A clustering method combining differential evolution with the K-means algorithm

The present paper considers the problem of partitioning a dataset into a known number of clusters using the sum of squared errors criterion (SSE). A new clustering method, called DE-KM, which combines differential evolution algorithm (DE) with the well known K-means procedure is described. In the method, the K-means algorithm is used to fine-tune each candidate solution obtained by mutation and crossover operators of DE. Additionally, a reordering procedure which allows the evolutionary algorithm to tackle the redundant representation problem is proposed. The performance of the DE-KM clustering method is compared to the performance of differential evolution, global K-means method, genetic K-means algorithm and two variants of the K-means algorithm. The experimental results show that if the number of clusters K is sufficiently large, DE-KM obtains solutions with lower SSE values than the other five algorithms.     

Fast and robust fuzzy c-means clustering algorithms incorporating local information for image segmentation

Fuzzy c-means (FCM) algorithms with spatial constraints (FCM_S) have been proven effective for image segmentation. However, they still have the following disadvantages: (1) although the introduction of local spatial information to the corresponding objective functions enhances their insensitiveness to noise to some extent, they still lack enough robustness to noise and outliers, especially in absence of prior knowledge of the noise; (2) in their objective functions, there exists a crucial parameter α used to balance between robustness to noise and effectiveness of preserving the details of the image, it is selected generally through experience; and (3) the time of segmenting an image is dependent on the image size, and hence the larger the size of the image, the more the segmentation time. In this paper, by incorporating local spatial and gray information together, a novel fast and robust FCM framework for image segmentation, i.e., fast generalized fuzzy c-means (FGFCM) clustering algorithms, is proposed. FGFCM can mitigate the disadvantages of FCM_S and at the same time enhances the clustering performance. Furthermore, FGFCM not only includes many existing algorithms, such as fast FCM and enhanced FCM as its special cases, but also can derive other new algorithms such as FGFCM_S1 and FGFCM_S2 proposed in the rest of this paper. The major characteristics of FGFCM are: (1) to use a new factor S_ij as a local (both spatial and gray) similarity measure aiming to guarantee both noise-immunity and detail-preserving for image, and meanwhile remove the empirically-adjusted parameter α; (2) fast clustering or segmenting image, the segmenting time is only dependent on the number of the gray-levels q rather than the size N(?q) of the image, and consequently its computational complexity is reduced from O(NcI₁) to O(qcI₂), where c is the number of the clusters, I₁ and are the numbers of iterations, respectively, in the standard FCM and our proposed fast segmentation method. The experiments on the synthetic and real-world images show that FGFCM algorithm is effective and efficient.     

New neutrosophic approach to image segmentation

Neutrosophic set (NS), a part of neutrosophy theory, studies the origin, nature and scope of neutralities, as well as their interactions with different ideational spectra. NS is a formal framework that has been recently proposed. However, NS needs to be specified from a technical point of view for a given application or field. We apply NS, after defining some concepts and operations, for image segmentation.The image is transformed into the NS domain, which is described using three membership sets: T, I and F. The entropy in NS is defined and employed to evaluate the indeterminacy. Two operations, α-mean and β-enhancement operations are proposed to reduce the set indeterminacy. Finally, the proposed method is employed to perform image segmentation using a γ-means clustering. We have conducted experiments on a variety of images. The experimental results demonstrate that the proposed approach can segment the images automatically and effectively. Especially, it can segment the “clean” images and the images having noise with different noise levels.     

A time-efficient pattern reduction algorithm for k-means clustering

This paper presents an efficient algorithm, called pattern reduction (PR), for reducing the computation time of k-means and k-means-based clustering algorithms. The proposed algorithm works by compressing and removing at each iteration patterns that are unlikely to change their membership thereafter. Not only is the proposed algorithm simple and easy to implement, but it can also be applied to many other iterative clustering algorithms such as kernel-based and population-based clustering algorithms. Our experiments—from 2 to 1000 dimensions and 150 to 10,000,000 patterns—indicate that with a small loss of quality, the proposed algorithm can significantly reduce the computation time of all state-of-the-art clustering algorithms evaluated in this paper, especially for large and high-dimensional data sets.     

Fuzzy C-means based clustering for linearly and nonlinearly separable data

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.     

Mean shift 模糊C 均值聚类图像分割算法

针对传统模糊C均值(FCM)聚类算法对结构复杂图像分割效果不理想且算法执行效率较低的缺陷,提出一种融合均值平移(mean shift)的FCM聚类算法.利用mean shift算法将图像分成若干同质区域,将此区域视为新的节点;通过图像局部信息熵描述新节点的空间和灰度特征;采用能较好模拟人眼非线性视觉响应的指数函数进行相似性测度.实验结果表明,对于复杂背景图像和含噪声图像,所提出的算法在目标提取效果和执行效率上具有较强的鲁棒性.     

Fast modified global k-means algorithm for incremental cluster construction

The k-means algorithm and its variations are known to be fast clustering algorithms. However, they are sensitive to the choice of starting points and are inefficient for solving clustering problems in large datasets. Recently, incremental approaches have been developed to resolve difficulties with the choice of starting points. The global k-means and the modified global k-means algorithms are based on such an approach. They iteratively add one cluster center at a time. Numerical experiments show that these algorithms considerably improve the k-means algorithm. However, they require storing the whole affinity matrix or computing this matrix at each iteration. This makes both algorithms time consuming and memory demanding for clustering even moderately large datasets. In this paper, a new version of the modified global k-means algorithm is proposed. We introduce an auxiliary cluster function to generate a set of starting points lying in different parts of the dataset. We exploit information gathered in previous iterations of the incremental algorithm to eliminate the need of computing or storing the whole affinity matrix and thereby to reduce computational effort and memory usage. Results of numerical experiments on six standard datasets demonstrate that the new algorithm is more efficient than the global and the modified global k-means algorithms.     

Improving the performance of k-means for color quantization

Color quantization is an important operation with many applications in graphics and image processing. Most quantization methods are essentially based on data clustering algorithms. However, despite its popularity as a general purpose clustering algorithm, k-means has not received much respect in the color quantization literature because of its high computational requirements and sensitivity to initialization. In this paper, we investigate the performance of k-means as a color quantizer. We implement fast and exact variants of k-means with several initialization schemes and then compare the resulting quantizers to some of the most popular quantizers in the literature. Experiments on a diverse set of images demonstrate that an efficient implementation of k-means with an appropriate initialization strategy can in fact serve as a very effective color quantizer.     

图像分割中改进空间约束贝叶斯网络模型的应用

针对马尔可夫链蒙特卡罗方法普遍存在的迭代收敛性问题,在具有空间平滑约束的高斯混合模型条件上提出改进空间约束贝叶斯网络模型并在图像分割领域进行具体应用。所提模型应用隐狄利克雷分布（LDA）概率密度模型和高斯-马尔可夫定理的随机域参数混合过程来实现参数平滑。所提方法根据空间信息先验平滑变换操作,在待处理像素点的上下文混合结构中引入LDA符合多项式分布,用来替换传统期望最大化算法中映射操作。LDA参数采用闭合形式将有利于准确估计最大后验概率（MAP）框架与上下文混合结构的相关比例。实验结果表明,应用PRI、VoI、GCE和BDE指标进行效果比较,该方法比联合系统工程组（JSEG）、当前变换矩阵（CTM）和最大后验概率-最大似然法（MM）方法的图像分割应用效果较好,高斯噪声对于该算法的鲁棒性影响较小。     

Robust spatially constrained fuzzy c-means algorithm for brain MR image segmentation

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.     

Fuzzy homogeneity and scale-space approach to color image segmentation

Image segmentation is the procedure in which the original image is partitioned into homogeneous regions, and has many applications. In this paper, a fuzzy homogeneity and scale-space approach to color image segmentation is proposed. A color image is transformed into fuzzy domain with maximum fuzzy entropy principle. The fuzzy homogeneity histogram is employed, and both global and local informations are considered when we process fuzzy homogeneity histogram. The scale-space filter is utilized for analyzing the fuzzy homogeneity histogram to find the appropriate segments of the homogeneity histogram bounded by the local extrema of the derivatives. A fuzzy region merging process is then implemented based on color difference and cluster sizes to avoid over-segmentation. The proposed method is compared with the space domain approach, and experimental results demonstrate the effectiveness of the proposed approach.     

Fast global k-means clustering using cluster membership and inequality

In this paper, we present a fast global k-means clustering algorithm by making use of the cluster membership and geometrical information of a data point. This algorithm is referred to as MFGKM. The algorithm uses a set of inequalities developed in this paper to determine a starting point for the jth cluster center of global k-means clustering. Adopting multiple cluster center selection (MCS) for MFGKM, we also develop another clustering algorithm called MFGKM+MCS. MCS determines more than one starting point for each step of cluster split; while the available fast and modified global k-means clustering algorithms select one starting point for each cluster split. Our proposed method MFGKM can obtain the least distortion; while MFGKM+MCS may give the least computing time. Compared to the modified global k-means clustering algorithm, our method MFGKM can reduce the computing time and number of distance calculations by a factor of 3.78-5.55 and 21.13-31.41, respectively, with the average distortion reduction of 5,487 for the Statlog data set. Compared to the fast global k-means clustering algorithm, our method MFGKM+MCS can reduce the computing time by a factor of 5.78-8.70 with the average reduction of distortion of 30,564 using the same data set. The performances of our proposed methods are more remarkable when a data set with higher dimension is divided into more clusters.     

A polynomial-time algorithm for computing K-terminal residual reliability of d-trapezoid graphs

Consider a probabilistic graph G   in which the edges of E(G)$E(G)$ are perfectly reliable, but the vertices of V(G)$V(G)$ may fail with some known probabilities. Given a subset K   of V(G)$V(G)$, the K-terminal residual reliability of G is the probability that all operational vertices in K are connected to each other. This problem can be considered to be a generalization of two well-known reliability problems – the K-terminal reliability problem and the residual connectedness reliability problem.     

Equally contributory privacy-preserving k-means clustering over vertically partitioned data

In recent years, there have been numerous attempts to extend the k-means clustering protocol for single database to a distributed multiple database setting and meanwhile keep privacy of each data site. Current solutions for (whether two or more) multiparty k-means clustering, built on one or more secure two-party computation algorithms, are not equally contributory, in other words, each party does not equally contribute to k-means clustering. This may lead a perfidious attack where a party who learns the outcome prior to other parties tells a lie of the outcome to other parties. In this paper, we present an equally contributory multiparty k-means clustering protocol for vertically partitioned data, in which each party equally contributes to k-means clustering. Our protocol is built on ElGamal's encryption scheme, Jakobsson and Juels's plaintext equivalence test protocol, and mix networks, and protects privacy in terms that each iteration of k-means clustering can be performed without revealing the intermediate values.     

区域化模糊均值高分辨率彩色遥感影像分割方法

基于像素模糊?? 均值算法(FCM) 及其改进算法难以解决高分辨率遥感影像中地物目标光谱测度相似性减弱和几何噪声增大带来的分割难题, 提出一种基于区域的FCM算法. 该方法利用Voronoi 几何划分将影像域划分为子区域, 并用子区域拟合地物目标的几何形状. 在此基础上, 定义区域FCM目标函数, 通过迭代最小化该目标函数实现高分辨率遥感影像分割. 实验结果表明, 与基于像素的FCM和增强FCM方法相比, 所提出方法可以更加精确地实现高分辨率遥感影像分割.

     

Global structure constrained local shape prior estimation for medical image segmentation

Organ shape plays an important role in clinical diagnosis, surgical planning and treatment evaluation. Shape modeling is a critical factor affecting the performance of deformable model based segmentation methods for organ shape extraction. In most existing works, shape modeling is completed in the original shape space, with the presence of outliers. In addition, the specificity of the patient was not taken into account. This paper proposes a novel target-oriented shape prior model to deal with these two problems in a unified framework. The proposed method measures the intrinsic similarity between the target shape and the training shapes on an embedded manifold by manifold learning techniques. With this approach, shapes in the training set can be selected according to their intrinsic similarity to the target image. With more accurate shape guidance, an optimized search is performed by a deformable model to minimize an energy functional for image segmentation, which is efficiently achieved by using dynamic programming. Our method has been validated on 2D prostate localization and 3D prostate segmentation in MRI scans. Compared to other existing methods, our proposed method exhibits better performance in both studies.     

A differential-processing extraction approach to text and image segmentation

To efficiently store the information found in paper documents, text and non-text regions need to be separated. Non-text regions include half-tone photographs and line diagrams. The text regions can be converted (via an optical character reader) to a computer-searchable form, and the non-text regions can be extracted and preserved in compressed form using image-compression algorithms. In this paper, an effective system for automatically segmenting a document image into regions of text and non-text is proposed. The system first performs an adaptive thresholding to obtain a binarized image. Subsequently the binarized image is smeared using a run-length differential algorithm. The smeared image is then subjected to a text characteristic filter to remove error smearing of non-text regions. Next, baseline cumulative blocking is used to rectangularize the smeared region. Finally, a text block growing algorithm is used to block out a text sentence. The recognition of text is carried out on a text sentence basis.