experiments it is observed that the proposed method outperforms the traditional fuzzy c-means and kernelized fuzzy c-means algorithms.     

结合类内和类间距离的可能聚类分割算法

目的 为了进一步提高噪声图像分割的抗噪性和准确性,提出一种结合类内距离和类间距离的改进可能聚类算法并将其应用于图像分割。方法 该算法避免了传统可能性聚类分割算法中仅仅考虑以样本点到聚类中心的距离作为算法的测度,将类内距离与类间距离相结合作为算法的新测度,即考虑了类内紧密程度又考虑了类间离散程度,以便对不同的聚类结构有较强的稳定性和更好的抗噪能力,并且将直方图融入可能模糊聚类分割算法中提出快速可能模糊聚类分割算法,使其对各种较复杂图像的分割具有即时性。结果 通过人工合成图像和实际遥感图像分割测试结果表明,本文改进可能聚类算法是有效的,其分割轮廓清晰,分类准确且噪声较小,其误分率相比其他算法至少降低了2个百分点,同时能获得更满意的分割效果。结论 针对模糊C-均值聚类分割算法和可能性聚类分割算法对于背景和目标颜色相近的图像分类不准确的缺陷,将类内距离与类间距离相结合作为算法的测度有效的解决了图像分割归类问题,并且结合直方图提出快速可能模糊聚类分割算法使其对于大篇幅复杂图像也具有适用性。