# A Robust Bias-Correction Fuzzy Weighted C-Ordered-Means Clustering Algorithm

Authors: Wenyuan Zhang; Tianyu Huang; Jun Chen
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

This paper proposes a modified fuzzy C-means (FCM) algorithm, which combines the local spatial information and the typicality of pixel data in a new fuzzy way. This new algorithm is called bias-correction fuzzy weighted C-ordered-means (BFWCOM) clustering algorithm. It can overcome the shortcomings of the existing FCM algorithm and improve clustering performance. The primary task of BFWCOM is the use of fuzzy local similarity measures (space and grayscale). Meanwhile, this new algorithm adds a typical analysis of data attributes to membership, in order to ensure noise insensitivity and the preservation of image details. Secondly, the local convergence of the proposed algorithm is mathematically proved, providing a theoretical preparation for fuzzy classification. Finally, data classification and real image experiments show the effectiveness of BFWCOM clustering algorithm, having a strong denoising and robust effect on noise images.


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## Body

## 1. Introduction

Clustering analysis is an important technique in data analysis, having a wide range of applications in data mining technique [1], image processing [2, 3], computer vision [4], and artificial intelligence [5]. Image segmentation is a type of image processing, dividing an image into specific nonoverlapping regions with unique characteristics, and putting forward goals of interest. It is a key step from image processing to image analysis. The existing image segmentation methods are mainly divided into the following categories: threshold-based segmentation method, region-based segmentation method, edge-based segmentation method, specific theory-based segmentation method, etc. Specifically, these methods include clustering [6, 7], region growing [8], Watershed Transformation [9], active contour model [10], MeanShift [11], Graph Cut [12], spectral clustering [13], Markov random field [14], Neural network [15], etc. The process of image segmentation is also a marking process, in which pixels belonging to the same area are given the same number. Image segmentation has always been one of the most challenging tasks in image processing and computer vision [16, 17]. Nowadays, there have been many methods for image segmentation [18–21]; however, they are not robust and effective enough for a large number of different images.How to segment a given image? And how to segment it in a rational way? These are the major problems which need to be solved by the clustering algorithm. A solution to such problems: under a certain measure, the data in classes is made to have as short distance within classes as possible and as large distance between classes as possible and meanwhile the summary of whole distance is minimum. FCM clustering algorithm was first proposed by Dunn [22, 23] and later developed by Bezdek [24–30]. It is a classical clustering algorithm, a combination of clustering analysis and fuzzy theory. It is an unsupervised machine learning algorithm, which can be divided by constantly modifying the clustering center and membership matrix. FCM algorithm is sensitive to noise due to its restricted condition that total membership of elements to all centers is 1. Thus, local spatial information is necessarily introduced into the objective function to improve the robustness of the FCM algorithm for image segmentation. This paper proposes a new algorithm to make the algorithm more robust by a detailed study of FCM clustering algorithm. Its formula is as (∗).


### 1.1. Introduction to FCM Clustering with Local Spatial Information

The traditional Hard C-means (HCM) clustering algorithm is very simple (the membership degree is only 0 or 1) [31], resulting in poor image segmentation, while FCM is more tolerant of ambiguity and retain more original image information. In addition, it is effective for images with simple texture and background. Therefore, FCM is better than HCM. The shortcoming of FCM is that it only takes the gray information into consideration while ignoring the spatial information. This algorithm cannot segment images with complex texture and background, because it cannot avoid the noise interference. In order to solve the problems mentioned above, many scholars have proposed to introduce local spatial information into the objective function. For example, Tolias [32] proposed to add spatial information to membership degree. Ahmed [33] (1999) and Pham [34] (2001) proposed a bias-correction FCM algorithm. Considering that the membership degree of the central pixel is affected by the pixels in the neighborhood, this algorithm adds the neighborhood pixel information to the objective function of the FCM algorithm as a penalty term and attempts to make neighborhood pixels and central pixels produce more consistent clustering results in the segmentation process. Liew [35] used a new neighborhood weighted dissimilarity measure for the Euclidean distance in the FCM objective function. Compared with FCM, this algorithm uses more spatial information, which can better realize the clustering and segmentation of image data. In order to make the FCM clustering algorithm perform better, Szilĺćgyi [36] introduced a new factor, starting from the standard FCM and its bias-corrected version. An enhanced fuzzy C-means clustering (EnFCM) algorithm was proposed which can reduce the amount of computation required, and provides a fast way to segment high quality brain images. Wu [37] replaced the Euclidean norm with an exponential function in C-means clustering, greatly improving the antinoise ability of the algorithm. Chen and Zhang [38] proposed a FCM clustering algorithm based on median filtering or mean filtering and a kernel method for improving the algorithm. A class of robust non-Euclidean distance measures is introduced to derive new objective functions, enhancing the robustness of the original clustering algorithm to noise and outliers. Yang et al. [39] proposed a Gaussian kernel-based FCM (GKFCM) algorithm, a generalization of FCM, BCFCM, KFCMS1, and KFCMS2 algorithms, having higher efficiency and robustness. Cai, Chen, and Zhang [40] proposed a fast generalized FCM clustering algorithm. It uses new factors as a measure for local similarity (space and grayscale), thus ensuring the noise immunity and details of images and removing the global parameter α. However, there is still a parameter that needs to be experimentally determined in the algorithm. Wang et al. [41] proposed local and nonlocal FCM clustering algorithms to reduce the effects of noise in the image segmentation process. By using a new dissimilarity index to replace the usual distance measures, this method incorporates local spatial context and nonlocal information into the standard FCM clustering algorithm, where the image local information takes into account the weighted distance between the pixel and the class center.The target criteria and differences of FCM, FCMS, FLICM, FCMS1, and FCMS2 are presented in Table1.Table 1
Noise immunity fuzzy clustering algorithm table.
Fuzzy  clustering The objective function FCM J m =  ∑ i = 1 C ∑ k = 1 N u i k m x k - v i 2 . (∗)  FCMS  FLICM  FCMS1   FCMS2 J m =  ∑ i = 1 c ∑ k = 1 n u i k m x k - v i 2 +  ∑ i = 1 c ∑ k = 1 n G i k .  (∗∗)   FCMS G i k = α N R u i k m  ∑ r ∈ N k x r - v i 2 Among them: α is a parameter used to control the effect of the neighborhood terms. NR is the cardinality of uik, xr is the neighbor of, Ni is the neighbor set in the window around.   FLICM G i k =  ∑ r ∈ N i i ≠ r 1 d i r + 1 1 - u r k m x r - v k 2 .  (∗∗∗)Among them: dir represents the spatial Euclidean distance between pixels vi and xr. Obviously, Gik is more complicated than in FCMS, so FLICM has a higher computational complexity than FCMS.  FCMS1  FCMS2 G i k = α u i k m x - k - v i 2 .  (∗∗∗∗)Among them: x-k is the average value or median value of adjacent pixels within the window surrounding. Gik have a more simplified form in FCMS1 and FCMS2 than the FCMS, and the clustering time can be reduced because α∑r∈Nixr-vk/NR is replaced by αx-k-vi2.Spatial neighborhood information is of great significance for image segmentation. FCMS and EnFCM [39] are good algorithms due to their short calculation time, because it performs a Gray Histogram-Based clustering rather than summing the pixels of images. The number of gray levels in an image is usually much smaller than that of pixels, and the calculation time is very short. However, only the neighborhood gray value information is added to the clustering process, while the distance information of the spatial neighborhood points is not taken into account. The effect of all neighborhood points on the central point is the same, which is obviously not in line with the actual situation. The neighborhood points that are closer to the central point should have more effect on the central point, while the neighborhood points that are farther away from the central point should have less effect on it.However, when EnFCM algorithm constructs linear weighted sum images, the lack of spatial information makes the image segmentation results sensitive to salt and pepper noise. In order to improve the segmentation results obtained by EnFCM, Cai et al. [40] proposed a fast generalized FCM algorithm (FGFCM). Inspired by the EnFCM algorithm, the FGFCM algorithm first redefines the linear weighted sum images by utilizing grayscale and spatial information of each pixel’s neighborhood window of the original image and then performs clustering on its gray histogram. This algorithm introduces a new factor as a local similarity measure, which is aimed to ensure the noise immunity and detail retention of image segmentation. Meanwhile, the empirical adjustment parameter α required in EnFCM is removed, and finally a gray histogram is clustered. The FGFCM algorithm can further improve the balance between noise and detail, enhance the robustness of the algorithm to noise, and improve the computational efficiency of FCM to some extent. However, it requires more parameters than EnFCM. The contribution strength of the spatial neighborhood information of FGFCM algorithm is hard to control and needs to be set manually. In addition, the strength value is a global variable, and the distribution of noise cannot be fully considered.In order to develop a new FCM algorithm, Krinidis and Chatzis [42] proposed a robust fuzzy local information C-mean (FLICM) clustering algorithm. The FLICM algorithm has the following advantages: first, Gij (fuzzy factors) not containing parameters other than necessary m and C is used. Therefore, there is no need to adjust parameters (such as parameters in the EnFCM algorithm) to balance image noise and image detail. In addition, compared with the traditional FCM algorithm, it can greatly improve the segmentation effect for images with noise by using spatial and grayscale information. It replaces parameters used in the EnFCM with a new fuzzy factor, which can be translated into an objective function that ensure the noise immunity and image detail preservation. FLICM overcomes the problem of parameter selection and improves image segmentation performance, but it is not robust enough for local information of different images and fixed spatial distance. Meanwhile, FLICM uses variance information in the pixel neighborhood to modify the fuzzy weight factors, thus improving the robustness and noise immunity of this algorithm. However, the image denoising effect is still poor for weak contrast images.Gong et al. [43] used variable local coefficients instead of fixed spatial distances and proposed a variant of FLICM (the formula as shown in Table 1  (∗∗∗)). A modified robust FLICM clustering algorithm is obtained by replacing Gij with a newly constructed fuzzy factor Gij′, which can utilize more local context information in the image. In order to further improve the segmentation performance of the FLICM algorithm, Gong et al. [43] proposed a fuzzy factor that links local spatial information with local gray information to obtain a new algorithm called kernel weighted fuzzy local information C-means (KWFLICM). In addition, the fuzzy weighting factor is modified by variance information in the pixel neighborhood to adaptively control the local spatial relationship, thus enhancing the robustness of FLICM to noise and outliers. The KWFLICM algorithm replaces the Euclidean distance measures in the FLICM algorithm with the nuclear distance measures and proposes a new local information weighted fuzzy factor to improve the antinoise performance of this algorithm. However, KWFLICM has a higher computational complexity than FLICM.Ahmed et al. [33] proposed an adaptively regularized kernel-based FCM framework (ARKFCM), aiming at the problem of the segmentation of brain magnetic resonance image. This algorithm utilizes the heterogeneity of grayscales in the neighborhood to obtain the local context information and replaces the Standard Euclidean distance with the Gaussian radial basis kernel functions. In this way, the robustness of preserving the image details is improved, the independence of cluster parameters is enhanced, and the computing costs are reduced. Lei T et al. [44] proposed an improved FCM algorithm based on morphological reconstruction and membership filtering (FRFCM), which is faster and more robust than FCM algorithm. By introducing morphological reconstruction operation, the local spatial information is incorporated into the FRFCM algorithm to ensure the antinoise ability and image detail preservation.Saranathan [45] introduced the nonlocal spatial information into the objective function by using variant parameters adaptive to the noise level of each pixel of the image, which has been extended to FCM to overcome the parameter selection to improve the robustness to noise. The FGFCM algorithm and the FLICM algorithm both use coordinate information to add spatial distance information to the clustering process. Zhao et al. [46] believe that the Euclidean distance cannot reflect the spatial information of the image well. Therefore, they proposed a neighborhood weighted fuzzy C-means (NWFCM) clustering algorithm, namely, Neighborhood Weight (NW) distance to reduce the running time of FLICM and KWFLICM and strengthen the robustness of the algorithm to noise. Although NWFCM uses more parameters, NWFCM is faster than FLICM and KWFLICM because image filtering is performed before the iteration begins.However, the NWFCM algorithm is still time-consuming due to the calculation of the NW distance and the parameter selection. Guo et al. [47] proposed an adaptive Noise Detection-based FCM (NDFCM). This algorithm overcomes the above-mentioned shortcoming, in which the trade-off parameters are automatically adjusted by measuring the local variance of the gray scale.


### 1.2. Introduction to Fuzzy C-Ordered-Means Clustering Algorithm

Fuzzy C-ordered-means (FCOM) [48] clustering algorithm combines FCM clustering with robust ordered statistics, in which Huber’s M-estimation [49] and Yager’s ordered weighted averaging (OWA) operators [50] are used together for fuzzy clustering to significantly improve its robustness. The typicality of each data item is calculated instead of attribute empowerment. The data items are sorted, and their typicality is updated in each iteration of the clustering process. The data closer to the cluster center are of high typicality, while those farther from the cluster center low typicality. In this way, outliers and noisy data items are classified as low typicality without damaging the clustering process. Thus this method has better performance in terms of noise and outlier data. Its formula is as shown in (1).Fuzzy weighted C-ordered-means (FWCOM) clustering algorithm [51] proposed a new fuzzy weighted C-ordered-means clustering algorithm based on FCOM. This algorithm looks for clusters within the fuzzy subspace of the original task space and assigns weights to dimensions (attributes) in each cluster. The weight is a number from the unit interval [0,1]. In order to enhance the robustness of the algorithm to outliers and noise, it is combined with sorting techniques.The formula of FWCOM [51] βik∈[0,1] is as follows:(1)JU,V=∑i=1c∑k=1nβikuikm∑d=1pxkd-vid2The restricted condition is as follows:(2)∀k∈X∑i=1cβikuik=fkIts membership degree formula is as follows:(3)uik=fi∑d=1pxid-vkd21/1-m∑i=1cβik∑d=1pxid-vkd21/1-mThe c-th clustering center of the d-th attribute is calculated using the following formula:(4)vcd=∑k=1nβckuikmheckdxkd∑k=1nβckuikmheckdFor the above-mentioned algorithm, this paper proposes an adaptive filtering based fuzzy clustering algorithm by combining the FWCOM clustering algorithm with the FCM clustering algorithm of spatial information. This algorithm first determines the nonlocal parameter balance factor according to the strength of the local spatial information, meanwhile calculating the typicality of each data item, sorting the data items, and updating their typicalities in each iteration of the clustering process, with an aim to more accurately reflect the spatial structure information contained in the image. Then it uses this balance factor to make an effective combination of the filtered image and the median filtered image of the original image. The obtained adaptive filtered image adaptively determines the degree of filtering according to the noise intensity, thus improving the algorithm’s ability to suppress noise and the robustness of the algorithm.Table2 lists symbols used in the paper.Table 2
Symbols used in the paper.
Symbol Meaning c Number of clusters X Set of data items,X=x1,x2,⋯,xn n Number of data items,n=X x i Data item,xi=xi1,xi2,⋯,xipT p Number of attributes x r the neighbor of,xi N R the cardinality of,uki N i the neighbor set in the window aroundxi α a parameter used to control the effect of the neighborhood terms U Membership matrixc×n u i k Membership of the i-th item to the k-th cluster V Matric of cluster centers,V=v1,v2,⋯,vcT v i Centre of i-th cluster,vi=vi1,vi2,…,vipT β i k Typicality of the k-th data item with respect to i-th cluster f k Global typicality of the k data item, cf. Eq.(42) m Weighting exponent for memberships e i d k Residual of d-th of k-th datum from the centre of i-th cluster ♢ S-norm, fc. Eq.(42) η k the bias field at the k-th pixel


## 1.1. Introduction to FCM Clustering with Local Spatial Information

The traditional Hard C-means (HCM) clustering algorithm is very simple (the membership degree is only 0 or 1) [31], resulting in poor image segmentation, while FCM is more tolerant of ambiguity and retain more original image information. In addition, it is effective for images with simple texture and background. Therefore, FCM is better than HCM. The shortcoming of FCM is that it only takes the gray information into consideration while ignoring the spatial information. This algorithm cannot segment images with complex texture and background, because it cannot avoid the noise interference. In order to solve the problems mentioned above, many scholars have proposed to introduce local spatial information into the objective function. For example, Tolias [32] proposed to add spatial information to membership degree. Ahmed [33] (1999) and Pham [34] (2001) proposed a bias-correction FCM algorithm. Considering that the membership degree of the central pixel is affected by the pixels in the neighborhood, this algorithm adds the neighborhood pixel information to the objective function of the FCM algorithm as a penalty term and attempts to make neighborhood pixels and central pixels produce more consistent clustering results in the segmentation process. Liew [35] used a new neighborhood weighted dissimilarity measure for the Euclidean distance in the FCM objective function. Compared with FCM, this algorithm uses more spatial information, which can better realize the clustering and segmentation of image data. In order to make the FCM clustering algorithm perform better, Szilĺćgyi [36] introduced a new factor, starting from the standard FCM and its bias-corrected version. An enhanced fuzzy C-means clustering (EnFCM) algorithm was proposed which can reduce the amount of computation required, and provides a fast way to segment high quality brain images. Wu [37] replaced the Euclidean norm with an exponential function in C-means clustering, greatly improving the antinoise ability of the algorithm. Chen and Zhang [38] proposed a FCM clustering algorithm based on median filtering or mean filtering and a kernel method for improving the algorithm. A class of robust non-Euclidean distance measures is introduced to derive new objective functions, enhancing the robustness of the original clustering algorithm to noise and outliers. Yang et al. [39] proposed a Gaussian kernel-based FCM (GKFCM) algorithm, a generalization of FCM, BCFCM, KFCMS1, and KFCMS2 algorithms, having higher efficiency and robustness. Cai, Chen, and Zhang [40] proposed a fast generalized FCM clustering algorithm. It uses new factors as a measure for local similarity (space and grayscale), thus ensuring the noise immunity and details of images and removing the global parameter α. However, there is still a parameter that needs to be experimentally determined in the algorithm. Wang et al. [41] proposed local and nonlocal FCM clustering algorithms to reduce the effects of noise in the image segmentation process. By using a new dissimilarity index to replace the usual distance measures, this method incorporates local spatial context and nonlocal information into the standard FCM clustering algorithm, where the image local information takes into account the weighted distance between the pixel and the class center.The target criteria and differences of FCM, FCMS, FLICM, FCMS1, and FCMS2 are presented in Table1.Table 1
Noise immunity fuzzy clustering algorithm table.
Fuzzy  clustering The objective function FCM J m =  ∑ i = 1 C ∑ k = 1 N u i k m x k - v i 2 . (∗)  FCMS  FLICM  FCMS1   FCMS2 J m =  ∑ i = 1 c ∑ k = 1 n u i k m x k - v i 2 +  ∑ i = 1 c ∑ k = 1 n G i k .  (∗∗)   FCMS G i k = α N R u i k m  ∑ r ∈ N k x r - v i 2 Among them: α is a parameter used to control the effect of the neighborhood terms. NR is the cardinality of uik, xr is the neighbor of, Ni is the neighbor set in the window around.   FLICM G i k =  ∑ r ∈ N i i ≠ r 1 d i r + 1 1 - u r k m x r - v k 2 .  (∗∗∗)Among them: dir represents the spatial Euclidean distance between pixels vi and xr. Obviously, Gik is more complicated than in FCMS, so FLICM has a higher computational complexity than FCMS.  FCMS1  FCMS2 G i k = α u i k m x - k - v i 2 .  (∗∗∗∗)Among them: x-k is the average value or median value of adjacent pixels within the window surrounding. Gik have a more simplified form in FCMS1 and FCMS2 than the FCMS, and the clustering time can be reduced because α∑r∈Nixr-vk/NR is replaced by αx-k-vi2.Spatial neighborhood information is of great significance for image segmentation. FCMS and EnFCM [39] are good algorithms due to their short calculation time, because it performs a Gray Histogram-Based clustering rather than summing the pixels of images. The number of gray levels in an image is usually much smaller than that of pixels, and the calculation time is very short. However, only the neighborhood gray value information is added to the clustering process, while the distance information of the spatial neighborhood points is not taken into account. The effect of all neighborhood points on the central point is the same, which is obviously not in line with the actual situation. The neighborhood points that are closer to the central point should have more effect on the central point, while the neighborhood points that are farther away from the central point should have less effect on it.However, when EnFCM algorithm constructs linear weighted sum images, the lack of spatial information makes the image segmentation results sensitive to salt and pepper noise. In order to improve the segmentation results obtained by EnFCM, Cai et al. [40] proposed a fast generalized FCM algorithm (FGFCM). Inspired by the EnFCM algorithm, the FGFCM algorithm first redefines the linear weighted sum images by utilizing grayscale and spatial information of each pixel’s neighborhood window of the original image and then performs clustering on its gray histogram. This algorithm introduces a new factor as a local similarity measure, which is aimed to ensure the noise immunity and detail retention of image segmentation. Meanwhile, the empirical adjustment parameter α required in EnFCM is removed, and finally a gray histogram is clustered. The FGFCM algorithm can further improve the balance between noise and detail, enhance the robustness of the algorithm to noise, and improve the computational efficiency of FCM to some extent. However, it requires more parameters than EnFCM. The contribution strength of the spatial neighborhood information of FGFCM algorithm is hard to control and needs to be set manually. In addition, the strength value is a global variable, and the distribution of noise cannot be fully considered.In order to develop a new FCM algorithm, Krinidis and Chatzis [42] proposed a robust fuzzy local information C-mean (FLICM) clustering algorithm. The FLICM algorithm has the following advantages: first, Gij (fuzzy factors) not containing parameters other than necessary m and C is used. Therefore, there is no need to adjust parameters (such as parameters in the EnFCM algorithm) to balance image noise and image detail. In addition, compared with the traditional FCM algorithm, it can greatly improve the segmentation effect for images with noise by using spatial and grayscale information. It replaces parameters used in the EnFCM with a new fuzzy factor, which can be translated into an objective function that ensure the noise immunity and image detail preservation. FLICM overcomes the problem of parameter selection and improves image segmentation performance, but it is not robust enough for local information of different images and fixed spatial distance. Meanwhile, FLICM uses variance information in the pixel neighborhood to modify the fuzzy weight factors, thus improving the robustness and noise immunity of this algorithm. However, the image denoising effect is still poor for weak contrast images.Gong et al. [43] used variable local coefficients instead of fixed spatial distances and proposed a variant of FLICM (the formula as shown in Table 1  (∗∗∗)). A modified robust FLICM clustering algorithm is obtained by replacing Gij with a newly constructed fuzzy factor Gij′, which can utilize more local context information in the image. In order to further improve the segmentation performance of the FLICM algorithm, Gong et al. [43] proposed a fuzzy factor that links local spatial information with local gray information to obtain a new algorithm called kernel weighted fuzzy local information C-means (KWFLICM). In addition, the fuzzy weighting factor is modified by variance information in the pixel neighborhood to adaptively control the local spatial relationship, thus enhancing the robustness of FLICM to noise and outliers. The KWFLICM algorithm replaces the Euclidean distance measures in the FLICM algorithm with the nuclear distance measures and proposes a new local information weighted fuzzy factor to improve the antinoise performance of this algorithm. However, KWFLICM has a higher computational complexity than FLICM.Ahmed et al. [33] proposed an adaptively regularized kernel-based FCM framework (ARKFCM), aiming at the problem of the segmentation of brain magnetic resonance image. This algorithm utilizes the heterogeneity of grayscales in the neighborhood to obtain the local context information and replaces the Standard Euclidean distance with the Gaussian radial basis kernel functions. In this way, the robustness of preserving the image details is improved, the independence of cluster parameters is enhanced, and the computing costs are reduced. Lei T et al. [44] proposed an improved FCM algorithm based on morphological reconstruction and membership filtering (FRFCM), which is faster and more robust than FCM algorithm. By introducing morphological reconstruction operation, the local spatial information is incorporated into the FRFCM algorithm to ensure the antinoise ability and image detail preservation.Saranathan [45] introduced the nonlocal spatial information into the objective function by using variant parameters adaptive to the noise level of each pixel of the image, which has been extended to FCM to overcome the parameter selection to improve the robustness to noise. The FGFCM algorithm and the FLICM algorithm both use coordinate information to add spatial distance information to the clustering process. Zhao et al. [46] believe that the Euclidean distance cannot reflect the spatial information of the image well. Therefore, they proposed a neighborhood weighted fuzzy C-means (NWFCM) clustering algorithm, namely, Neighborhood Weight (NW) distance to reduce the running time of FLICM and KWFLICM and strengthen the robustness of the algorithm to noise. Although NWFCM uses more parameters, NWFCM is faster than FLICM and KWFLICM because image filtering is performed before the iteration begins.However, the NWFCM algorithm is still time-consuming due to the calculation of the NW distance and the parameter selection. Guo et al. [47] proposed an adaptive Noise Detection-based FCM (NDFCM). This algorithm overcomes the above-mentioned shortcoming, in which the trade-off parameters are automatically adjusted by measuring the local variance of the gray scale.


## 1.2. Introduction to Fuzzy C-Ordered-Means Clustering Algorithm

Fuzzy C-ordered-means (FCOM) [48] clustering algorithm combines FCM clustering with robust ordered statistics, in which Huber’s M-estimation [49] and Yager’s ordered weighted averaging (OWA) operators [50] are used together for fuzzy clustering to significantly improve its robustness. The typicality of each data item is calculated instead of attribute empowerment. The data items are sorted, and their typicality is updated in each iteration of the clustering process. The data closer to the cluster center are of high typicality, while those farther from the cluster center low typicality. In this way, outliers and noisy data items are classified as low typicality without damaging the clustering process. Thus this method has better performance in terms of noise and outlier data. Its formula is as shown in (1).Fuzzy weighted C-ordered-means (FWCOM) clustering algorithm [51] proposed a new fuzzy weighted C-ordered-means clustering algorithm based on FCOM. This algorithm looks for clusters within the fuzzy subspace of the original task space and assigns weights to dimensions (attributes) in each cluster. The weight is a number from the unit interval [0,1]. In order to enhance the robustness of the algorithm to outliers and noise, it is combined with sorting techniques.The formula of FWCOM [51] βik∈[0,1] is as follows:(1)JU,V=∑i=1c∑k=1nβikuikm∑d=1pxkd-vid2The restricted condition is as follows:(2)∀k∈X∑i=1cβikuik=fkIts membership degree formula is as follows:(3)uik=fi∑d=1pxid-vkd21/1-m∑i=1cβik∑d=1pxid-vkd21/1-mThe c-th clustering center of the d-th attribute is calculated using the following formula:(4)vcd=∑k=1nβckuikmheckdxkd∑k=1nβckuikmheckdFor the above-mentioned algorithm, this paper proposes an adaptive filtering based fuzzy clustering algorithm by combining the FWCOM clustering algorithm with the FCM clustering algorithm of spatial information. This algorithm first determines the nonlocal parameter balance factor according to the strength of the local spatial information, meanwhile calculating the typicality of each data item, sorting the data items, and updating their typicalities in each iteration of the clustering process, with an aim to more accurately reflect the spatial structure information contained in the image. Then it uses this balance factor to make an effective combination of the filtered image and the median filtered image of the original image. The obtained adaptive filtered image adaptively determines the degree of filtering according to the noise intensity, thus improving the algorithm’s ability to suppress noise and the robustness of the algorithm.Table2 lists symbols used in the paper.Table 2
Symbols used in the paper.
Symbol Meaning c Number of clusters X Set of data items,X=x1,x2,⋯,xn n Number of data items,n=X x i Data item,xi=xi1,xi2,⋯,xipT p Number of attributes x r the neighbor of,xi N R the cardinality of,uki N i the neighbor set in the window aroundxi α a parameter used to control the effect of the neighborhood terms U Membership matrixc×n u i k Membership of the i-th item to the k-th cluster V Matric of cluster centers,V=v1,v2,⋯,vcT v i Centre of i-th cluster,vi=vi1,vi2,…,vipT β i k Typicality of the k-th data item with respect to i-th cluster f k Global typicality of the k data item, cf. Eq.(42) m Weighting exponent for memberships e i d k Residual of d-th of k-th datum from the centre of i-th cluster ♢ S-norm, fc. Eq.(42) η k the bias field at the k-th pixel


## 2. A New Algorithm BFWCOM with Incorporating Spatial Information and Fuzzy Weights

A robust algorithm should have the following properties: (1) It should have fairly good precision in the assumed model. (2) Small deviations from the model assumption should only cause damage to a small amount of performance. (3) Large deviations from the model assumption should not cause disaster.The standard FCM objective function for partitioningX=xii=1n into C cluster: this method is also used for clustering in the paper.


### 2.1. Criterion Function for BFWCOM Algorithm

The FCM clustering algorithm uses a quadratic loss function as a different measure between the data and the cluster center. The reason for using this method is for simplicity and low computational burden. However, this method is sensitive to noise and outliers. In many literatures, there are many suggestions for robust loss functions.For example, Huber proposed a mathematical formula [49, 50]:(5)τHUBe=e2δ2,e≤δeδ2,e>δwhereδ>0 represents a parameter. Another well-known robust loss function is a logarithmic function. (6)τHUBe=0,e=0log1+e2,e≠0UseD(xk,vi)=τxk-vi as a discrepancy measure between the k-th data and the i-th prototype center and an additional weighting. The criterion function for BFWCOM is as follows: (7)JU,V=∑i=1c∑k=1nβikuikmDxk,vi+∑i=1c∑k=1nαNRuikm∑r∈Nixr-vi2N R is the cardinality of uki, xr is the neighbor of xi, Ni is the neighbor set in the window around xi. α is a parameter used to control the effect of the neighborhood terms. The relative importance of the adjustment terms is inversely proportional to the signal-to-noise ratio (SNR). A lower signal-to-noise ratio requires a higher parameter value.The area of application to which the criterion function faces is as follows:(8)Mfnc=U∈RcN∣∀uik∈0,1;∑i=1cβikuik=fk;0<∑k=1Nuik<N


### 2.2. The Laplace Algorithm Used to Determine the Membership Degree and the Central Formula

#### 2.2.1. The Membership Degree Formula

The Laplace algorithm used to give the specific formula is as follows:(9)GU,V=∑i=1c∑k=1nβikuikmDxk,vi+∑i=1c∑k=1nαNRuikmSDxr,vi-λ∑i=1cβikuik-fkwhereSD(xr,vi)=∑r∈Nixr-vi, xr is the neighbor of xi, and D(xk,vi) is the distance from the neighbor point xk to the center vi.Take the partial derivative of formula (9) with respect to uik: (10)∂GU,V∂uik=mβikuikm-1Dxk,vi+mαNRuikm-1SDxr,vi-λβik=0(11)λm=uikm-1Dxk,vi+α/NRSDxr,viβikThe membership degreeuik is derived: (12)uik=λm1/m-1Dxk,vi+α/NRSDxr,viβik1/1-m(13)Dxk,vi=∑d=1pxkd-vid2Available from formula (8), (14)∑i=1cβikuik=fkSubstitute formula (12) into formula (14): (15)fk=∑i=1cβikuik=∑i=1cβikλm1/m-1Dxk,vi+α/NRSDxr,viβik1/1-mThe following formula is derived from formula (15): (16)λm1/m-1=fk∑i=1cβikDxk,vi+α/NRSDxr,viβik1/1-m-1The obtained membership degree formula is as follows:(17)uik=fkDxk,vi+α/NRSDxr,viβik-11/1-m∑i=1cβikDxk,vi+α/NRSDxr,viβik-11/1-m


#### 2.2.2. The Clustering Center Formula

The distance formulaD(xk,vi) can also be written as (18)Dxk,vi=xk-viTAxk-viFormula (7) can be written as follows: (19)JU,V=∑i=1c∑k=1nβikuikmxk-viTAxk-vi+∑i=1c∑k=1nαNRuikm∑r∈Nixr-viTAxr-viTake the derivative of formula (19) with respect to vi: (20)∂J∂vi=-2A∑k=1nβikuikmxk-vi-2A∑k=1nαNRuikm∑r∈Nixr-vi=0The clustering center formula as follows can be obtained by deriving formula (20): (21)vi=∑k=1nuikmβikxk+α/NR∑r∈Nixr∑k=1nuikmβik+α


### 2.3. Clustering Prototype Updating

The objective functionJU,V can be minimized in a manner similar to the standard FCM algorithm. The first-order derivatives of the objective function JU,V with respect to uki, vi, and ηk are set 0, resulting in three necessary but insufficient conditions for local extrema. In the following subsections, we will obtain these three conditions.According to literature [33], the additive bias field formula yk=xk+ηk, ∀k∈{1,2,…,N} is given, where xk and yk are the true and observed log-transformed intensities at the kth pixel, respectively, and ηk is the bias field at the kth pixel. (22)JmU,V=∑i=1c∑k=1nβikuikmDik+αNRuikmγiwhere(23)Dik=yk-ηk-vi2(24)γi=∑r∈Niyr-ηr-vi2Use Laplace’s theorem:(25)Gm=∑i=1c∑k=1nβikuikmDik+αNRuikmγi-λ∑i=1cβikuik-fkTake the derivative ofuik, 1≤i≤n; 1≤k≤c: (26)∂Gm∂uik=mβikuikm-1Dik+mαNRuikm-1γi-λβik=0Sort out the formula:(27)uik=λm1/m-1Dik+αγiNRβik1/1-mSort out the formula by substituting∑i=1cβikuik=fk: (28)∑i=1cβikλm1/m-1Dik+αγiNRβik1/1-m=fk(29)λm1/m-1=fk∑i=1cβikDik+αγiNRβik1/1-m-1The following formula is available from formula (27) and by substituting formula (29): (30)uik=fkDik+αγi/NRβik1/1-m∑i=1cβikDik+αγi/NRβik1/1-mAvailable from formulas (23) and (24), formula (19) can also be written as follows: (31)Jm=∑i=1c∑k=1nβikuikmDyk-ηk,vi+αNRuikm∑r∈NiDyr-ηr,viamong (32)Dyk-ηk,vi=yk-ηk-viTAyk-ηk-viDyr-ηr,vi=yr-ηr-viTAyr-ηr-viUse the Laplace’s method: (33)Gm=∑i=1c∑k=1nβikuikmDyk-ηk,vi+αNRuikm∑r∈NiDyr-ηr,vi+λ∑i=1cβikuik-fkTake the derivative of(34)∂Gm∂vi=-2A∑k=1nβikuikmyk-ηk-vi-2A∑k=1nαNRuikm∑r∈Niyr-ηr-vi=0The clustering center formula as follows can be obtained by deriving formula (34): (35)vi=∑k=1nuikmβikyk-ηk+α/NR∑r∈Niyr-ηr∑k=1nuikmβik+α


### 2.4. Bias Field Estimation

In a similar manner, take the derivative ofJm with respect to ηk, and the result is set to 0.(36)∑i=1c∂∂ηk∑k=1nβikuikmDik+αNRuikmγi=0Because only the k-th term in the second sum depends on, then we can have(37)∑i=1c∂∂ηkβikuikmyk-ηk-vi2=0Differentiate the distance function:(38)∑i=1cβikuikmyk-ηk-vi=0Therefore, the zero gradient condition of the bias field estimator is expressed as(39)ηk=yk-∑i=1cβikuikmvi∑i=1cβikuikmHere we should first discuss the ordering of data items. For each data item in each cluster, the properties of each attribute are computed separately. To do this, the data items are sorted by their distance from the cluster center. The d-th attribute of the closest data item is marked with the ordinal number 1, while that of the farthest data item n (n stands for the number of data items). The valueβckd represents the characteristic of the d-th attribute relative to the k-th data item of the c-th cluster and is calculated with the function:(40)βckd=pcn-χckd2pln+12∧1∨0It is called piecewise linear ordered weighted averaging (PLOWA) or(41)βckd=11+exp⁡2.944/panχckd-pcnAmong them,∧ and ∨, respectively, represent the minimum and maximum operations. These two functions, which can be called weighting functions, are not increased relative to the parameter χckd∈{1,2,…,n}. For χckd=pcn, both functions are equal to 0.5. Parameters pl>0 and pa>0 affect the slope. In the case of a piecewise linear function, for χckd∈pcn-pln,pcn+pln, its value is linearly reduced from 1 to 0 (see Figure 2 for details). In the case of the S-function, for χckd∈pcn-pan,pcn+pan, its value dropped from 0.95 to 0.05. The functions defined by formula (40) and formula (41) are, respectively, called S-ordered weighted average (SOWA) and piecewise linear ordered weighted averaging (PLOWA) in the rest of the work. If residual ordering is not applied, this is equivalent to using a uniform weighting function Based on OWA-UOWA for all χckd,βckd=1. Then we call this case an unordered cluster (or no weighting function).It is called S-ordered weighted average (SOWA) [50]. In these two functions, χckd represents the k-th data item index after reordering the distance relative to the c-th cluster of the d-th attribute. The two functions are shown in Figure 1, n = 100 and, pa=0.2, pc=0.5, and pl=0.2,Figure 1
The PLOWA and SOWA weighting functions forn=100, pa=0.2, pc=0.5, and pl=0.2.Figure 2
Comparison of segmentation results on the synthetic image from denoising base on salt & pepper. (a) Original image; (b) the image added salt & pepper (zero mean and 20% variance); (c) FCMS result; (d) FCMS1 result; (e) FCMS2 result; (f) FLICM result; (g) ARKFCM result; (h) FRFCM result; (i) BFWCOM result.
(a) (b) (c) (d) (e) (f) (g) (h) (i)The importance of this data item affects the central location of the cluster. The global characteristic of the k-th data item is calculated using s-norm (♢) of all cluster data item characteristics: (42)fi=β1i♢β2i♢⋯♢βciWe use the largest operator as the s-norm operator.


## 2.1. Criterion Function for BFWCOM Algorithm

The FCM clustering algorithm uses a quadratic loss function as a different measure between the data and the cluster center. The reason for using this method is for simplicity and low computational burden. However, this method is sensitive to noise and outliers. In many literatures, there are many suggestions for robust loss functions.For example, Huber proposed a mathematical formula [49, 50]:(5)τHUBe=e2δ2,e≤δeδ2,e>δwhereδ>0 represents a parameter. Another well-known robust loss function is a logarithmic function. (6)τHUBe=0,e=0log1+e2,e≠0UseD(xk,vi)=τxk-vi as a discrepancy measure between the k-th data and the i-th prototype center and an additional weighting. The criterion function for BFWCOM is as follows: (7)JU,V=∑i=1c∑k=1nβikuikmDxk,vi+∑i=1c∑k=1nαNRuikm∑r∈Nixr-vi2N R is the cardinality of uki, xr is the neighbor of xi, Ni is the neighbor set in the window around xi. α is a parameter used to control the effect of the neighborhood terms. The relative importance of the adjustment terms is inversely proportional to the signal-to-noise ratio (SNR). A lower signal-to-noise ratio requires a higher parameter value.The area of application to which the criterion function faces is as follows:(8)Mfnc=U∈RcN∣∀uik∈0,1;∑i=1cβikuik=fk;0<∑k=1Nuik<N


## 2.2. The Laplace Algorithm Used to Determine the Membership Degree and the Central Formula

### 2.2.1. The Membership Degree Formula

The Laplace algorithm used to give the specific formula is as follows:(9)GU,V=∑i=1c∑k=1nβikuikmDxk,vi+∑i=1c∑k=1nαNRuikmSDxr,vi-λ∑i=1cβikuik-fkwhereSD(xr,vi)=∑r∈Nixr-vi, xr is the neighbor of xi, and D(xk,vi) is the distance from the neighbor point xk to the center vi.Take the partial derivative of formula (9) with respect to uik: (10)∂GU,V∂uik=mβikuikm-1Dxk,vi+mαNRuikm-1SDxr,vi-λβik=0(11)λm=uikm-1Dxk,vi+α/NRSDxr,viβikThe membership degreeuik is derived: (12)uik=λm1/m-1Dxk,vi+α/NRSDxr,viβik1/1-m(13)Dxk,vi=∑d=1pxkd-vid2Available from formula (8), (14)∑i=1cβikuik=fkSubstitute formula (12) into formula (14): (15)fk=∑i=1cβikuik=∑i=1cβikλm1/m-1Dxk,vi+α/NRSDxr,viβik1/1-mThe following formula is derived from formula (15): (16)λm1/m-1=fk∑i=1cβikDxk,vi+α/NRSDxr,viβik1/1-m-1The obtained membership degree formula is as follows:(17)uik=fkDxk,vi+α/NRSDxr,viβik-11/1-m∑i=1cβikDxk,vi+α/NRSDxr,viβik-11/1-m


### 2.2.2. The Clustering Center Formula

The distance formulaD(xk,vi) can also be written as (18)Dxk,vi=xk-viTAxk-viFormula (7) can be written as follows: (19)JU,V=∑i=1c∑k=1nβikuikmxk-viTAxk-vi+∑i=1c∑k=1nαNRuikm∑r∈Nixr-viTAxr-viTake the derivative of formula (19) with respect to vi: (20)∂J∂vi=-2A∑k=1nβikuikmxk-vi-2A∑k=1nαNRuikm∑r∈Nixr-vi=0The clustering center formula as follows can be obtained by deriving formula (20): (21)vi=∑k=1nuikmβikxk+α/NR∑r∈Nixr∑k=1nuikmβik+α


## 2.2.1. The Membership Degree Formula

The Laplace algorithm used to give the specific formula is as follows:(9)GU,V=∑i=1c∑k=1nβikuikmDxk,vi+∑i=1c∑k=1nαNRuikmSDxr,vi-λ∑i=1cβikuik-fkwhereSD(xr,vi)=∑r∈Nixr-vi, xr is the neighbor of xi, and D(xk,vi) is the distance from the neighbor point xk to the center vi.Take the partial derivative of formula (9) with respect to uik: (10)∂GU,V∂uik=mβikuikm-1Dxk,vi+mαNRuikm-1SDxr,vi-λβik=0(11)λm=uikm-1Dxk,vi+α/NRSDxr,viβikThe membership degreeuik is derived: (12)uik=λm1/m-1Dxk,vi+α/NRSDxr,viβik1/1-m(13)Dxk,vi=∑d=1pxkd-vid2Available from formula (8), (14)∑i=1cβikuik=fkSubstitute formula (12) into formula (14): (15)fk=∑i=1cβikuik=∑i=1cβikλm1/m-1Dxk,vi+α/NRSDxr,viβik1/1-mThe following formula is derived from formula (15): (16)λm1/m-1=fk∑i=1cβikDxk,vi+α/NRSDxr,viβik1/1-m-1The obtained membership degree formula is as follows:(17)uik=fkDxk,vi+α/NRSDxr,viβik-11/1-m∑i=1cβikDxk,vi+α/NRSDxr,viβik-11/1-m


## 2.2.2. The Clustering Center Formula

The distance formulaD(xk,vi) can also be written as (18)Dxk,vi=xk-viTAxk-viFormula (7) can be written as follows: (19)JU,V=∑i=1c∑k=1nβikuikmxk-viTAxk-vi+∑i=1c∑k=1nαNRuikm∑r∈Nixr-viTAxr-viTake the derivative of formula (19) with respect to vi: (20)∂J∂vi=-2A∑k=1nβikuikmxk-vi-2A∑k=1nαNRuikm∑r∈Nixr-vi=0The clustering center formula as follows can be obtained by deriving formula (20): (21)vi=∑k=1nuikmβikxk+α/NR∑r∈Nixr∑k=1nuikmβik+α


## 2.3. Clustering Prototype Updating

The objective functionJU,V can be minimized in a manner similar to the standard FCM algorithm. The first-order derivatives of the objective function JU,V with respect to uki, vi, and ηk are set 0, resulting in three necessary but insufficient conditions for local extrema. In the following subsections, we will obtain these three conditions.According to literature [33], the additive bias field formula yk=xk+ηk, ∀k∈{1,2,…,N} is given, where xk and yk are the true and observed log-transformed intensities at the kth pixel, respectively, and ηk is the bias field at the kth pixel. (22)JmU,V=∑i=1c∑k=1nβikuikmDik+αNRuikmγiwhere(23)Dik=yk-ηk-vi2(24)γi=∑r∈Niyr-ηr-vi2Use Laplace’s theorem:(25)Gm=∑i=1c∑k=1nβikuikmDik+αNRuikmγi-λ∑i=1cβikuik-fkTake the derivative ofuik, 1≤i≤n; 1≤k≤c: (26)∂Gm∂uik=mβikuikm-1Dik+mαNRuikm-1γi-λβik=0Sort out the formula:(27)uik=λm1/m-1Dik+αγiNRβik1/1-mSort out the formula by substituting∑i=1cβikuik=fk: (28)∑i=1cβikλm1/m-1Dik+αγiNRβik1/1-m=fk(29)λm1/m-1=fk∑i=1cβikDik+αγiNRβik1/1-m-1The following formula is available from formula (27) and by substituting formula (29): (30)uik=fkDik+αγi/NRβik1/1-m∑i=1cβikDik+αγi/NRβik1/1-mAvailable from formulas (23) and (24), formula (19) can also be written as follows: (31)Jm=∑i=1c∑k=1nβikuikmDyk-ηk,vi+αNRuikm∑r∈NiDyr-ηr,viamong (32)Dyk-ηk,vi=yk-ηk-viTAyk-ηk-viDyr-ηr,vi=yr-ηr-viTAyr-ηr-viUse the Laplace’s method: (33)Gm=∑i=1c∑k=1nβikuikmDyk-ηk,vi+αNRuikm∑r∈NiDyr-ηr,vi+λ∑i=1cβikuik-fkTake the derivative of(34)∂Gm∂vi=-2A∑k=1nβikuikmyk-ηk-vi-2A∑k=1nαNRuikm∑r∈Niyr-ηr-vi=0The clustering center formula as follows can be obtained by deriving formula (34): (35)vi=∑k=1nuikmβikyk-ηk+α/NR∑r∈Niyr-ηr∑k=1nuikmβik+α


## 2.4. Bias Field Estimation

In a similar manner, take the derivative ofJm with respect to ηk, and the result is set to 0.(36)∑i=1c∂∂ηk∑k=1nβikuikmDik+αNRuikmγi=0Because only the k-th term in the second sum depends on, then we can have(37)∑i=1c∂∂ηkβikuikmyk-ηk-vi2=0Differentiate the distance function:(38)∑i=1cβikuikmyk-ηk-vi=0Therefore, the zero gradient condition of the bias field estimator is expressed as(39)ηk=yk-∑i=1cβikuikmvi∑i=1cβikuikmHere we should first discuss the ordering of data items. For each data item in each cluster, the properties of each attribute are computed separately. To do this, the data items are sorted by their distance from the cluster center. The d-th attribute of the closest data item is marked with the ordinal number 1, while that of the farthest data item n (n stands for the number of data items). The valueβckd represents the characteristic of the d-th attribute relative to the k-th data item of the c-th cluster and is calculated with the function:(40)βckd=pcn-χckd2pln+12∧1∨0It is called piecewise linear ordered weighted averaging (PLOWA) or(41)βckd=11+exp⁡2.944/panχckd-pcnAmong them,∧ and ∨, respectively, represent the minimum and maximum operations. These two functions, which can be called weighting functions, are not increased relative to the parameter χckd∈{1,2,…,n}. For χckd=pcn, both functions are equal to 0.5. Parameters pl>0 and pa>0 affect the slope. In the case of a piecewise linear function, for χckd∈pcn-pln,pcn+pln, its value is linearly reduced from 1 to 0 (see Figure 2 for details). In the case of the S-function, for χckd∈pcn-pan,pcn+pan, its value dropped from 0.95 to 0.05. The functions defined by formula (40) and formula (41) are, respectively, called S-ordered weighted average (SOWA) and piecewise linear ordered weighted averaging (PLOWA) in the rest of the work. If residual ordering is not applied, this is equivalent to using a uniform weighting function Based on OWA-UOWA for all χckd,βckd=1. Then we call this case an unordered cluster (or no weighting function).It is called S-ordered weighted average (SOWA) [50]. In these two functions, χckd represents the k-th data item index after reordering the distance relative to the c-th cluster of the d-th attribute. The two functions are shown in Figure 1, n = 100 and, pa=0.2, pc=0.5, and pl=0.2,Figure 1
The PLOWA and SOWA weighting functions forn=100, pa=0.2, pc=0.5, and pl=0.2.Figure 2
Comparison of segmentation results on the synthetic image from denoising base on salt & pepper. (a) Original image; (b) the image added salt & pepper (zero mean and 20% variance); (c) FCMS result; (d) FCMS1 result; (e) FCMS2 result; (f) FLICM result; (g) ARKFCM result; (h) FRFCM result; (i) BFWCOM result.
(a) (b) (c) (d) (e) (f) (g) (h) (i)The importance of this data item affects the central location of the cluster. The global characteristic of the k-th data item is calculated using s-norm (♢) of all cluster data item characteristics: (42)fi=β1i♢β2i♢⋯♢βciWe use the largest operator as the s-norm operator.


## 3. Pseudocode, Complexity Calculation, and Convergence of BFWCOM Algorithm

### 3.1. Pseudocode and Complexity Calculation of BFWCOM Algorithm

By writing the pseudocode mentioned in Algorithm1, it can be seen that it is a 4-fold loop with a complexity of ON4.Algorithm 1:Pseudocode of BFWCOM clustering algorithm.
Procedure BFWCOM (X,c,m,l) X:Data item matrix c:The number of clusters m:Membership degree weighted exponent l:the number of iterations n:=the number of data items in X; 1:Initialize all β′s with 1s; 2:Calculate data iteme characteristics using formula (42); 3:Set up the initial assignment of the random number matrix U, and normalize it using the constraint formula (14); 4:Calculate the clustering prototype V using formula (35); 5:for iter:=1 to Ndo 6:Update U and normalize it using formula (30); 7:for c: =1 to cdo 8:for d: =1 to pdo 9:for for k: =1 to ndo 10:ecdk:=|xkd-vcd|; 11:end for 12:sort residual; 13:mark each residual with the number of se- quences X the ordered sequence; 14:for k: =1 to ndo 15:calculate the characteristics of βckd using formulas (40) (41), or uniform weighting; 16:end for 17:end for 18:for k: =1 to ndo 19:Calculate the characteristics of βck with the maximum of the multiple values obtained informula (40),(41); 20:end for 21:Update the prototype using formula (35); 22:end for 23:end for 24:Return U, V; 25:End procedure;


### 3.2. Proof of the Convergence of BFWCOM Algorithm

Algorithm1 starts from initializing the membership degree matrix U0. The local minimum point or saddle point of the fuzzy C line (FCL) is its solution point. The BFWCOM algorithm can converge to a local minimum point.Definition 1.
In formula (22), JmU,V,m∈[1,∞]. Let BFWCOM algorithm obtain the best clustering result U∗,V∗ by minimizing (43)∀U∈Mfnc,JmU∗,V∗≤JmU,V∗(44)∀V∈Rsc,JmU∗,V∗≤JmU∗,V
Ω is called the solution set of the problem FCL.Definition 2.
LetJ:Mfnc→Rsc;JU=V=(V1,V2,⋯,Vc). Calculate Vi using formula (35).
LetF:Rsc→PMfnc; FV=V1,V2,⋯,Vc={U∈Mfnc∣U,Vsubject to (17)and (35)}.
BFWCOM operator (point-to-set mapping)Tm:Mfnc×Rsc→PMfnc×Rsc is defined as (45)Tm=M2∘M1where (46)M1:Mfnc×Rsc→PMfnc;M1U,V=U,V∣U∈FVM2:PMfnc→PMfnc×Rsc;M2U,V=U,JUThen (47)TmU,V=M2∘M1U,V=U′,V′∣U′∈FV,V′=JUDefinition 3.
When anyUk,Vkk=1l∈Mfnc×Rsc, there is an initial value of U0,V0, l is the number of iterations, Uk,Vk=TUk-1,Vk-1, and then Uk,Vk is called the iterative sequence of BFWCOM algorithm.
If the nested implicit relationship among the membership function and the clustering prototype and the parameter m is not considered, take the partial derivative of formula (22) with respect to m: (48)∂JmU,V∂m=∑i=1c∑k=1nβikuikmlog⁡uikDik+αNRuikmlog⁡uikγi=∑i=1c∑k=1nβikuikmDik+αNRuikmγilog⁡uik<0
It can be seen from formula (48) that the increase of JmU,V is monotonously decreasing and JmU,V≥0. According to the monotone bounded sequence convergence theorem, Uk,Vk must converge to a point in the solution set. Meanwhile, there are different optimal fuzzy C partitions for different m values. Therefore, the BFWCOM algorithm has a clustering validity problem for the weighted index m. In other words, it is necessary to determine the value of m, so that the corresponding clustering result is the most effective and reasonable. The parameter m has important influence on FCM algorithm.
According to the above statement, the core theorem of BFWCOM algorithm is as follows.
Core theorem: letX=x1,x2,⋯,xn, xi∈Rp; c is the number of clusters 1<c<n, for the formula(49)JmU,V=∑i=1c∑k=1nβikuikmDik+αNRuikmγi
whereDik=yk-ηk-vi2; γi=∑r∈Niyr-ηr-vi2; V=V1,V2,⋯,Vc⊂Rp; m(1<m<∞) is weight factor; U=[uij]c×n∈Mfnc. Let V(0)∈Rsc and U(0)=F(V(0)); U(0),V(0) is the initial value of the mapping T. The iterative sequence of BFWCOM terminates at a point in the solution set of min⁡(Jm) (local convergence theorem).


## 3.1. Pseudocode and Complexity Calculation of BFWCOM Algorithm

By writing the pseudocode mentioned in Algorithm1, it can be seen that it is a 4-fold loop with a complexity of ON4.Algorithm 1:Pseudocode of BFWCOM clustering algorithm.
Procedure BFWCOM (X,c,m,l) X:Data item matrix c:The number of clusters m:Membership degree weighted exponent l:the number of iterations n:=the number of data items in X; 1:Initialize all β′s with 1s; 2:Calculate data iteme characteristics using formula (42); 3:Set up the initial assignment of the random number matrix U, and normalize it using the constraint formula (14); 4:Calculate the clustering prototype V using formula (35); 5:for iter:=1 to Ndo 6:Update U and normalize it using formula (30); 7:for c: =1 to cdo 8:for d: =1 to pdo 9:for for k: =1 to ndo 10:ecdk:=|xkd-vcd|; 11:end for 12:sort residual; 13:mark each residual with the number of se- quences X the ordered sequence; 14:for k: =1 to ndo 15:calculate the characteristics of βckd using formulas (40) (41), or uniform weighting; 16:end for 17:end for 18:for k: =1 to ndo 19:Calculate the characteristics of βck with the maximum of the multiple values obtained informula (40),(41); 20:end for 21:Update the prototype using formula (35); 22:end for 23:end for 24:Return U, V; 25:End procedure;


## 3.2. Proof of the Convergence of BFWCOM Algorithm

Algorithm1 starts from initializing the membership degree matrix U0. The local minimum point or saddle point of the fuzzy C line (FCL) is its solution point. The BFWCOM algorithm can converge to a local minimum point.Definition 1.
In formula (22), JmU,V,m∈[1,∞]. Let BFWCOM algorithm obtain the best clustering result U∗,V∗ by minimizing (43)∀U∈Mfnc,JmU∗,V∗≤JmU,V∗(44)∀V∈Rsc,JmU∗,V∗≤JmU∗,V
Ω is called the solution set of the problem FCL.Definition 2.
LetJ:Mfnc→Rsc;JU=V=(V1,V2,⋯,Vc). Calculate Vi using formula (35).
LetF:Rsc→PMfnc; FV=V1,V2,⋯,Vc={U∈Mfnc∣U,Vsubject to (17)and (35)}.
BFWCOM operator (point-to-set mapping)Tm:Mfnc×Rsc→PMfnc×Rsc is defined as (45)Tm=M2∘M1where (46)M1:Mfnc×Rsc→PMfnc;M1U,V=U,V∣U∈FVM2:PMfnc→PMfnc×Rsc;M2U,V=U,JUThen (47)TmU,V=M2∘M1U,V=U′,V′∣U′∈FV,V′=JUDefinition 3.
When anyUk,Vkk=1l∈Mfnc×Rsc, there is an initial value of U0,V0, l is the number of iterations, Uk,Vk=TUk-1,Vk-1, and then Uk,Vk is called the iterative sequence of BFWCOM algorithm.
If the nested implicit relationship among the membership function and the clustering prototype and the parameter m is not considered, take the partial derivative of formula (22) with respect to m: (48)∂JmU,V∂m=∑i=1c∑k=1nβikuikmlog⁡uikDik+αNRuikmlog⁡uikγi=∑i=1c∑k=1nβikuikmDik+αNRuikmγilog⁡uik<0
It can be seen from formula (48) that the increase of JmU,V is monotonously decreasing and JmU,V≥0. According to the monotone bounded sequence convergence theorem, Uk,Vk must converge to a point in the solution set. Meanwhile, there are different optimal fuzzy C partitions for different m values. Therefore, the BFWCOM algorithm has a clustering validity problem for the weighted index m. In other words, it is necessary to determine the value of m, so that the corresponding clustering result is the most effective and reasonable. The parameter m has important influence on FCM algorithm.
According to the above statement, the core theorem of BFWCOM algorithm is as follows.
Core theorem: letX=x1,x2,⋯,xn, xi∈Rp; c is the number of clusters 1<c<n, for the formula(49)JmU,V=∑i=1c∑k=1nβikuikmDik+αNRuikmγi
whereDik=yk-ηk-vi2; γi=∑r∈Niyr-ηr-vi2; V=V1,V2,⋯,Vc⊂Rp; m(1<m<∞) is weight factor; U=[uij]c×n∈Mfnc. Let V(0)∈Rsc and U(0)=F(V(0)); U(0),V(0) is the initial value of the mapping T. The iterative sequence of BFWCOM terminates at a point in the solution set of min⁡(Jm) (local convergence theorem).


## 4. Algorithm Experiment

The experiment consists of three steps.First, IRIS data are classified using FCM, FCMS, FWCOM, ARKFCM, FRFCM, and BFWCOM. Then, the advantages and disadvantages of the proposed algorithm in this paper are compared with other algorithms according to the running results.Second, these algorithms are all applied to the specific composite image processing. By image denoising, the advantages and disadvantages of the new algorithm are compared with other ones.Third, the advantages and disadvantages of the proposed algorithm are verified by the real image processing.Experimental environment: software environments: Windows 7 Ultimate and Software MATLAB; hardware environments: (1) processor: Intel(R) Pentium(R) CPU G2020@2.90GHz 2.90GHz; (2) installed memory: 8.00GB; (3) system type: 64-bit operating system.In all experiments, the fuzzy parameter value is uniformly set: m=2. For FCMS, FCMS1, FCMS2, FLICM, ARKFCM, FRFCM, and BFWCOM, the iteration stops when the Frobenius norm of continuous V matrix difference is less than10-4. The loss function adopts ε=0.5, α=6.0, β=1.0, and δ=1.0. The weighted functions (40)-(41) adopt pc=0.5, pl=0.2, and pa=0.2. For the calculated terminal prototype, we use the Frobenius norm of the difference between the real center matrix and the terminal prototype matrix to measure the clustering performance.


### 4.1. IRIS Data Used for Comparison of Classification Effectiveness

Iris data set, collected by Fisher in 1936, is a commonly used data set for classification experiment. Iris, also called Iris flower data set, is a data set for multivariate analysis. It contains 150 data sets, divided into 3 classes, 50 data per class, each data containing 4 attributes. Aiming at the commonly used IRIS data, six algorithms, namely, FCM, FCMS, FWCOM, ARKFCM, FRFCM, and BFWCOM, are used for comparison. The comparison process is as follows.


#### 4.1.1. Classification Error Rate (CER)

The CER is referred to as the probability that a sample that should belong to one of the class c is incorrectly assigned to other classes. In this paper,X=xij∣1≤i≤n;1≤j≤c; that is to say, the data are divided into c spaces, each space being a class. Regardless of how it is divided, there is always a misclassification phenomenon that an individual of a class is assigned to other classes. Therefore, the good and bad of cluster analysis algorithm can be equivalent to the division of the lowest average misclassification rate. (50)CER=1-NumberofcorrectlyclassifiedpixelsTotalnumber×100%


#### 4.1.2. Deviation Degree (DD) Calculation

Definition 4.
Sum of squares for error is also called residual sum of squares (RSS) or sum of squares within groups. After fitting the appropriate model based on n observations, the remaining that is underfitting (ei=xi-x-) is called the residual, of which x- represents the average value of n observations x-=∑i=1nxi. The sum of the squares of all the n residuals is called residual sum of squares (RRS) or sum of squares for error (SSE), SSE=∑i=1nxi-x-2.Definition 5.
In the set,X=xij∣1≤i≤n;1≤j≤c, xi=xi1,xi2,⋯,xipT∈Rp, j is the concrete class to which xi belongs to. The mean of sum of the j-th sample is as follows: x-j=1/nj∑l=1njxlj, of which ni is the total number of data in class j. The mean of sum of squares of total deviation for all the data samples, also called the mean of the residual sum of squares, is as shown below: (51)S=1p∑i=1p1n∑j=1c∑l=1nixlij-x-ji2S, the mean of sum of squares of total deviation, can reflect the deviation degree between all measured data and the mean value of the data it classifies. The function above is called Deviation degree (DD).It is obviously seen that the smaller the deviation degree, the better the clustering. In Table2, CER is calculated with function (50), DD is calculated with function (51), and the clustering center is calculated with corresponding center formulas of algorithms.Table3 shows that BFWCOM, the new algorithm, which performs better than FCM, FCMS, FWCOM, ARKFCM, and FRFCM in classification, for the residual sum of squares is the smallest, and the misclassification rate is the lowest.Table 3
Classification effect of FCM algorithm, FCMS algorithm, and BFWCOM algorithm on IRIS data.
Algorithm the number of misclassification CER Clustering center DD FCM 16 10.7% v 1 = ( 5.0040,3.4141,1.4828,0.2535 ) 0.08 v 2 = 5.8887,2.7610,4.3636,1.3972 v 3 = ( 6.7748,3.0523,5.6465,2.0534 )  FCMS 10 6.7% v 1 = ( 4.9940,3.3649,1.4833,0.2537 ) 0.06 v 2 = 5.7787,2.7851,4.2636,1.3882 v 3 = ( 6.6648,3.0573,5.6485,2.0564 )  FWCOM 8 5.3% v 1 = 4.9940,3.3649,1.4833,0.2537 0.04 v 2 = 5.7787,2.7851,4.2636,1.3882 v 3 = ( 6.6648,3.0573,5.6485,2.0564 )  ARKFCM 7 4.7% v 1 = 4.9840,3.3549,1.4533,0.2497 0.04 v 2 = ( 5.7817,2.8051,4.2539,1.3082 ) v 3 = ( 6.3948,3.0583,5.3485,2.0821 )  FRFCM 5 3.3% v 1 = 4.9941,3.3650,1.4863,0.2577 0.037 v 2 = ( 5.7767,2.7853,4.2615,1.2782 ) v 3 = ( 6.4648,3.0893,5.3485,2.1564 )  BFWCOM 5 3.3% v 1 = 4.9969,3.3649,1.4623,0.2158 0.035 v 2 = ( 5.7723,2.7852,4.1582,1.2782 ) v 3 = ( 6.4490,2.9253,5.4026,2.0110 )


### 4.2. Comparison of Experimental Results Using Composite Images

In this part, synthetic images are used to verify the performance of the proposed algorithm. In addition to BFWCOM, the other six algorithms are also taken into consideration, namely, FCMS, FCMS1, FCMS2, FLICM, ARKFCM, and FRFCM. For all fuzzy algorithms, the size of the neighboring window is 3×3. In order to compare the segmentation performance, segmentation accuracy (SA) is used as a quantitative index [52]. (52)SA=NumberofcorrectlyclassifiedpixelsTotalnumber×100%As shown in Figure2, the composite image contains five regions, namely, five classes. The gray values of these five regions are, respectively, 0, 60, 120, 180, and 255. Figure 2 shows the different clustering results of several methods under the condition of images with Gaussian noise with the variance: σg=20. It can be observed that BFWFCM produces a better segmentation than the other six fuzzy clustering algorithms. Visually, they can achieve acceptable segmentation results under Gaussian noise. Figure 3 shows the segmentation accuracies (SAs) of different methods for composite images under different noises, of which σg is the variance of Gaussian noise. As can be seen from the first line of Table 4, for synthetic images with different noise levels, the segmentation accuracy of our method is always higher than that of other methods.Table 4
Comparison of segmentation accuracies (%) on synthetic images degraded by different noise.
Noise FCMS FCMS1 FCMS2 FLICM ARKFCM FRFCM BFWCOM Gaussian (σg=10) 99.91 99.93 99.96 99.95 99.97 99.98 99.98 Gaussian (σg=20) 96.15 96.40 96.44 97.68 97.85 98.89 99.35 Gaussian (σg=25) 71.50 94.74 94.90 98.84 98.97 99.32 99.68 Salt & pepper (5%) 96.64 96.64 96.64 97.34 98.44 99.28 99.87 Salt & pepper (10%) 93.17 93.17 93.17 94.23 95.84 97.84 99.77 Salt & pepper (20%) 86.07 86.17 87.05 93.68 94.56 95.78 98.12Figure 3
Comparison of segmentation results on the synthetic image from denoising base on salt & pepper. (a) Original image; (b) the image added salt & pepper (zero mean and 20% variance); (c) FCMS result; (d) FCMS1 result; (e) FCMS2 result; (f) FLICM result; (g) ARKFCM result; (h) FRFCM result; (i) BFWCOM result.
(a) (b) (c) (d) (e) (f) (g) (h) (i)As shown in Figure3, “salt & pepper” noise is added to the synthetic images above to carry out the denoising experiment. This figure shows the clustering denoising results of images with 20% “salt & pepper” noise. It can be observed that, under this condition, BFWFCM produces a better segmentation than the other six fuzzy clustering algorithms. Visually, they can achieve acceptable segmentation results under the “salt & pepper” noise. Table 4 shows the segmentation accuracies (SAs) of different methods for synthetic images under different “salt & pepper” noises. From Table 4, it can be seen that, for synthetic images with different levels of “salt & pepper” noise, the segmentation accuracy of our method is always higher than that of other methods.


### 4.3. Comparison of Experimental Results Using Brain MR and Lena Images

In this section, we experimented with real images, particularly magnetic resonance (MR) images and Lena images, in order to prove the performance of the proposed BFWCOM. Meanwhile, some well-known image segmentation methods, such as FCMS, FCMS1, FCMS2, FLICM, ARKFCM, and FRFCM, were used as comparison methods. In Figures2 and 3, the denosing results of synthetic images have been compared. In addition to this, in Figures 4 and 5, we first use gauss and “salt & pepper” mixture noise to erode the real images [53] and then use these 7 algorithms to denoise real images with noise. Eventually, the comparison is made on the denoising results of real images. The selection of such real images is mainly carried out from two aspects. One is to select MR image for testing; the other is to select the head image of Lena for testing. Such selection also reflects the comparison between medical image and natural portrait.Figure 4
Comparison of segmentation results on the brain CT image from denoising base on salt & pepper. (a) Ground truth; (b) the image added salt & pepper (zero mean and 20% variance); (c) FCMS result; (d) FCMS1 result; (e) FCMS2 result; (f) FLICM result; (g) BFWCOM result.
(a) (b) (c) (d) (e) (f) (g) (h) (i)Figure 5
Comparison of segmentation results on the brain CT image from denoising base on Gaussian. (a) Ground truth; (b) the image added Gaussian (zero mean and 20% variance); (c) FCMS result; (d) FCMS1 result; (e) FCMS2 result; (f) FLICM result; (g)FRFCM result; (i) BFWCOM result.
(a) (b) (c) (d) (e) (f) (g) (h) (i)


#### 4.3.1. Denoising Experimental Results of MR Image

Figures4(c)–4(i) show the denoising segmentation results of different algorithms for images with 20% “salt & pepper” noise. Figures 5(c)–5(i) show the Gaussian denoising segmentation results of various algorithms with. The segmentation results show that the proposed BFWCOM algorithm can achieve excellent segmentation, while the other fuzzy methods are more or less affected by noise. The FRFCM algorithm is a relatively excellent method; however, it merely considers the influence of traditional fuzzy c-means clustering algorithm and spatial neighborhood, but fails to consider the typicality of data attributes. Thus, it is less delicate than BFWCOM algorithm in image denoising.


#### 4.3.2. Denoising Experimental Results of Lena’s Head Images

The parameters in the BFWCOM include the fuzzy control parameter m, size of the neighborhood N, and similarity window S. Roughly speaking, the larger the size of N, the smoother the final results, the larger the similarity window, and the more structural information. However, it might use considerably large neighborhood windows and similarity windows for oversmoothing. In the experiment, we find that the 3×3 windows of N and S are applicable to brain MR image. For the real Lena images (Figure6), a larger window is needed to measure the similarity. Under this circumstance, 5×5 or 7×7 windows of S are applicable. In this paper, 5×5 windows are adopted.Figure 6
Comparison of segmentation results on the Lena image from denoising base on Gaussian. (a) Original Lena image; (b) gray image; (c) the image added Gaussian (zero mean and 20% variance); (d) FCMS result; (e) FCMS1 result; (f) FCMS2 result; (g) FLICM result; (h) ARKFCM result; (i) FRRCM result; (j) BFWCOM result.
(a) (b) (c) (d) (e) (f) (g) (h) (i) (j)Figure6 is a comparison of denoising results on the Lena image with added Gaussian noise with 20% variance. As can be seen from the figure above, the proposed BFWCOM algorithm has more obviously advantage than FCMS, FCMS1, FCMS2, FLICM, and ARKFCM. The FRRCM algorithm achieves a very good performance in the processing effect but is not better than BFWCOM algorithm in detail.In order to further verify the denoising effect of the BFWCOM algorithm proposed in this paper, the performance of the proposed algorithm is compared and analyzed based on the objective data. Mean Squared Error (MSE), Peak Signal-to-Noise Ratio (PSNR) [54, 55], and signal-to-noise ratio (SNR) [56] are used for numerical calculation of the image after denoising.The function for MSE is shown as below:(53)MSE=1M×N∑i=1M∑j=1Nfj,i-f^ji2The functions for PSNR and SNR are shown as below:(54)PSNR=10×log102552MSE=20·log10255MSE(55)SNR=10×log10⁡∑i=1M∑j=1Nfj,i2M×N·MSEof whichM×Nis the size of test image; fj,i and f^ji represent the noise-free image and denoised image, respectively. In order to compare the performance of the proposed algorithm more comprehensively, the MSE function (53), the PSNR function (54) and SNR function (55) are used for comparison of the image denoised by algorithms in Figure 6. The calculation results are shown in Table 5.Table 5
Noise-free Lena image denoising effect evaluation index (n=3;δ=0.2).
Denosing method MSE PSNR/db SNR/db Image with noise 640.5461 20.0653 13.4480 FCMS 138.6119 26.7128 21.0557 FCMS1 179.6752 26.1314 20.0553 FCMS2 182.4737 25.5188 19.8616 FLICM 107.1739 26.1670 20.5099 ARKFCM 91.4922 28.8986 22.2414 FRRCM 67.5066 29.5425 23.8853 BFWCOM 54.9023 30.7349 24.6808The MSE here is referred to as the mean of the squared sum of gray level difference values of pixel points corresponding to the original image and denoised image. The smaller the value is, the better the denoised image quality is. The unit of PSNR is dB. The higher the value of PSNR is, the better the denoising effect is. According to function (53) and function (54) and by limitingfj,i-f^ji2≥1, PSNR∈0,48.12 after calculation. When the PSNR value here is above 40, it has been close to the original image. Therefore, 30.7349, the PSNR value of the image denoised by the BFWCOM algorithm, is quite good. SNR means signal-to-noise ratio. The larger its value, the better. By the comparison in Table 4, it can be seen that the denoising effect of BFWCOM algorithm in Lena image is better than that of other algorithms.


## 4.1. IRIS Data Used for Comparison of Classification Effectiveness

Iris data set, collected by Fisher in 1936, is a commonly used data set for classification experiment. Iris, also called Iris flower data set, is a data set for multivariate analysis. It contains 150 data sets, divided into 3 classes, 50 data per class, each data containing 4 attributes. Aiming at the commonly used IRIS data, six algorithms, namely, FCM, FCMS, FWCOM, ARKFCM, FRFCM, and BFWCOM, are used for comparison. The comparison process is as follows.


### 4.1.1. Classification Error Rate (CER)

The CER is referred to as the probability that a sample that should belong to one of the class c is incorrectly assigned to other classes. In this paper,X=xij∣1≤i≤n;1≤j≤c; that is to say, the data are divided into c spaces, each space being a class. Regardless of how it is divided, there is always a misclassification phenomenon that an individual of a class is assigned to other classes. Therefore, the good and bad of cluster analysis algorithm can be equivalent to the division of the lowest average misclassification rate. (50)CER=1-NumberofcorrectlyclassifiedpixelsTotalnumber×100%


### 4.1.2. Deviation Degree (DD) Calculation

Definition 4.
Sum of squares for error is also called residual sum of squares (RSS) or sum of squares within groups. After fitting the appropriate model based on n observations, the remaining that is underfitting (ei=xi-x-) is called the residual, of which x- represents the average value of n observations x-=∑i=1nxi. The sum of the squares of all the n residuals is called residual sum of squares (RRS) or sum of squares for error (SSE), SSE=∑i=1nxi-x-2.Definition 5.
In the set,X=xij∣1≤i≤n;1≤j≤c, xi=xi1,xi2,⋯,xipT∈Rp, j is the concrete class to which xi belongs to. The mean of sum of the j-th sample is as follows: x-j=1/nj∑l=1njxlj, of which ni is the total number of data in class j. The mean of sum of squares of total deviation for all the data samples, also called the mean of the residual sum of squares, is as shown below: (51)S=1p∑i=1p1n∑j=1c∑l=1nixlij-x-ji2S, the mean of sum of squares of total deviation, can reflect the deviation degree between all measured data and the mean value of the data it classifies. The function above is called Deviation degree (DD).It is obviously seen that the smaller the deviation degree, the better the clustering. In Table2, CER is calculated with function (50), DD is calculated with function (51), and the clustering center is calculated with corresponding center formulas of algorithms.Table3 shows that BFWCOM, the new algorithm, which performs better than FCM, FCMS, FWCOM, ARKFCM, and FRFCM in classification, for the residual sum of squares is the smallest, and the misclassification rate is the lowest.Table 3
Classification effect of FCM algorithm, FCMS algorithm, and BFWCOM algorithm on IRIS data.
Algorithm the number of misclassification CER Clustering center DD FCM 16 10.7% v 1 = ( 5.0040,3.4141,1.4828,0.2535 ) 0.08 v 2 = 5.8887,2.7610,4.3636,1.3972 v 3 = ( 6.7748,3.0523,5.6465,2.0534 )  FCMS 10 6.7% v 1 = ( 4.9940,3.3649,1.4833,0.2537 ) 0.06 v 2 = 5.7787,2.7851,4.2636,1.3882 v 3 = ( 6.6648,3.0573,5.6485,2.0564 )  FWCOM 8 5.3% v 1 = 4.9940,3.3649,1.4833,0.2537 0.04 v 2 = 5.7787,2.7851,4.2636,1.3882 v 3 = ( 6.6648,3.0573,5.6485,2.0564 )  ARKFCM 7 4.7% v 1 = 4.9840,3.3549,1.4533,0.2497 0.04 v 2 = ( 5.7817,2.8051,4.2539,1.3082 ) v 3 = ( 6.3948,3.0583,5.3485,2.0821 )  FRFCM 5 3.3% v 1 = 4.9941,3.3650,1.4863,0.2577 0.037 v 2 = ( 5.7767,2.7853,4.2615,1.2782 ) v 3 = ( 6.4648,3.0893,5.3485,2.1564 )  BFWCOM 5 3.3% v 1 = 4.9969,3.3649,1.4623,0.2158 0.035 v 2 = ( 5.7723,2.7852,4.1582,1.2782 ) v 3 = ( 6.4490,2.9253,5.4026,2.0110 )


## 4.1.1. Classification Error Rate (CER)

The CER is referred to as the probability that a sample that should belong to one of the class c is incorrectly assigned to other classes. In this paper,X=xij∣1≤i≤n;1≤j≤c; that is to say, the data are divided into c spaces, each space being a class. Regardless of how it is divided, there is always a misclassification phenomenon that an individual of a class is assigned to other classes. Therefore, the good and bad of cluster analysis algorithm can be equivalent to the division of the lowest average misclassification rate. (50)CER=1-NumberofcorrectlyclassifiedpixelsTotalnumber×100%


## 4.1.2. Deviation Degree (DD) Calculation

Definition 4.
Sum of squares for error is also called residual sum of squares (RSS) or sum of squares within groups. After fitting the appropriate model based on n observations, the remaining that is underfitting (ei=xi-x-) is called the residual, of which x- represents the average value of n observations x-=∑i=1nxi. The sum of the squares of all the n residuals is called residual sum of squares (RRS) or sum of squares for error (SSE), SSE=∑i=1nxi-x-2.Definition 5.
In the set,X=xij∣1≤i≤n;1≤j≤c, xi=xi1,xi2,⋯,xipT∈Rp, j is the concrete class to which xi belongs to. The mean of sum of the j-th sample is as follows: x-j=1/nj∑l=1njxlj, of which ni is the total number of data in class j. The mean of sum of squares of total deviation for all the data samples, also called the mean of the residual sum of squares, is as shown below: (51)S=1p∑i=1p1n∑j=1c∑l=1nixlij-x-ji2S, the mean of sum of squares of total deviation, can reflect the deviation degree between all measured data and the mean value of the data it classifies. The function above is called Deviation degree (DD).It is obviously seen that the smaller the deviation degree, the better the clustering. In Table2, CER is calculated with function (50), DD is calculated with function (51), and the clustering center is calculated with corresponding center formulas of algorithms.Table3 shows that BFWCOM, the new algorithm, which performs better than FCM, FCMS, FWCOM, ARKFCM, and FRFCM in classification, for the residual sum of squares is the smallest, and the misclassification rate is the lowest.Table 3
Classification effect of FCM algorithm, FCMS algorithm, and BFWCOM algorithm on IRIS data.
Algorithm the number of misclassification CER Clustering center DD FCM 16 10.7% v 1 = ( 5.0040,3.4141,1.4828,0.2535 ) 0.08 v 2 = 5.8887,2.7610,4.3636,1.3972 v 3 = ( 6.7748,3.0523,5.6465,2.0534 )  FCMS 10 6.7% v 1 = ( 4.9940,3.3649,1.4833,0.2537 ) 0.06 v 2 = 5.7787,2.7851,4.2636,1.3882 v 3 = ( 6.6648,3.0573,5.6485,2.0564 )  FWCOM 8 5.3% v 1 = 4.9940,3.3649,1.4833,0.2537 0.04 v 2 = 5.7787,2.7851,4.2636,1.3882 v 3 = ( 6.6648,3.0573,5.6485,2.0564 )  ARKFCM 7 4.7% v 1 = 4.9840,3.3549,1.4533,0.2497 0.04 v 2 = ( 5.7817,2.8051,4.2539,1.3082 ) v 3 = ( 6.3948,3.0583,5.3485,2.0821 )  FRFCM 5 3.3% v 1 = 4.9941,3.3650,1.4863,0.2577 0.037 v 2 = ( 5.7767,2.7853,4.2615,1.2782 ) v 3 = ( 6.4648,3.0893,5.3485,2.1564 )  BFWCOM 5 3.3% v 1 = 4.9969,3.3649,1.4623,0.2158 0.035 v 2 = ( 5.7723,2.7852,4.1582,1.2782 ) v 3 = ( 6.4490,2.9253,5.4026,2.0110 )


## 4.2. Comparison of Experimental Results Using Composite Images

In this part, synthetic images are used to verify the performance of the proposed algorithm. In addition to BFWCOM, the other six algorithms are also taken into consideration, namely, FCMS, FCMS1, FCMS2, FLICM, ARKFCM, and FRFCM. For all fuzzy algorithms, the size of the neighboring window is 3×3. In order to compare the segmentation performance, segmentation accuracy (SA) is used as a quantitative index [52]. (52)SA=NumberofcorrectlyclassifiedpixelsTotalnumber×100%As shown in Figure2, the composite image contains five regions, namely, five classes. The gray values of these five regions are, respectively, 0, 60, 120, 180, and 255. Figure 2 shows the different clustering results of several methods under the condition of images with Gaussian noise with the variance: σg=20. It can be observed that BFWFCM produces a better segmentation than the other six fuzzy clustering algorithms. Visually, they can achieve acceptable segmentation results under Gaussian noise. Figure 3 shows the segmentation accuracies (SAs) of different methods for composite images under different noises, of which σg is the variance of Gaussian noise. As can be seen from the first line of Table 4, for synthetic images with different noise levels, the segmentation accuracy of our method is always higher than that of other methods.Table 4
Comparison of segmentation accuracies (%) on synthetic images degraded by different noise.
Noise FCMS FCMS1 FCMS2 FLICM ARKFCM FRFCM BFWCOM Gaussian (σg=10) 99.91 99.93 99.96 99.95 99.97 99.98 99.98 Gaussian (σg=20) 96.15 96.40 96.44 97.68 97.85 98.89 99.35 Gaussian (σg=25) 71.50 94.74 94.90 98.84 98.97 99.32 99.68 Salt & pepper (5%) 96.64 96.64 96.64 97.34 98.44 99.28 99.87 Salt & pepper (10%) 93.17 93.17 93.17 94.23 95.84 97.84 99.77 Salt & pepper (20%) 86.07 86.17 87.05 93.68 94.56 95.78 98.12Figure 3
Comparison of segmentation results on the synthetic image from denoising base on salt & pepper. (a) Original image; (b) the image added salt & pepper (zero mean and 20% variance); (c) FCMS result; (d) FCMS1 result; (e) FCMS2 result; (f) FLICM result; (g) ARKFCM result; (h) FRFCM result; (i) BFWCOM result.
(a) (b) (c) (d) (e) (f) (g) (h) (i)As shown in Figure3, “salt & pepper” noise is added to the synthetic images above to carry out the denoising experiment. This figure shows the clustering denoising results of images with 20% “salt & pepper” noise. It can be observed that, under this condition, BFWFCM produces a better segmentation than the other six fuzzy clustering algorithms. Visually, they can achieve acceptable segmentation results under the “salt & pepper” noise. Table 4 shows the segmentation accuracies (SAs) of different methods for synthetic images under different “salt & pepper” noises. From Table 4, it can be seen that, for synthetic images with different levels of “salt & pepper” noise, the segmentation accuracy of our method is always higher than that of other methods.


## 4.3. Comparison of Experimental Results Using Brain MR and Lena Images

In this section, we experimented with real images, particularly magnetic resonance (MR) images and Lena images, in order to prove the performance of the proposed BFWCOM. Meanwhile, some well-known image segmentation methods, such as FCMS, FCMS1, FCMS2, FLICM, ARKFCM, and FRFCM, were used as comparison methods. In Figures2 and 3, the denosing results of synthetic images have been compared. In addition to this, in Figures 4 and 5, we first use gauss and “salt & pepper” mixture noise to erode the real images [53] and then use these 7 algorithms to denoise real images with noise. Eventually, the comparison is made on the denoising results of real images. The selection of such real images is mainly carried out from two aspects. One is to select MR image for testing; the other is to select the head image of Lena for testing. Such selection also reflects the comparison between medical image and natural portrait.Figure 4
Comparison of segmentation results on the brain CT image from denoising base on salt & pepper. (a) Ground truth; (b) the image added salt & pepper (zero mean and 20% variance); (c) FCMS result; (d) FCMS1 result; (e) FCMS2 result; (f) FLICM result; (g) BFWCOM result.
(a) (b) (c) (d) (e) (f) (g) (h) (i)Figure 5
Comparison of segmentation results on the brain CT image from denoising base on Gaussian. (a) Ground truth; (b) the image added Gaussian (zero mean and 20% variance); (c) FCMS result; (d) FCMS1 result; (e) FCMS2 result; (f) FLICM result; (g)FRFCM result; (i) BFWCOM result.
(a) (b) (c) (d) (e) (f) (g) (h) (i)


### 4.3.1. Denoising Experimental Results of MR Image

Figures4(c)–4(i) show the denoising segmentation results of different algorithms for images with 20% “salt & pepper” noise. Figures 5(c)–5(i) show the Gaussian denoising segmentation results of various algorithms with. The segmentation results show that the proposed BFWCOM algorithm can achieve excellent segmentation, while the other fuzzy methods are more or less affected by noise. The FRFCM algorithm is a relatively excellent method; however, it merely considers the influence of traditional fuzzy c-means clustering algorithm and spatial neighborhood, but fails to consider the typicality of data attributes. Thus, it is less delicate than BFWCOM algorithm in image denoising.


### 4.3.2. Denoising Experimental Results of Lena’s Head Images

The parameters in the BFWCOM include the fuzzy control parameter m, size of the neighborhood N, and similarity window S. Roughly speaking, the larger the size of N, the smoother the final results, the larger the similarity window, and the more structural information. However, it might use considerably large neighborhood windows and similarity windows for oversmoothing. In the experiment, we find that the 3×3 windows of N and S are applicable to brain MR image. For the real Lena images (Figure6), a larger window is needed to measure the similarity. Under this circumstance, 5×5 or 7×7 windows of S are applicable. In this paper, 5×5 windows are adopted.Figure 6
Comparison of segmentation results on the Lena image from denoising base on Gaussian. (a) Original Lena image; (b) gray image; (c) the image added Gaussian (zero mean and 20% variance); (d) FCMS result; (e) FCMS1 result; (f) FCMS2 result; (g) FLICM result; (h) ARKFCM result; (i) FRRCM result; (j) BFWCOM result.
(a) (b) (c) (d) (e) (f) (g) (h) (i) (j)Figure6 is a comparison of denoising results on the Lena image with added Gaussian noise with 20% variance. As can be seen from the figure above, the proposed BFWCOM algorithm has more obviously advantage than FCMS, FCMS1, FCMS2, FLICM, and ARKFCM. The FRRCM algorithm achieves a very good performance in the processing effect but is not better than BFWCOM algorithm in detail.In order to further verify the denoising effect of the BFWCOM algorithm proposed in this paper, the performance of the proposed algorithm is compared and analyzed based on the objective data. Mean Squared Error (MSE), Peak Signal-to-Noise Ratio (PSNR) [54, 55], and signal-to-noise ratio (SNR) [56] are used for numerical calculation of the image after denoising.The function for MSE is shown as below:(53)MSE=1M×N∑i=1M∑j=1Nfj,i-f^ji2The functions for PSNR and SNR are shown as below:(54)PSNR=10×log102552MSE=20·log10255MSE(55)SNR=10×log10⁡∑i=1M∑j=1Nfj,i2M×N·MSEof whichM×Nis the size of test image; fj,i and f^ji represent the noise-free image and denoised image, respectively. In order to compare the performance of the proposed algorithm more comprehensively, the MSE function (53), the PSNR function (54) and SNR function (55) are used for comparison of the image denoised by algorithms in Figure 6. The calculation results are shown in Table 5.Table 5
Noise-free Lena image denoising effect evaluation index (n=3;δ=0.2).
Denosing method MSE PSNR/db SNR/db Image with noise 640.5461 20.0653 13.4480 FCMS 138.6119 26.7128 21.0557 FCMS1 179.6752 26.1314 20.0553 FCMS2 182.4737 25.5188 19.8616 FLICM 107.1739 26.1670 20.5099 ARKFCM 91.4922 28.8986 22.2414 FRRCM 67.5066 29.5425 23.8853 BFWCOM 54.9023 30.7349 24.6808The MSE here is referred to as the mean of the squared sum of gray level difference values of pixel points corresponding to the original image and denoised image. The smaller the value is, the better the denoised image quality is. The unit of PSNR is dB. The higher the value of PSNR is, the better the denoising effect is. According to function (53) and function (54) and by limitingfj,i-f^ji2≥1, PSNR∈0,48.12 after calculation. When the PSNR value here is above 40, it has been close to the original image. Therefore, 30.7349, the PSNR value of the image denoised by the BFWCOM algorithm, is quite good. SNR means signal-to-noise ratio. The larger its value, the better. By the comparison in Table 4, it can be seen that the denoising effect of BFWCOM algorithm in Lena image is better than that of other algorithms.


## 4.3.1. Denoising Experimental Results of MR Image

Figures4(c)–4(i) show the denoising segmentation results of different algorithms for images with 20% “salt & pepper” noise. Figures 5(c)–5(i) show the Gaussian denoising segmentation results of various algorithms with. The segmentation results show that the proposed BFWCOM algorithm can achieve excellent segmentation, while the other fuzzy methods are more or less affected by noise. The FRFCM algorithm is a relatively excellent method; however, it merely considers the influence of traditional fuzzy c-means clustering algorithm and spatial neighborhood, but fails to consider the typicality of data attributes. Thus, it is less delicate than BFWCOM algorithm in image denoising.


## 4.3.2. Denoising Experimental Results of Lena’s Head Images

The parameters in the BFWCOM include the fuzzy control parameter m, size of the neighborhood N, and similarity window S. Roughly speaking, the larger the size of N, the smoother the final results, the larger the similarity window, and the more structural information. However, it might use considerably large neighborhood windows and similarity windows for oversmoothing. In the experiment, we find that the 3×3 windows of N and S are applicable to brain MR image. For the real Lena images (Figure6), a larger window is needed to measure the similarity. Under this circumstance, 5×5 or 7×7 windows of S are applicable. In this paper, 5×5 windows are adopted.Figure 6
Comparison of segmentation results on the Lena image from denoising base on Gaussian. (a) Original Lena image; (b) gray image; (c) the image added Gaussian (zero mean and 20% variance); (d) FCMS result; (e) FCMS1 result; (f) FCMS2 result; (g) FLICM result; (h) ARKFCM result; (i) FRRCM result; (j) BFWCOM result.
(a) (b) (c) (d) (e) (f) (g) (h) (i) (j)Figure6 is a comparison of denoising results on the Lena image with added Gaussian noise with 20% variance. As can be seen from the figure above, the proposed BFWCOM algorithm has more obviously advantage than FCMS, FCMS1, FCMS2, FLICM, and ARKFCM. The FRRCM algorithm achieves a very good performance in the processing effect but is not better than BFWCOM algorithm in detail.In order to further verify the denoising effect of the BFWCOM algorithm proposed in this paper, the performance of the proposed algorithm is compared and analyzed based on the objective data. Mean Squared Error (MSE), Peak Signal-to-Noise Ratio (PSNR) [54, 55], and signal-to-noise ratio (SNR) [56] are used for numerical calculation of the image after denoising.The function for MSE is shown as below:(53)MSE=1M×N∑i=1M∑j=1Nfj,i-f^ji2The functions for PSNR and SNR are shown as below:(54)PSNR=10×log102552MSE=20·log10255MSE(55)SNR=10×log10⁡∑i=1M∑j=1Nfj,i2M×N·MSEof whichM×Nis the size of test image; fj,i and f^ji represent the noise-free image and denoised image, respectively. In order to compare the performance of the proposed algorithm more comprehensively, the MSE function (53), the PSNR function (54) and SNR function (55) are used for comparison of the image denoised by algorithms in Figure 6. The calculation results are shown in Table 5.Table 5
Noise-free Lena image denoising effect evaluation index (n=3;δ=0.2).
Denosing method MSE PSNR/db SNR/db Image with noise 640.5461 20.0653 13.4480 FCMS 138.6119 26.7128 21.0557 FCMS1 179.6752 26.1314 20.0553 FCMS2 182.4737 25.5188 19.8616 FLICM 107.1739 26.1670 20.5099 ARKFCM 91.4922 28.8986 22.2414 FRRCM 67.5066 29.5425 23.8853 BFWCOM 54.9023 30.7349 24.6808The MSE here is referred to as the mean of the squared sum of gray level difference values of pixel points corresponding to the original image and denoised image. The smaller the value is, the better the denoised image quality is. The unit of PSNR is dB. The higher the value of PSNR is, the better the denoising effect is. According to function (53) and function (54) and by limitingfj,i-f^ji2≥1, PSNR∈0,48.12 after calculation. When the PSNR value here is above 40, it has been close to the original image. Therefore, 30.7349, the PSNR value of the image denoised by the BFWCOM algorithm, is quite good. SNR means signal-to-noise ratio. The larger its value, the better. By the comparison in Table 4, it can be seen that the denoising effect of BFWCOM algorithm in Lena image is better than that of other algorithms.


## 5. Conclusion

This paper proposes a fast and robust BFWCOM algorithm for image segmentation to improve image segmentation quality and reduce the impact of image noise. The following things are done:(1)
Improve segmentation effect by using the local spatial information of the image and the typicality of the data item.(2)
BFWCOM uses the membership filtering algorithm to exploit the local spatial constraint. Since noise points will have low compatibility in all clusters, the membership functions obtained by this algorithm more approach to the concept of typicality, making their impact on clustering negligible. Therefore, this algorithm is naturally more immune to noise.(3)
This algorithm has additional advantages in calculation. It is a natural mechanism, assigning “fuzzy labels” to data in each iteration. Therefore, it can be used for more complex pattern recognition.(4)
The convergence of BFWCOM is proved by mathematical theory, providing a theoretical preparation for the algorithm.(5)
Experimental results show that the proposed BFWCOM can provide better segmentation results without adjusting parameters for different grayscale.



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*Source: 5984649-2019-06-18.xml*