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Abstract: Objective: Accurate brain tissue segmentation from magnetic resonance (MR) images is an essential step in quantitative brain image analysis, and hence has attracted extensive research attention. However, due to the existence of noise and intensity inhomogeneity in brain MR images, many segmentation algorithms suffer from limited robustness to outliers, over-smoothness for segmentations and limited segmentation accuracy for image details. To further improve the accuracy for brain MR image segmentation, a robust spatially constrained fuzzy c-means (RSCFCM) algorithm is proposed in this paper. Method: Firstly, a novel spatial factor is proposed to overcome the impact of noise in the images. By incorporating the spatial information amongst neighborhood pixels, the proposed spatial factor is constructed based on the posterior probabilities and prior probabilities, and takes the spatial direction into account. It plays a role as linear filters for smoothing and restoring images corrupted by noise. Therefore, the proposed spatial factor is fast and easy to implement, and can preserve more details. Secondly, the negative log-posterior is utilized as dissimilarity function by taking the prior probabilities into account, which can further improve the ability to identify the class for each pixel. Finally, to overcome the impact of intensity inhomogeneity, we approximate the bias field at the pixel-by-pixel level by using a linear combination of orthogonal polynomials. The fuzzy objective function is then integrated with the bias field estimation model to overcome the intensity inhomogeneity in the image and segment the brain MR images simultaneously. Results: To demonstrate the performances of the proposed algorithm for the images with/without skull stripping, the first group of experiments is carried out in clinical 3T-weighted brain MR images which contain quite serious intensity inhomogeneity and noise. Then we quantitatively compare our algorithm to state-of-the-art segmentation approaches by using Jaccard similarity on benchmark images obtained from IBSR and BrainWeb with different level of noise and intensity inhomogeneity. The comparison results demonstrate that the proposed algorithm can produce higher accuracy segmentation and has stronger ability of denoising, especially in the area with abundant textures and details. Conclusion: In this paper, the RSCFCM algorithm is proposed by utilizing the negative log-posterior as the dissimilarity function, introducing a novel factor and integrating the bias field estimation model into the fuzzy objective function. This algorithm successfully overcomes the drawbacks of existing FCM-type clustering schemes and EM-type mixture models. Our statistical results (mean and standard deviation of Jaccard similarity for each tissue) on both synthetic and clinical images show that the proposed algorithm can overcome the difficulties caused by noise and bias fields, and is capable of improving over 5% segmentation accuracy comparing with several state-of-the-art algorithms. [Copyright &y& Elsevier]

Abstract: Fuzzy c-means (FCM) is one of the most popular techniques for data clustering. Since FCM tends to balance the number of data points in each cluster, centers of smaller clusters are forced to drift to larger adjacent clusters. For datasets with unbalanced clusters, the partition results of FCM are usually unsatisfactory. Cluster size insensitive FCM (csiFCM) dealt with “cluster-size sensitivity” problem by dynamically adjusting the condition value for the membership of each data point based on cluster size after the defuzzification step in each iterative cycle. However, the performance of csiFCM is sensitive to both the initial positions of cluster centers and the “distance” between adjacent clusters. In this paper, we present a cluster size insensitive integrity-based FCM method called siibFCM to improve the deficiency of csiFCM. The siibFCM method can determine the membership contribution of every data point to each individual cluster by considering cluster's integrity, which is a combination of compactness and purity. “Compactness” represents the distribution of data points within a cluster while “purity” represents how far a cluster is away from its adjacent cluster. We tested our siibFCM method and compared with the traditional FCM and csiFCM methods extensively by using artificially generated datasets with different shapes and data distributions, synthetic images, real images, and Escherichia coli dataset. Experimental results showed that the performance of siibFCM is superior to both traditional FCM and csiFCM in terms of the tolerance for “distance” between adjacent clusters and the flexibility of selecting initial cluster centers when dealing with datasets with unbalanced clusters. [Copyright &y& Elsevier]

• We proposed a Nakagami-Fuzzy imaging framework for medical image segmentation • This paper would be the first application of Nakagami-Fuzzy protocol to MRI images • We enhanced images by Nakagami and segmented lesions by modified fuzzy 2-means • We achieved 92.61% dice score for the main clinical experiment we conducted • Dice scores are computed as 91.88%/89.25% for BraTS 2012/2020 dataset experiments Nakagami distribution and related imaging methods are very efficient in diagnostic ultrasonography for visualization and characterization of tissues for years. Abnormalities in tissues are distinguished from surrounding cells by application of the distribution ruled by the Nakagami m-parameter. The potential of discrimination in ultrasonography enables intelligent segmentation of lesions by other diagnostic tools and the imaging technique is very promising in other areas of medicine, like magnetic resonance imaging (MRI) for brain lesion identification, as presented in this paper. Therefore, we propose a novel Nakagami-Fuzzy imaging framework for intelligent and fully automated suspicious region segmentation from axial FLAIR MRI images exhibiting brain tumor characteristics to satisfy ground truth images with different precision levels. The images from MRI data set are processed by applying Nakagami distribution from pre-Rayleigh to post-Rayleigh for adjusting m-parameter. Amorphous and non-homogenous suspicious regions revealed by Nakagami imaging are segmented using customized Fuzzy 2-means to compare with two types of binary ground truths. The framework we propose is an outstanding example of fuzzy-based expert systems providing an average of 92.61% dice score for the main clinical experiment we conducted using the images and two types of ground truths provided by University of Hospital, Hradec Kralove. We also tested our framework by the BraTS 2012 and BraTS 2020 datasets and achieved an average of 91.88% and 89.25% dice scores respectively, which are competitive among the relevant researches. [ABSTRACT FROM AUTHOR]

Abstract: Most existing fuzzy clustering approaches group objects in a dataset based on either a feature-vector representation of each object, or pairwise relationship representation between each pair of objects. However, when both forms of data representations from different descriptions are available for a given dataset, we believe that a dual and cooperative analysis of feature-vectors (vector data) and pair-wise relationships (relational data) is likely to gain a more comprehensive understanding on the characteristics of the dataset, based on which a better clustering result may be achieved. In this paper, we develop a new fuzzy clustering approach called LinkFCM, which integrates pair-wise relationships into fuzzy c-means vector data clustering. The objective function of LinkFCM consists of two different ways to measure the compactness of clusters with respect to vector data and relational data, respectively, so that clusters are formed by utilizing these two forms of data descriptions. Our experimental study shows that LinkFCM is able to produce good clustering results for real-world document datasets by effectively making use of both content of documents and links among documents. This demonstrates the great potential of the proposed approach for data clustering, where pair-wise relationships are available together with features that describe each object. [Copyright &y& Elsevier]

Abstract: The objective function of the original (fuzzy) c-mean method is modified by a regularizing functional in the form of total variation (TV) with regard to gradient sparsity, and a regularization parameter is used to balance clustering and smoothing. An alternating direction method of multipliers in conjunction with the fast discrete cosine transform is used to solve the TV-regularized optimization problem. The new algorithm is tested on both synthetic and real data, and is demonstrated to be effective and robust in treating images with noise and missing data (incomplete data). [Copyright &y& Elsevier]

Abstract: Identifying the optimal cluster number and generating reliable clustering results are necessary but challenging tasks in cluster analysis. The effectiveness of clustering analysis relies not only on the assumption of cluster number but also on the clustering algorithm employed. This paper proposes a new clustering analysis method that identifies the desired cluster number and produces, at the same time, reliable clustering solutions. It first obtains many clustering results from a specific algorithm, such as Fuzzy C-Means (FCM), and then integrates these different results as a judgement matrix. An iterative graph-partitioning process is implemented to identify the desired cluster number and the final result. The proposed method is a robust approach as it is demonstrated its effectiveness in clustering 2D data sets and multi-dimensional real-world data sets of different shapes. The method is compared with cluster validity analysis and other methods such as spectral clustering and cluster ensemble methods. The method is also shown efficient in mesh segmentation applications. The proposed method is also adaptive because it not only works with the FCM algorithm but also other clustering methods like the k-means algorithm. [Copyright &y& Elsevier]

Abstract: We present in this paper a fuzzy clustering algorithm which can handle spatially constraint problems often encountered in pattern recognition. The proposed method is based on the notions of hyperplanes, the fuzzy c-means, and spatial constraints. By adding a spatial regularizer into the fuzzy hyperplane-based objective function, the proposed method can take into account additionally important information of inherently spatial data. Experimental results have demonstrated that the proposed algorithm achieves superior results to some other popular fuzzy clustering models, and has potential for cluster analysis in spatial domain. [Copyright &y& Elsevier]

Abstract: In this paper, in order to improve both the performance and the efficiency of the conventional Gaussian Mixture Models (GMMs), generalized GMMs are firstly introduced by integrating the conventional GMMs and the active curve axis GMMs for fitting non-linear datasets, and then two types of Fuzzy Gaussian Mixture Models (FGMMs) with a faster convergence process are proposed based on the generalized GMMs, inspired from the mechanism of Fuzzy C-means (FCMs) which introduces the degree of fuzziness on the dissimilarity function based on distances. One is named as probability based FGMMs defining the dissimilarity as the multiplicative inverse of probability density function, and the other is distance based FGMMs which define the dissimilarity function focusing the degree of fuzziness only on the distances between points and component centres. Different from FCMs, both of the proposed dissimilarity functions are based on the exponential function of the distance. The FGMMs are compared with the conventional GMMs and the generalized GMMs in terms of the fitting degree and convergence speed. The experimental results show that the proposed FGMMs not only possess the non-linearity to fit datasets with curve manifolds but also have a much faster convergence process saving more than half computational cost than GMMs''. [Copyright &y& Elsevier]

Abstract: The weighting exponent m is called the fuzzifier that can influence the performance of fuzzy c-means (FCM). It is generally suggested that m∈[1.5,2.5]. On the basis of a robust analysis of FCM, a new guideline for selecting the parameter m is proposed. We will show that a large m value will make FCM more robust to noise and outliers. However, considerably large m values that are greater than the theoretical upper bound will make the sample mean a unique optimizer. A simple and efficient method to avoid this unexpected case in fuzzy clustering is to assign a cluster core to each cluster. We will also discuss some clustering algorithms that extend FCM to contain the cluster cores in fuzzy clusters. For a large theoretical upper bound case, we suggest the implementation of the FCM with a suitable large m value. Otherwise, we suggest implementing the clustering methods with cluster cores. When the data set contains noise and outliers, the fuzzifier m=4 is recommended for both FCM and cluster-core-based methods in a large theoretical upper bound case. [Copyright &y& Elsevier]

Abstract: In this paper we present a new distance metric that incorporates the distance variation in a cluster to regularize the distance between a data point and the cluster centroid. It is then applied to the conventional fuzzy C-means (FCM) clustering in data space and the kernel fuzzy C-means (KFCM) clustering in a high-dimensional feature space. Experiments on two-dimensional artificial data sets, real data sets from public data libraries and color image segmentation have shown that the proposed FCM and KFCM with the new distance metric generally have better performance on non-spherically distributed data with uneven density for linear and nonlinear separation. [Copyright &y& Elsevier]

Abstract: Image segmentation is an important tool in image processing and can serve as an efficient front end to sophisticated algorithms and thereby simplify subsequent processing. In this paper, we present a color image segmentation using pixel wise support vector machine (SVM) classification. Firstly, the pixel-level color feature and texture feature of the image, which is used as input of SVM model (classifier), are extracted via the local homogeneity model and Gabor filter. Then, the SVM model (classifier) is trained by using FCM with the extracted pixel-level features. Finally, the color image is segmented with the trained SVM model (classifier). This image segmentation not only can fully take advantage of the local information of color image, but also the ability of SVM classifier. Experimental evidence shows that the proposed method has a very effective segmentation results and computational behavior, and decreases the time and increases the quality of color image segmentation in comparison with the state-of-the-art segmentation methods recently proposed in the literature. [Copyright &y& Elsevier]

Abstract: This paper presents a novel histogram thresholding – fuzzy C-means hybrid (HTFCM) approach that could find different application in pattern recognition as well as in computer vision, particularly in color image segmentation. The proposed approach applies the histogram thresholding technique to obtain all possible uniform regions in the color image. Then, the Fuzzy C-means (FCM) algorithm is utilized to improve the compactness of the clusters forming these uniform regions. Experimental results have demonstrated that the low complexity of the proposed HTFCM approach could obtain better cluster quality and segmentation results than other segmentation approaches that employing ant colony algorithm. [Copyright &y& Elsevier]

Abstract: Cluster validity indices are used for estimating the quality of partitions produced by clustering algorithms and for determining the number of clusters in data. Cluster validation is difficult task, because for the same data set more partitions exists regarding the level of details that fit natural groupings of a given data set. Even though several cluster validity indices exist, they are inefficient when clusters widely differ in density or size. We propose a clustering validity index that addresses these issues. It is based on compactness and overlap measures. The overlap measure, which indicates the degree of overlap between fuzzy clusters, is obtained by calculating the overlap rate of all data objects that belong strongly enough to two or more clusters. The compactness measure, which indicates the degree of similarity of data objects in a cluster, is calculated from membership values of data objects that are strongly enough associated to one cluster. We propose ratio and summation type of index using the same compactness and overlap measures. The maximal value of index denotes the optimal fuzzy partition that is expected to have a high compactness and a low degree of overlap among clusters. Testing many well-known previously formulated and proposed indices on well-known data sets showed the superior reliability and effectiveness of the proposed index in comparison to other indices especially when evaluating partitions with clusters that widely differ in size or density. [Copyright &y& Elsevier]

Abstract: This paper proposes an adaptive unsupervised scheme that could find diverse applications in pattern recognition as well as in computer vision, particularly in color image segmentation. The algorithm, named Ant Colony–Fuzzy C-means Hybrid Algorithm (AFHA), adaptively clusters image pixels viewed as three dimensional data pieces in the RGB color space. The Ant System (AS) algorithm is applied for intelligent initialization of cluster centroids, which endows clustering with adaptivity. Considering algorithmic efficiency, an ant subsampling step is performed to reduce computational complexity while keeping the clustering performance close to original one. Experimental results have demonstrated AFHA clustering''s advantage of smaller distortion and more balanced cluster centroid distribution over FCM with random and uniform initialization. Quantitative comparisons with the X-means algorithm also show that AFHA makes a better pre-segmentation scheme over X-means. We further extend its application to natural image segmentation, taking into account the spatial information and conducting merging steps in the image space. Extensive tests were taken to examine the performance of the proposed scheme. Results indicate that compared with classical segmentation algorithms such as mean shift and normalized cut, our method could generate reasonably good or better image partitioning, which illustrates the method''s practical value. [Copyright &y& Elsevier]

Abstract: To find the best partition of a large and complex network into a small number of clusters has been addressed in many different ways. However, the probabilistic setting in which each node has a certain probability of belonging to a certain cluster has been scarcely discussed. In this paper, a fuzzy partitioning formulation, which is extended from a deterministic framework for network partition based on the optimal prediction of a random walker Markovian dynamics, is derived to solve this problem. The algorithms are constructed to minimize the objective function under this framework. It is demonstrated by the simulation experiments that our algorithms can efficiently determine the probabilities with which a node belongs to different clusters during the learning process. Moreover, they are successfully applied to two real-world networks, including the social interactions between members of a karate club and the relationships of some books on American politics bought from Amazon.com. [Copyright &y& Elsevier]

Abstract: An important goal in cluster analysis is the internal validation of results using an objective criterion. Of particular relevance in this respect is the estimation of the optimum number of clusters capturing the intrinsic structure of your data. This paper proposes a method to determine this optimum number based on the evaluation of fuzzy partition stability under bootstrap resampling. The method is first characterized on synthetic data with respect to hyper-parameters, like the fuzzifier, and spatial clustering parameters, such as feature space dimensionality, clusters degree of overlap, and number of clusters. The method is then validated on experimental datasets. Furthermore, the performance of the proposed method is compared to that obtained using a number of traditional fuzzy validity rules based on the cluster compactness-to-separation criteria. The proposed method provides accurate and reliable results, and offers better generalization capabilities than the classical approaches. [Copyright &y& Elsevier]

Abstract: This paper proposes an automatic text-independent writer identification framework that integrates an industrial handwriting recognition system, which is used to perform an automatic segmentation of an online handwritten document at the character level. Subsequently, a fuzzy c-means approach is adopted to estimate statistical distributions of character prototypes on an alphabet basis. These distributions model the unique handwriting styles of the writers. The proposed system attained an accuracy of 99.2% when retrieved from a database of 120 writers. The only limitation is that a minimum length of text needs to be present in the document in order for sufficient accuracy to be achieved. We have found that this minimum length of text is about 160 characters or approximately equivalent to 3 lines of text. In addition, the discriminative power of different alphabets on the accuracy is also reported. [Copyright &y& Elsevier]

Abstract: Coding-based methods are among the most promising palmprint recognition methods because of their small feature size, fast matching speed and high verification accuracy. The competitive coding scheme, one representative coding-based method, first convolves the palmprint image with a bank of Gabor filters with different orientations and then encodes the dominant orientation into its bitwise representation. Despite the effectiveness of competitive coding, few investigations have been given to study the influence of the number of Gabor filters and the orientation of each Gabor filter. In this paper, based on the statistical orientation distribution and the orientation separation characteristics, we propose a modified fuzzy C-means cluster algorithm to determine the orientation of each Gabor filter. Since the statistical orientation distribution is based on a set of real palmprint images, the proposed method is more suitable for palmprint recognition. Experimental results indicate that the proposed method achieves higher verification accuracy while compared with that of the original competitive coding scheme and several state-of-the-art methods, such as ordinal measure and RLOC. Considering both the computational complexity and the verification accuracy, competitive code with six orientations would be the optimal choice for palmprint recognition. [Copyright &y& Elsevier]

Abstract: Vector quantization is a useful approach for multi-dimensional data compression and pattern classification. One of the most popular techniques for vector quantization design is the LBG (Linde, Buzo, Gray) algorithm. To address the problem of producing poor estimate of vector centroids which are subjected to biased data in vector quantization; we propose a fuzzy declustering strategy for the LBG algorithm. The proposed technique calculates appropriate declustering weights to adjust the global data distribution. Using the result of fuzzy declustering-based vector quantization design, we incorporate the notion of fuzzy partition entropy into the distortion measures that can be useful for classification of spectral features. Experimental results obtained from simulated and real data sets demonstrate the effective performance of the proposed approach. [Copyright &y& Elsevier]

Abstract: Cluster validity indexes can be used to evaluate the fitness of data partitions produced by a clustering algorithm. Validity indexes are usually independent of clustering algorithms. However, the values of validity indexes may be heavily influenced by noise and outliers. These noise and outliers may not influence the results from clustering algorithms, but they may affect the values of validity indexes. In the literature, there is little discussion about the robustness of cluster validity indexes. In this paper, we analyze the robustness of a validity index using the function of M-estimate and then propose several robust-type validity indexes. Firstly, we discuss the validity measure on a single data point and focus on those validity indexes that can be categorized as the mean type of validity indexes. We then propose median-type validity indexes that are robust to noise and outliers. Comparative examples with numerical and real data sets show that the proposed median-type validity indexes work better than the mean-type validity indexes. [Copyright &y& Elsevier]

Abstract: In this paper, a remote sensing image segmentation procedure that utilizes a single point iterative weighted fuzzy C-means clustering algorithm is proposed based upon the prior information. This method can solve the fuzzy C-means algorithm''s problem that the clustering quality is greatly affected by the data distributing and the stochastic initializing the centrals of clustering. After the probability statistics of original data, the weights of data attribute are designed to adjust original samples to the uniform distribution, and added in the process of cyclic iteration, which could be suitable for the character of fuzzy C-means algorithm so as to improve the precision. Furthermore, appropriate initial clustering centers adjacent to the actual final clustering centers can be found by the proposed single point adjustment method, which could promote the convergence speed of the overall iterative process and drastically reduce the calculation time. Otherwise, the modified algorithm is updated from multidimensional data analysis to color images clustering. Moreover, with the comparison experiments of the UCI data sets, public Berkeley segmentation dataset and the actual remote sensing data, the real validity of proposed algorithm is proved. [Copyright &y& Elsevier]

Abstract: Erich Novak and Klaus Ritter developed in 1996 a global optimization algorithm that uses hyperbolic cross points (HCPs). In this paper we develop a hybrid algorithm for clustering called CMHCP that uses a modified version of this HCP algorithm for global search and the alternating optimization for local search. The program has been tested extensively with very promising results and high efficiency. This provides a nice addition to the arsenal of global optimization in clustering. In the process, we also analyze the smoothness of some reformulated objective functions. [Copyright &y& Elsevier]

Abstract: This paper presents a new algorithm to produce a near optimal mixture of experts model (MEM) architecture for a continuous mapping. The MEM is applied to a new method incorporating photon scatter for designing compensators for intensity modulated radiation therapy. The algorithm utilizes the fuzzy C-means clustering algorithm to partition data before training commences. A reduction in the size of training sets also allows the Levenberg–Marquardt algorithm to be implemented. As a result, both training time and validation error are reduced. A 71% reduction in prediction error compared with that of a single neural network is achieved. [Copyright &y& Elsevier]

Abstract: In this paper, we offer a simple and accurate clustering algorithm which was derived as a closed-form analytical solution to a cluster fit function minimization problem. As a result, the algorithm finds the global minimum of the fit function, and combines exceptional efficiency with optimal clustering results. [Copyright &y& Elsevier]

Abstract: In fuzzy clustering, the fuzzy c-means (FCM) clustering algorithm is the best known and used method. Since the FCM memberships do not always explain the degrees of belonging for the data well, Krishnapuram and Keller proposed a possibilistic approach to clustering to correct this weakness of FCM. However, the performance of Krishnapuram and Keller''s approach depends heavily on the parameters. In this paper, we propose another possibilistic clustering algorithm (PCA) which is based on the FCM objective function, the partition coefficient (PC) and partition entropy (PE) validity indexes. The resulting membership becomes the exponential function, so that it is robust to noise and outliers. The parameters in PCA can be easily handled. Also, the PCA objective function can be considered as a potential function, or a mountain function, so that the prototypes of PCA can be correspondent to the peaks of the estimated function. To validate the clustering results obtained through a PCA, we generalized the validity indexes of FCM. This generalization makes each validity index workable in both fuzzy and possibilistic clustering models. By combining these generalized validity indexes, an unsupervised possibilistic clustering is proposed. Some numerical examples and real data implementation on the basis of the proposed PCA and generalized validity indexes show their effectiveness and accuracy. [Copyright &y& Elsevier]

Abstract: By using a kernel function, data that are not easily separable in the original space can be clustered into homogeneous groups in the implicitly transformed high-dimensional feature space. Kernel k-means algorithms have recently been shown to perform better than conventional k-means algorithms in unsupervised classification. However, few reports have examined the benefits of using a kernel function and the relative merits of the various kernel clustering algorithms with regard to the data distribution. In this study, we reformulated four representative clustering algorithms based on a kernel function and evaluated their performances for various data sets. The results indicate that each kernel clustering algorithm gives markedly better performance than its conventional counterpart for almost all data sets. Of the kernel clustering algorithms studied in the present work, the kernel average linkage algorithm gives the most accurate clustering results. [Copyright &y& Elsevier]

Clustering is an important research topic that has practical applications in many fields. It has been demonstrated that fuzzy clustering, using algorithms such as the fuzzy C-means (FCM), has clear advantages over crisp and probabilistic clustering methods. Like most clustering algorithms, however, FCM and its derivatives need the number of clusters in the given data set as one of their initializing parameters. The main goal of this paper is to develop an effective fuzzy algorithm for automatically determining the number of clusters. After a brief review of the relevant literature, we present a new algorithm for determining the number of clusters in a given data set and a new validity index for measuring the “goodness” of clustering. Experimental results and comparisons are given to illustrate the performance of the new algorithm. [Copyright &y& Elsevier]

A new cluster validity index is proposed that determines the optimal partition and optimal number of clusters for fuzzy partitions obtained from the fuzzy c-means algorithm. The proposed validity index exploits an overlap measure and a separation measure between clusters. The overlap measure, which indicates the degree of overlap between fuzzy clusters, is obtained by computing an inter-cluster overlap. The separation measure, which indicates the isolation distance between fuzzy clusters, is obtained by computing a distance between fuzzy clusters. A good fuzzy partition is expected to have a low degree of overlap and a larger separation distance. Testing of the proposed index and nine previously formulated indexes on well-known data sets showed the superior effectiveness and reliability of the proposed index in comparison to other indexes. [Copyright &y& Elsevier]

This study is concerned with clustering carried out in presence of labeled patterns. An objective of this optimization is to reconcile between the structure residing in data (and being primarily discovered by the underlying clustering mechanism) and the labels of the patterns forming such structure. In this sense, one can consider the supervised fuzzy clustering to be a framework of preliminary data analysis providing with a thorough insight into the structure of the data and supporting the ensuing design of detailed classifiers. The proposed method augments the standard fuzzy C-means algorithm by extending the original objective function by the supervision component (labeled patterns). Experimental results illustrate the approach and discuss the use of this type of clustering in vector quantization. [Copyright &y& Elsevier]

Fuzzy clustering algorithms are becoming the major technique in cluster analysis. In this paper, we consider the fuzzy clustering based on objective functions. They can be divided into two categories: possibilistic and probabilistic approaches leading to two different function families depending on the conditions required to state that fuzzy clusters are a fuzzy c-partition of the input data. Recently, we have presented in Menard and Eboueya (Fuzzy Sets and Systems, 27, to be published) an axiomatic derivation of the Possibilistic and Maximum Entropy Inference (MEI) clustering approaches, based upon an unifying principle of physics, that of extreme physical information (EPI) defined by Frieden (Physics from Fisher information, A unification, Cambridge University Press, Cambridge, 1999). Here, using the same formalism, we explicitly give a new criterion in order to provide a theoretical justification of the objective functions, constraint terms, membership functions and weighting exponent m used in the probabilistic and possibilistic fuzzy clustering. Moreover, we propose an unified framework including the two procedures. This approach is inspired by the work of Frieden and Plastino and Plastino and Miller (Physics A 235, 577) extending the principle of extremal information in the framework of the non-extensive thermostatistics. Then, we show how, with the help of EPI, one can propose extensions of the FcM and Possibilistic algorithms. [Copyright &y& Elsevier]

In this paper we propose a new metric to replace the Euclidean norm in c-means clustering procedures. On the basis of the robust statistic and the influence function, we claim that the proposed new metric is more robust than the Euclidean norm. We then create two new clustering methods called the alternative hard c-means (AHCM) and alternative fuzzy c-means (AFCM) clustering algorithms. These alternative types of c-means clustering have more robustness than c-means clustering. Numerical results show that AHCM has better performance than HCM and AFCM is better than FCM. We recommend AFCM for use in cluster analysis. Recently, this AFCM algorithm has successfully been used in segmenting the magnetic resonance image of Ophthalmology to differentiate the abnormal tissues from the normal tissues. [Copyright &y& Elsevier]

In this paper, fuzzy possibilistic c-means (FPCM) approach based on penalized and compensated constraints are proposed to vector quantization (VQ) in discrete cosine transform (DCT) for image compression. These approaches are named penalized fuzzy possibilistic c-means (PFPCM) and compensated fuzzy possibilistic c-means (CFPCM). The main purpose is to modify the FPCM strategy with penalized or compensated constraints so that the cluster centroids can be updated with penalized or compensated terms iteratively in order to find near-global solution in optimal problem. The information transformed by DCT was separated into DC and AC coefficients. Then, the AC coefficients are trained by using the proposed methods to generate better codebook based on VQ. The compression performances using the proposed approaches are compared with FPCM and conventional VQ method. From the experimental results, the promising performances can be obtained using the proposed approaches. [Copyright &y& Elsevier]

A fuzzy clustering problem consists of assigning a set of patterns to a given number of clusters with respect to some criteria such that each of them may belong to more than one cluster with different degrees of membership. In order to solve it, we first propose a new local search heuristic, called Fuzzy J-Means, where the neighbourhood is defined by all possible centroid-to-pattern relocations. The “integer” solution is then moved to a continuous one by an alternate step, i.e., by finding centroids and membership degrees for all patterns and clusters. To alleviate the difficulty of being stuck in local minima of poor value, this local search is then embedded into the Variable Neighbourhood Search metaheuristic. Results on five standard test problems from the literature are reported and compared with those obtained with the well-known Fuzzy C-Means heuristic. It appears that solutions of substantially better quality are obtained with the proposed methods than with this former one. [Copyright &y& Elsevier]

• A novel locality preserving based fuzzy C-means clustering method (LPFCM) is presented. • An orthogonally projected space, which preserves the locality of structural properties, can be generated in LPFCM. • The capability of FCM for handling high-dimensional data can be enhanced. • The ideas of fuzzy clustering, geometric structure preservation and feature extraction are seamlessly integrated. • Experimental results on some benchmark data sets show the effectiveness of LPFCM. Traditional partition-based clustering algorithms, hard or fuzzy version of C-means, could not deal with high-dimensional data sets effectively as redundant features may impact the computation of distances and local spatial structures among patterns are rarely considered. High dimensionality of space gives rise to so-called concentration effect that is detrimental. In this paper, a novel locality preserving based fuzzy C-means (LPFCM) clustering method and its optimization are presented. An orthogonally projected space, which preserves the locality of structural properties, can be generated in LPFCM, thus enhancing the capability of fuzzy C-means (FCM) for handling high-dimensional data. It is the first time to introduce projection techniques to the FCM optimization objective function directly, and the ideas of fuzzy clustering, geometric structure preservation and feature extraction are seamlessly integrated. LPFCM is also regarded as a unified model that combines two separate stages of spectral clustering. Experimental results on some benchmark data sets show the effectiveness of LPFCM in comparison with FCM and some state-of-the-art methods. [ABSTRACT FROM AUTHOR]

This paper comments on the published work dealing with "Enhanced soft subspace clustering integrating within-cluster and between-cluster information" (Pattern Recognition, vol. 43, pp. 767–781, 2010) proposed by Z. Deng et al. Their clustering approach is based on a new mathematical model with two groups of variables: cluster centers and clusters memberships. They proposed an iterative algorithm to obtain the solution of this model. In each iteration, they fixed cluster centers and used some conditions to obtain the optimal clusters memberships. In this paper, we further analysis their approach and show that the mentioned conditions are necessary and sufficient condition for optimality of clusters memberships when cluster centers are fixed. [ABSTRACT FROM AUTHOR]

Conventional, soft-partition clustering approaches, such as fuzzy c -means (FCM), maximum entropy clustering (MEC) and fuzzy clustering by quadratic regularization (FC-QR), are usually incompetent in those situations where the data are quite insufficient or much polluted by underlying noise or outliers. In order to address this challenge, the quadratic weights and Gini-Simpson diversity based fuzzy clustering model (QWGSD-FC), is first proposed as a basis of our work. Based on QWGSD-FC and inspired by transfer learning, two types of cross-domain, soft-partition clustering frameworks and their corresponding algorithms, referred to as type-I/type-II knowledge-transfer-oriented c -means (TI-KT-CM and TII-KT-CM), are subsequently presented, respectively. The primary contributions of our work are four-fold: (1) The delicate QWGSD-FC model inherits the most merits of FCM, MEC and FC-QR. With the weight factors in the form of quadratic memberships, similar to FCM, it can more effectively calculate the total intra-cluster deviation than the linear form recruited in MEC and FC-QR. Meanwhile, via Gini-Simpson diversity index, like Shannon entropy in MEC, and equivalent to the quadratic regularization in FC-QR, QWGSD-FC is prone to achieving the unbiased probability assignments, (2) owing to the reference knowledge from the source domain, both TI-KT-CM and TII-KT-CM demonstrate high clustering effectiveness as well as strong parameter robustness in the target domain, (3) TI-KT-CM refers merely to the historical cluster centroids, whereas TII-KT-CM simultaneously uses the historical cluster centroids and their associated fuzzy memberships as the reference. This indicates that TII-KT-CM features more comprehensive knowledge learning capability than TI-KT-CM and TII-KT-CM consequently exhibits more perfect cross-domain clustering performance and (4) neither the historical cluster centroids nor the historical cluster centroid based fuzzy memberships involved in TI-KT-CM or TII-KT-CM can be inversely mapped into the raw data. This means that both TI-KT-CM and TII-KT-CM can work without disclosing the original data in the source domain, i.e. they are of good privacy protection for the source domain. In addition, the convergence analyses regarding both TI-KT-CM and TII-KT-CM are conducted in our research. The experimental studies thoroughly evaluated and demonstrated our contributions on both synthetic and real-life data scenarios.