Rough C-means Research Articles

Diabetic Retinopathy Diagnosis Using ResNet with Fuzzy Rough C-Means Clustering

Diabetic Retinopathy (DR) is a vision disease due to the long-term prevalence of Diabetes Mellitus. It affects the retina of the eye and causes severe damage to the vision. If not treated on time it may lead to permanent vision loss in diabetic patients. Today’s development in science has no medication to cure Diabetic Retinopathy. However, if diagnosed at an early stage it can be controlled and permanent vision loss can be avoided. Compared to the diabetic population, experts to diagnose Diabetic Retinopathy are very less in particular to local areas. Hence an automatic computer-aided diagnosis for DR detection is necessary. In this paper, we propose an unsupervised clustering technique to automatically cluster the DR into one of its five development stages. The deep learning based unsupervised clustering is made to improve itself with the help of fuzzy rough c-means clustering where cluster centers are updated by fuzzy rough c-means clustering algorithm during the forward pass and the deep learning model representations are updated by Stochastic Gradient Descent during the backward pass of training. The proposed method was implemented using python and the results were taken on DGX server with Tesla V100 GPU cards. An experimental result on the publically available Kaggle dataset shows an overall accuracy of 88.7%. The proposed model improves the accuracy of DR diagnosis compared to the existing unsupervised algorithms like k-means, FCM, auto-encoder, and FRCM with alexnet.

Noise Rejection Approaches for Various Rough Set-BasedC-Means Clustering

In the field of cluster analysis, rough set-based extensions of hardC-means (HCM;k-means) including roughC-means (RCM), rough setC-means (RSCM), and rough membershipC-means (RMCM) are promising approaches for dealing with the certainty, possibility, uncertainty of belonging of object to clusters. SinceC-means-type methods are strongly affected by noise, noise clustering approaches have been proposed. In noise clustering approaches, noise objects, which are far from any cluster center, are rejected for robust estimation. In this paper, we introduce noise rejection approaches for rough set-basedC-means based on probabilistic memberships and propose noise RCM with membership normalization (NRCM-MN), noise RSCM with membership normalization (NRSCM-MN), and noise RMCM (NRMCM). In addition, visualization demonstration of the cluster boundaries on the two-dimensional plane of the proposed methods is carried out to confirm the characteristics of each method. Furthermore, the clustering performance is verified by numerical experiments using real-world datasets.

Objective function-based rough membership C-means clustering

Hard C-means (HCM) is one of the most widely used partitive clustering methods and was extended to rough C-means (RCM) by referencing to the perspective of rough set theory to deal with the certain, possible, and uncertain belonging of object to clusters. Furthermore, rough set C-means (RSCM) and rough membership C-means (RMCM) have been proposed as clustering models on an approximation space considering the granularity of the universe (object space) based on binary relations. Although these rough set-based C-means methods are practical, they are not formulated based on objective functions, but are built on heuristic schemes. Objective function-based methods can be a basis for discussion of the validity of clustering and further theoretical developments. In this paper, we propose a novel RMCM framework, which is called RMCM version 2 (RMCM2), based on an objective function. The objective function is designed to derive the same updating rule for cluster centers as in RMCM. We demonstrate the characteristics of RMCM2 by visualizing cluster boundaries on a grid point dataset. Furthermore, we verify the clustering performance of RMCM2 through numerical experiments by using real-world datasets.

A unified approach for cluster-wise and general noise rejection approaches for k-means clustering.

Hard C-means (HCM; k-means) is one of the most widely used partitive clustering techniques. However, HCM is strongly affected by noise objects and cannot represent cluster overlap. To reduce the influence of noise objects, objects distant from cluster centers are rejected in some noise rejection approaches including general noise rejection (GNR) and cluster-wise noise rejection (CNR). Generalized rough C-means (GRCM) can deal with positive, negative, and boundary belonging of object to clusters by reference to rough set theory. GRCM realizes cluster overlap by the linear function threshold-based object-cluster assignment. In this study, as a unified approach for GNR and CNR in HCM, we propose linear function threshold-based C-means (LiFTCM) by relaxing GRCM. We show that the linear function threshold-based assignment in LiFTCM includes GNR, CNR, and their combinations as well as rough assignment of GRCM. The classification boundary is visualized so that the characteristics of LiFTCM in various parameter settings are clarified. Numerical experiments demonstrate that the combinations of rough clustering or the combinations of GNR and CNR realized by LiFTCM yield satisfactory results.

Rough Set-Based Clustering Utilizing Probabilistic Memberships

Representing the positive, possible, and boundary regions of clusters, rough set-based C-means clustering methods, such as generalized rough C-means (GRCM) and rough set C-means (RSCM), are promising for analyzing vague cluster shapes and realizing reliable classification. In this study, we consider rough set-based clustering approaches that utilize probabilistic memberships as variants of GRCM and RSCM, including π generalized rough C-means (πGRCM), π rough set C-means (πRSCM), and rough membership C-means (RMCM). πGRCM and πRSCM assign equal probabilities of cluster belonging according to Laplace’s principle of indifference, whereas RMCM assigns the probabilities according to rough memberships, which represent conditional probabilities based on the object’s neighborhood derived from a binary relation. In addition, we discuss the theoretical validity of our RMCM approach and compare it with other methods considered in this study. Furthermore, we conducted numerical experiments for evaluating the classification performances of the abovementioned methods. Based on our experimental results, the methods were found to be effective.

Characteristics of Rough SetC-Means Clustering

HardC-means (HCM), which is one of the most popular clustering techniques, has been extended by using soft computing approaches such as fuzzy theory and rough set theory. FuzzyC-means (FCM) and roughC-means (RCM) are respectively fuzzy and rough set extensions of HCM. RCM can detect the positive and the possible regions of clusters by using the lower and the upper areas which are respectively analogous to the lower and the upper approximations in rough set theory. RCM-type methods have the problem that the original definitions of the lower and the upper approximations are not actually used. In this paper, rough setC-means (RSCM), which is an extension of HCM based on the original rough set definition, is proposed as a rough set-based counterpart of RCM. Specifically, RSCM is proposed as a clustering model on an approximation space considering a space granulated by a binary relation and uses the lower and the upper approximations of temporal clusters. For this study, we investigated the characteristics of the proposed RSCM through basic considerations, visual demonstrations, and comparative experiments. We observed the geometric characteristics of the examined methods by using visualizations and numerical experiments conducted for the problem of classifying patients as having benign or malignant disease based on a medical dataset. We compared the classification performance by viewing the trade-off between the classification accuracy in the positive region and the fraction of objects classified as being in the positive region.

Special Issue on Recent Methodological Developments in Fuzzy Clustering and Related Topics

Various applications of data analysis and their effects have been reported recently. With the remarkable progress in classification methods, one example being support vector machines, clustering as the main method of unsupervised classification has also been studied extensively. Consequently, fuzzy methods of clustering is becoming a standard technique. However, unsolved theoretical and methodological problems in fuzzy clustering remain and have to be studied more deeply. This issue collects five papers concerned with fuzzy clustering and related fields, and in all of them the main interest is methodology. Kondo and Kanzawa consider fuzzy clustering with a new objective function using q-divergence, which is a generalization of the well-known Kullback-Leibler divergence. Among different data types, they focus on categorical data. They also show the relations of different methods of fuzzy c-means. Thus, this study tends to further generalize methods of fuzzy clustering, trying to find the methodological boundaries of the capabilities of fuzzy clustering models. Kitajima, Endo, and Hamasuna propose a method of controlling cluster sizes so that the resulting clusters have an even size, which is different from the optimizing of cluster sizes dealt with in other studies. This technique enhances application fields of clustering in which cluster sizes are more important than cluster shapes. Hamasuna et al. study the validity measures of clusters for network data. Cluster validity measures are generally proposed for points in Euclidean spaces, but the authors consider the application of validity measures to network data. Several validity measures are modified and adapted to network data, and their effectiveness is examined using simple network examples. Ubukata et al. propose a new method of c-means related to rough sets, a method based on a different idea from well-known rough c-means by Lingras. Finally, Kusunoki, Wakou, and Tatsumi study the maximum margin model for the nearest prototype classifier that leads to the optimization of the difference of convex functions. All papers include methodologically important ideas that have to be further investigated and applied to real-world problems.

Adaptive density trajectory cluster based on time and space distance

There are some hotspot problems remaining in trajectory cluster for discovering mobile behavior regularity, such as the computation of distance between sub trajectories, the setting of parameter values in cluster algorithm and the uncertainty/boundary problem of data set. As a result, based on the time and space, this paper tries to define the calculation method of distance between sub trajectories. The significance of distance calculation for sub trajectories is to clearly reveal the differences in moving trajectories and to promote the accuracy of cluster algorithm. Besides, a novel adaptive density trajectory cluster algorithm is proposed, in which cluster radius is computed through using the density of data distribution. In addition, cluster centers and number are selected by a certain strategy automatically, and uncertainty/boundary problem of data set is solved by designed weighted rough c-means. Experimental results demonstrate that the proposed algorithm can perform the fuzzy trajectory cluster effectively on the basis of the time and space distance, and obtain the optimal cluster centers and rich cluster results information adaptably for excavating the features of mobile behavior in mobile and sociology network.

Soft fuzzy rough set-based MR brain image segmentation

Fuzzy sets, rough sets are efficient tools to handle uncertainty and vagueness in the medical images and are widely used for medical image segmentation. Soft sets are a new mathematical approach to uncertainty and vagueness. In this paper, a hybrid segmentation algorithm based on soft sets namely soft fuzzy rough c-means is proposed to extract the white matter, gray matter and the cerebro spinal fluid from MR brain image with bias field correction. In this algorithm, soft fuzzy rough approximations are applied to obtain the rough regions of image. These approximations are free from defining thresholds, weight parameters and are less complex compared to the existing rough set based algorithms. Soft sets use similarity coefficients to find the similarity of the clusters formed in present and previous step. The proposed algorithm does not involve any negative region, hence all the pixels participate in clustering avoiding clustering mistakes. Also, the histogram based centroids choose the centroids close to the ground truth that in turn effect the definition of approximations, standardizing the clusters. The proposed algorithm evaluated through simulation, compared it with existing k-means, rough k-means, fuzzy c-means and other hybrid algorithms. The soft fuzzy rough c-means algorithm outperforms the considered algorithms in all analyzed scenarios even in extracting the tumor from the brain tissue.

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).

A modified rough c-means clustering algorithm based on hybrid imbalanced measure of distance and density

Traditional c-means clustering partitions a group of objects into a number of non-overlapping sets. Rough sets provide more flexible and objective representation than classical sets with hard partition and fuzzy sets with subjective membership function for a given dataset. Rough c-means clustering and its extensions were introduced and successfully applied in many real life applications in recent years. Each cluster is represented by a reasonable pair of lower and upper approximations. However, the most available algorithms pay no attention to the influence of the imbalanced spatial distribution within a cluster. The limitation of the mean iterative calculation function, with the same weight for all the data objects in a lower or upper approximation, is analyzed. A hybrid imbalanced measure of distance and density for the rough c-means clustering is defined, and a modified rough c-means clustering algorithm is presented in this paper. To evaluate the proposed algorithm, it has been applied to several real world data sets from UCI. The validity of this algorithm is demonstrated by the results of comparative experiments.

A Comparative Study of Color Image Segmentation Using Hard, Fuzzy,Rough Set Based Clustering Techniques

Image segmentation is the process of subdividing an image into its constituent parts and extracting these parts of interest, which are the objects. Colour image segmentation emerges as a new area of research. It can solve many contemporary problems in medical imaging, mining and mineral imaging, bioinformatics, and material sciences. Naturally, color image segmentation demands well defined borders of different objects in an image. So, there is a fundamental demand of accuracy. The segmented regions or components should not be further away from the true object than one or a few pixels. So, there is a need for improved image segmentation technique that can segment different components precisely. Image data may have corrupted values due to the usual limitations or artifacts of imaging devices. Noisy data, data sparsity, and high dimensionality of data create difficulties in image pixel clustering. As a result, image pixel clustering becomes a harder problem than other form of data. Taking into account all the above considerations we propose an unsupervised image segmentation method using Rough-Fuzzy C-Mean a hybrid model for segmenting RGB image by reducing cluster centers using rough sets and Fuzzy C-Means Method, and also compare the effectiveness of the clustering methods such as Hard C Means (HCM), Fuzzy C Means (FCM), Fuzzy K Means (FKM), Rough C Means (RCM) with cluster validity index such as DB Index, XB Index and Dunn Index. A good clustering procedure should make the value of DB index as low as possible, for Dunn Index high value, and for XB Index low value.

An extension to Rough c-means clustering based on decision-theoretic Rough Sets model

Rough c-means algorithm has gained increasing attention in recent years. However, the assignment scheme of Rough c-means algorithm does not incorporate any information about the neighbors of the data point to be assigned and may cause undesirable solutions in practice. This paper proposes an extended Rough c-means clustering algorithm based on the concepts of decision-theoretic Rough Sets model. In the risk calculation, a new kind of loss function is utilized to capture the loss information of the neighbors. The assignment scheme of the present multi-category decision-theoretic Rough Sets model is also adjusted to deal with the potentially high computational cost. Experimental results are provided to validate the effectiveness of the proposed approach.

Rough C-means and Fuzzy Rough C-means for Colour Quantisation

Colour quantisation algorithms are essential for displaying true colour images using a limited palette of distinct colours. The choice of a good colour palette is crucial as it directly determines the quality of the resulting image. Colour quantisation can also be seen as a clustering problem where the task is to identify those clusters that best represent the colours in an image. In this paper we propose rough c-means and fuzzy rough c-means clustering algorithms for colour quantisation of images. Both approaches utilise the concept of lower and upper approximations of clusters to define palette colours. While in the rough c-means approach cluster centroids are refined iteratively through a linear combination of elements of the lower and upper approximations, the fuzzy rough c-means technique assigns variable membership values to the elements in the boundary region which in turn are incorporated into the calculation of cluster centres. Experimental results on a standard set of images show that these approaches performs significantly better than other, purpose built colour quantisation algorithms.

Rough C-means and Fuzzy Rough C-means for Colour Quantisation

Colour quantisation algorithms are essential for displaying true colour images using a limited palette of distinct colours. The choice of a good colour palette is crucial as it directly determines ...

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.

Rough colour quantisation

Colour quantisation algorithms are essential for displaying true colour images using a limited palette of distinct colours. The choice of a good colour palette is crucial as it directly determines the quality of the resulting image. Colour quantisation can also be seen as a clustering problem where the task is to identify those clusters that best represent the colours in an image. In this paper, we use a rough c-means clustering algorithm for colour quantisation of images. Experimental results on a standard set of images show that this rough colour quantisation approach performs significantly better than other, purpose built colour reduction algorithms.

Shadowed sets in the characterization of rough-fuzzy clustering

In this study, we develop a technique of an automatic selection of a threshold parameter, which determines approximation regions in rough set-based clustering. The proposed approach exploits a concept of shadowed sets. All patterns (data) to be clustered are placed into three categories assuming a certain perspective established by an optimization process. As a result, a lack of knowledge about global relationships among objects caused by the individual absolute distance in rough C-means clustering or individual membership degree in rough-fuzzy C-means clustering can be circumvented. Subsequently, relative approximation regions of each cluster are detected and described. By integrating several technologies of Granular Computing including fuzzy sets, rough sets, and shadowed sets, we show that the resulting characterization leads to an efficient description of information granules obtained through the process of clustering including their overlap regions, outliers, and boundary regions. Comparative experimental results reported for synthetic and real-world data illustrate the essence of the proposed idea.

Semi-supervised outlier detection based on fuzzy rough C-means clustering

This paper presents a fuzzy rough semi-supervised outlier detection (FRSSOD) approach with the help of some labeled samples and fuzzy rough C-means clustering. This method introduces an objective function, which minimizes the sum squared error of clustering results and the deviation from known labeled examples as well as the number of outliers. Each cluster is represented by a center, a crisp lower approximation and a fuzzy boundary by using fuzzy rough C-means clustering and only those points located in boundary can be further discussed the possibility to be reassigned as outliers. As a result, this method can obtain better clustering results for normal points and better accuracy for outlier detection. Experiment results show that the proposed method, on average, keep, or improve the detection precision and reduce false alarm rate as well as reduce the number of candidate outliers to be discussed.

An evolutionary rough partitive clustering

An evolutionary rough c-means clustering algorithm is proposed. Genetic algorithms are employed to tune the threshold, and relative importance of upper and lower approximations of the rough sets modeling the clusters. The Davies–Bouldin clustering validity index is used as the fitness function, that is minimized while arriving at an optimal partitioning. A comparative study of its performance is made with related partitive algorithms. The effectiveness of the algorithm is demonstrated on real and synthetic datasets, including microarray gene expression data from Bioinformatics.