Fuzzy C-partition Research Articles

A Framework for Multi-Threshold Image Segmentation of Low Contrast Medical Images

Accurate medical images segmentation plays a vital role in contouring during diagnosis and treatment planning. To improve the segmentation accuracy in low contrast images, we propose a method by combining Hill entropy and fuzzy c-partition. Here, using membership function, an image is first transformed into fuzzy domain. Subsequently, the fuzzy Hill entropies are defined for foreground (object) and background. Next, the total fuzzy Hill entropy is maximized to compute the accurate threshold; this process is employed to calculate a proper parameter combination of membership function. This Hill entropy is then optimized to acquire an image threshold by Differential Evolution “DE” optimization algorithm. The key benefit of the presented approach is that it considers the information of background and object as well as interactions between them in threshold selection mechanism. The results and performance evaluations show the better accuracy of our technique over other existing approaches.

Brain tumor segmentation using neutrosophic expert maximum fuzzy-sure entropy and other approaches

Abstract Glioblastoma is the most aggressive and most common primary brain tumor in adult individuals. Magnetic resonance imagery (MRI) is widely used in the brain tumor diagnosis. This study proposes an approach called neutrosophic set – expert maximum fuzzy-sure entropy (NS-EMFSE), which is a successful edge detection approach, by combining two powerful approaches such as neutrosophic set (NS) and expert maximum fuzzy-sure entropy (EMFSE). Thus, a high performance approach is designed for Glioblastoma, which is the most difficult brain tumor segmentation and edge finding process. The proposed NS-EMFSE approach was designed to detect enhancing part of the tumor in brain MRI image. Using maximum fuzzy entropy and fuzzy c-partition methods, EMFSE determines the necessary threshold value to convert images into binary format. NS has been recently proposed as an efficient approach based on neutrosophy theory, and yields remarkably successful results for indeterminate situations. The proposed algorithm was compared to NS with Otsu thresholding (NS-Otsu), support vector machine (SVM), fuzzy c-means (FCM), Darwinian particle swarm optimization (DPSO). SVM, FCM, DPSO algorithms have been so far used for edge detection and segmentation in various fields. In this study, figure of merid (FOM) and jaccard index (JI) tests were carried out to evaluate the performances of these 5 edge detection approaches on 100 MRI images. These tests indicate which approach yields the best performance in enhancing part detection of the tumor in MRI image. Analysis of variance (ANOVA) was performed on FOM and JI data. As a result, the maximum values of FOM and JI results for the NS-EMFSE are 0.984000, and 0.965000, the mean values are 0.933440 and 0.912000, and the minimum values are 0.699000 and 0.671000, respectively. When these statistical results are compared with the statistical results of other 4 approaches, it is understood that the proposed method yields higher FOM and JI results. In addition, other statistical analysis results proved that the proposed NS-EMFSE performed better than other 4 methods.

Fuzzy maximum likelihood change-point algorithms for identifying the time of shifts in process data

Process data provide important information for monitoring product quality. A common problem of process readings is their deviation from in-control values due to systematic errors or instrument biases. Timely detection and modification of instrument fault is crucial for process manipulation. Monitoring changes in process mean and variance simultaneously is important because special causes can evoke changes in both at once. Many existing methods identify either the mean or variance only; some even inaccurately assumes that the in-control parameters are known. We propose a new method called fuzzy maximum likelihood change-point (FMLCP) algorithm that allows detection of the time of shifts in mean and variance simultaneously without knowing the in-control parameters. The fuzzy c-partition concept is embedded into change-point formulation to deal with the vagueness of boundaries between adjacent segments. An FMLCP procedure is constructed and suitable for processes following any distribution. The FMLCP algorithm can be applied to both phase I and II processes in quality control without any information of in-control parameters; multiple change points are allowed and the shifts in each individual segment can be estimated simultaneously. Our experimental results demonstrate the preferred utility of FMLCP over traditional statistical maximum likelihood approaches. We used real datasets to demonstrate the effectiveness of FMLCP. Using FMLCP to detect small shifts is particularly beneficial for identifying root causes quickly and correctly in phase II applications where small changes occur more often and the average run length tends to be long.

Fuzzy Similarity Measure Based Spectral Clustering Framework for Noisy Image Segmentation

In recent times, graph based spectral clustering algorithms have received immense attention in many areas like, data mining, object recognition, image analysis and processing. The commonly used similarity measure in the clustering algorithms is the Gaussian kernel function which uses sensitive scaling parameter and when applied to the segmentation of noise contaminated images leads to unsatisfactory performance because of neglecting the spatial pixel information. The present work introduces a novel framework for spectral clustering which embodied local spatial information and fuzzy based similarity measure to tackle the above mentioned issues. In our approach, firstly we filter the noise components from original image by using the spatial and gray–level information. The similarity matrix is then constructed by employing a similarity measure which takes into account the fuzzy c-partition matrix and vectors of the cluster centers obtained by fuzzy c-means clustering algorithm. In the last step, spectral clustering technique is realized on derived similarity matrix to obtain the desired segmentation result. Experimental results on segmentation of synthetic and Berkeley benchmark images with noise demonstrates the effectiveness and robustness of the proposed method, giving it an edge over the clustering based segmentation method reported in the literature.

Unsupervised hierarchical image segmentation through fuzzy entropy maximization

The fuzzy c-partition entropy has been widely adopted as a global optimization technique for finding the optimal thresholds when performing multilevel gray image segmentation. Nevertheless, existing fuzzy c-partition entropy approaches generally have two limitations, i.e., partition number c needs to be manually tuned for different input and the methods can process grayscale images only. To address these two limitations, an unsupervised multilevel segmentation algorithm is presented in this paper. The core step of our algorithm is a bi-level segmentation operator, which uses binary graph cuts to maximize both fuzzy 2-partition entropy and segmentation smoothness. By iteratively performing this bi-level segmentation operator, multilevel image segmentation is achieved in a hierarchical manner: Starting from the input color image, our algorithm first picks the color channel that can best segment the image into two labels, and then iteratively selects channels to further split each labels until convergence. The experimental results demonstrate the presented hierarchical segmentation scheme can efficiently segment both grayscale and color images. Quantitative evaluations over classic gray images and the Berkeley Segmentation Database show that our method is comparable to the state-of-the-art multi-scale segmentation methods, yet has the advantage of being unsupervised, efficient, and easy to implement.

A novel image enhancement method using fuzzy Sure entropy

Image enhancement is a very significant issue in image processing and analysis. In practice, many images (e.g.images captured from X-ray systems) are of low quality, such a slow-luminance and low-contrast, which must be enhanced before further processing. Fuzzy set theory is a useful tool for handling the ambiguity or uncertainty. Many researchers use the maximum Shannon entropy and fuzzy complement for image enhancement. But these methods are easy to be over-enhanced or under-enhanced or time-consuming. In this paper, a flexible method is proposed, which utilizes the maximum fuzzy Sure entropy, fuzzy c-partition and fuzzy complement (MSRM). Furthermore, a positive threshold value selection algorithm is developed to tune the enhancement performance of the proposed method. A variety of highly degraded images have been experimented by the proposed method. The comparisons of those experimental results show that the performance of our method overwhelms those of the existing ones.

Measuring the congruence of fuzzy partitions in fuzzy c-means clustering

We propose an internal cluster validity index for a fuzzy c-means algorithm which combines a mathematical model for the fuzzy c-partition and a heuristic search for the number of clusters in the data. Our index resorts to information theoretic principles, and aims to assess the congruence between such a model and the data that have been observed. The optimal cluster solution represents a trade-off between discrepancy and the complexity of the underlying fuzzy c-partition. We begin by testing the effectiveness of the proposed index using two sets of synthetic data, one comprising a well-defined cluster structure and the other containing only noise. Then we use datasets arising from real life problems. Our results are compared to those provided by several available indices and their goodness is judged by an external measure of similarity. We find substantial evidence supporting our index as a credible alternative to the cluster validation problem, especially when it concerns structureless data.

Change-point detection for shifts in control charts using fuzzy shift change-point algorithms

Knowing the real time of changes, called change-point, in a process is essential for quickly identifying and removing special causes. Many change-point methods in statistical process control assume the distribution and the in-control parameters of the process known, however, they are rarely known accurately. Small errors accompanied with estimated parameters may lead to unfavorable change-point estimates. In this paper, a new method, called fuzzy shift change-point algorithm, which does not require the knowledge of the distribution nor the parameter of the process, is proposed to detect change-points for shifts in process mean. The fuzzy c-partition concept is embedded into change-point formulation in which any possible collection of change-points is considered as a partitioning of data with a fuzzy membership. These memberships are then transferred into the pseudo memberships of observations belonging to each individual cluster, so the fuzzy c-means clustering can be used to obtain the estimates for shifts. Subsequently, the fuzzy c-means algorithm is used again to obtain new iterates of change-point collection memberships by minimizing an objective function concerning the deviations between observations and the corresponding cluster means. The proposed algorithm is nonparametric and applicable to normal and non-normal processes in both phase I and II. The performance of the proposed fuzzy shift change-point algorithm is discussed in comparison with powerful statistical methods through extensive simulation studies. The results demonstrate the superiority and usefulness of our proposed method.

A new perspective of modified partition coefficient

We begin by showing that the modified partition coefficient (MPC) is an average Euclidean distance between membership degrees and the centre of the fuzzy c-partition. Subsequently, we construct alternative MPCs using several other measures of dissimilarity and examine how differently they perform when compared with the original proposal. Empirical evidence shows that the MPC based on a Chernoff’s measure of divergence is more robust to the initial conditions of the fuzzy c-means algorithm.

Fuzzy Classification Maximum Likelihood Clustering with T-Distributions

Classification maximum likelihood (CML) procedure is a maximum likelihood mixture approach to clustering. In 1993, Yang first extended the CML to a so-called fuzzy CML (FCML), by combining fuzzy c-partitions with the CML function for a normal mixture model. However, normal distribution is not robust for outliers. In this paper we consider FCML with t-distributions and create a clustering algorithm, called FCMLT. Numerical examples and real data applications with comparisons are given to demonstrate the effectiveness and superiority of the proposed method.

A Comparison of Evolutionary Algorithms: PSO, DE and GA for Fuzzy C-Partition

fuzzy c-partition entropy technique for threshold selection is one of the best image thresholding techniques, but its complexity increases with the number of thresholds. In this paper, the selection of thresholds (fuzzy parameters) was seen as an optimization problem and solved using particle swarm optimization (PSO), differential evolution (DE), genetic (GA) algorithms. The proposed fast approaches have been tested on many images. For example, the processing time of four-level thresholding using PSO, DE and GA is reduced to less than 0.4s. PSO, DE and GA show equal performance when the number of thresholds is small. When the number of thresholds is greater, the PSO algorithm performs better than GA and DE in terms of precision and robustness. But the GA algorithm is the most efficient with respect to the execution time.

Efficient multilevel image segmentation through fuzzy entropy maximization and graph cut optimization

The fuzzy c-partition entropy has been widely adopted as a global optimization technique for finding the optimized thresholds for multilevel image segmentation. However, it involves expensive computation as the number of thresholds increases and often yields noisy segmentation results since spatial coherence is not enforced. In this paper, an iterative calculation scheme is presented for reducing redundant computations in entropy evaluation. The efficiency of threshold selection is further improved through utilizing the artificial bee colony algorithm as the optimization technique. Finally, instead of performing thresholding for each pixel independently, the presented algorithm oversegments the input image into small regions and uses the probabilities of fuzzy events to define the costs of different label assignments for each region. The final segmentation results is computed using graph cut, which produces smooth segmentation results. The experimental results demonstrate the presented iterative calculation scheme can greatly reduce the running time and keep it stable as the number of required thresholds increases. Quantitative evaluations over 20 classic images also show that the presented algorithm outperforms existing multilevel segmentation approaches.

A ROBUST FUZZY CLASSIFICATION MAXIMUM LIKELIHOOD CLUSTERING FRAMEWORK

In 1993, Yang first extended the classification maximum likelihood (CML) to a so-called fuzzy CML, by combining fuzzy c-partitions with the CML function. Fuzzy c-partitions are generally an extension of hard c-partitions. It was claimed that this was more robust. However, the fuzzy CML still lacks some robustness as a clustering algorithm, such as its in-ability to detect different volumes of clusters, its heavy dependence on parameter initializations and the necessity to provide an a priori cluster number. In this paper, we construct a robust fuzzy CML clustering framework that has a robust clustering method. The eigenvalue decomposition of a covariance matrix is firstly considered using the fuzzy CML model. The Bayesian information criterion (BIC) is then used for model selection, in order to choose the best model with the optimal number of clusters. Therefore, the proposed robust fuzzy CML clustering framework exhibits clustering characteristics that are robust in terms of the parameter initialization, robust in terms of the cluster number and also in terms of its capability to detect different volumes of clusters. Numerical examples and real data applications with comparisons are provided, which demonstrate the effectiveness and superiority of the proposed method.

A fast recursive algorithm based on fuzzy 2-partition entropy approach for threshold selection

The fuzzy c-partition entropy approach for threshold selection is an effective approach for image segmentation. The approach models the image with a fuzzy c-partition, which is obtained using parameterized membership functions. The ideal threshold is determined by searching an optimal parameter combination of the membership functions such that the entropy of the fuzzy c-partition is maximized. It involves large computation when the number of parameters needed to determine the membership function increases. In this paper, a recursive algorithm is proposed for fuzzy 2-partition entropy method, where the membership function is selected as S-function and Z-function with three parameters. The proposed recursive algorithm eliminates many repeated computations, thereby reducing the computation complexity significantly. The proposed method is tested using several real images, and its processing time is compared with those of basic exhaustive algorithm, genetic algorithm (GA), particle swarm optimization (PSO), ant colony optimization (ACO) and simulated annealing (SA). Experimental results show that the proposed method is more effective than basic exhaustive search algorithm, GA, PSO, ACO and SA.

Retrieval of Web Documents Using a Fuzzy Hierarchical Clustering

ABSTRACT The World Wide Web has huge amount of information that is retrieved using information retrieval tool like Search Engine. Page repository of Search Engine contains the web documents downloaded by the crawler. This repository contains variety of web documents from different domains. In this paper, a technique called “Retrieval of Web documents using a fuzzy hierarchical clustering” is being proposed that creates the clusters of web documents using fuzzy hierarchical clustering. Keywords Search Engine, Web documents, Fuzzy Hierarchical Clustering Indexer 1. Introduction WWW [01, 02 and 04] is a huge repository of information consisting of hyperlinked documents spread over the internet. The size of the web as on February 2010 stands at around 35 million pages. For a user, it is practically impossible to search through this extremely large database for the information needed by him. Hence the need for Search Engine (see fig. 1) arises. The search engine uses crawlers to gather information and stores it in database maintained at search engine side. For a given user's query the search engine searches in the local database and very quickly displays the results. The huge amount of information [05] is retrieved using data mining tools. Classification, Clustering and Association tools etc. are used for data mining technique. Clustering plays a key role in searching for structures in data. As the number of available documents nowadays is large, hierarchical approaches are better suited because they permit categories to be defined at different pensiveness work about levels. The problem of clustering in finite set of data is to find several cluster centers that can properly characterize relevant classes of finite set of data such that degree of association is strong for data within blocks of the partition and weak for data in different blocks. When the weakness of a crisp partition of finite set of data is replaced with a fuzzy partition, this area is known as fuzzy clustering. Fuzzy clustering is a relevant technique for information retrieval. As a document might be relevant to multiple queries, this document should be given in the corresponding response sets, otherwise, the users would not be aware of it. Fuzzy clustering seems a natural technique for document categorization. There are two basic methods of fuzzy clustering, one which is based on fuzzy c-partitions, is called a fuzzy c-means clustering method and the other, based on the fuzzy equivalence relations, is called a fuzzy equivalence clustering method.

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English

Threshold Selection Based on Fuzzy Tsallis Entropy and Particle Swarm Optimization

Tsallis entropy is a generalization of Shannon entropy and can describe physical system with long range interactions, long time memories and fractal-type structures. In this paper, a novel threshold selection technique for image segmentation is proposed by combining Tsallis entropy and fuzzy c-partition. The image to be segmented is firstly transformed into fuzzy domain using membership function. Then, the fuzzy Tsallis entropies for object and background are defined. The threshold is selected by finding a proper parameter combination of membership function such that the total fuzzy Tsallis entropy is maximized. To reduce the computational complexity, particle swarm optimization (PSO) is used to search the optimal parameter combination. The main advantage of the proposed method is that it considers not only the information of object and background but also interactions between them in threshold selection procedure. Experimental results show that the proposed method can give better segmentation performance than methods based on traditional Shannon entropy.

Identification of upwelling areas on sea surface temperature images using fuzzy clustering

Sixteen sea surface temperature (SST) images obtained over the coastal ocean of Portugal during the period September 1992–September 2003 were used aiming to identify automatically the areas covered by upwelling waters. Suitable high resolution colour scales were applied to the SST images in order to enhance the thermal patterns and easily identify the waters with a coastal upwelling origin. The automatic identification of the areas covered by upwelling waters was developed by the authors in a previous work, through the application of fuzzy clustering and validation indexes, and here is explored as an oceanographic application to the Portuguese coastal upwelling. The fuzzy c-means (FCM) algorithm showed to be able to find partitions that closely defined the upwelling areas and the visualization of the fuzzy c-partitions was achieved through the application of a colour scale. The Xie-Beni validation index was used to select the c-partition that best represented the stage of the upwelling event and showed an agreement with the oceanographic interpretation in 10 of the 14 SST segmented images used in this work. Two SST images without upwelling were also used in order to check the response of the algorithm to the absence of the phenomena. The computation of the matching rate between a c-partition and the two areas split by the hand-contoured upwelling boundary also allowed the evaluation of how closely the obtained segmentation reproduced the shape of the areas covered by upwelling waters. This method successfully identified the upwelling boundary regions in 10 of the 14 SST images. The values obtained for the matching rate were higher than 0.77, thus indicating the good quality of the fuzzy partitions. The segmented images with 3 or 4 clusters were the most suitable ones to reproduce the areas covered by upwelling waters, but it was also shown that, for some cases, the upwelling areas could be reasonably well reproduced by the FCM 2-partition images. While in the latter, the area covered with upwelling waters was coincident with the first cluster, in the former, the segmented image showed two clusters within the upwelling area: the first cluster coincided with the area occupied by the most recently upwelled waters near the coast, while the second cluster was coincident with the area occupied by the “older” upwelling waters with its extensions offshore, the so-called cold filaments. The FCM algorithm revealed to be a promising technique in the automatic identification of upwelling areas on SST images.

An expert system based on fuzzy entropy for automatic threshold selection in image processing

In pattern recognition and image processing, the selection of appropriate threshold is a very significant issue. Especially, the selecting gray-level thresholds is a critical issue for many pattern recognition applications. Here, the maximum fuzzy entropy and fuzzy c-partition methods are used for the aim of the gray-level automatic threshold selection method. The fuzzy theory has been successfully applied to many areas, such as image processing, pattern recognition, computer vision, medicine, control, etc. The images have some fuzziness in nature. In this study, expert maximum fuzzy-Sure entropy (EMFSE) method for the maximum fuzzy entropy and fuzzy c-partition processes in automatic threshold selection is proposed. The experimental studies were conducted on many images by testing maximum fuzzy-Sure entropy against maximum fuzzy-Shannon entropy (MFSHE), maximum fuzzy-Havrada and Charvat entropy (MFHCE) methods for selecting optimum 2-level threshold value, respectively. The obtained experimental results show that the used MFSE method is superior to other MFSHE and MFHCE methods on selecting the 2-level threshold value automatically and effectively.

Categorical data fuzzy clustering: An analysis of local search heuristics

The fuzzy c partition of a set of qualitative data is the problem of selecting the optimal c centroids that are the most representative of the whole population. Moreover, a set of weights w ij must be determined, describing the fuzzy membership function of pattern i to the cluster represented by centroid j. Both problems are formulated by a single mathematical programming problem, that is an extension of the classic p-median models often used for clustering. The new objective function is neither concave nor convex and the application requires the clustering of many thousands of data, therefore heuristic methods are to be developed to find the best fuzzy partition. In this contribution, four methods are selected, that are implementations of meta-heuristics tested to solve p-median problems. Here, they are implemented and tested to find the optimal fuzzy c-partition. All heuristics implement neighborhood search with different strategies of visiting neighboring solutions: they are random restart method (RR), that is used in many commercial softwares and suggested in textbooks, tabu search (TS) that tries to find the best move to escape from a local optimum, variable neighborhood search (VNS), that explores quickly the solution space, candidate list search (CLS), that explores only interesting starting solutions. It is found that there is not a clear best method, but their performance depends on some parameter. TS is usually accurate, but time consuming. When c is small, VNS can be a reliable alternative, while, when c is large and there are many data to cluster, CLS provides good results. We point out that the simple RR method, that is sometimes used in commercial codes is of very poor quality: the implementation of one of the neighbor search algorithms leads to substantial improvements.