Image segmentation algorithm based on neutrosophic fuzzy clustering with non‐local information
2019; Institution of Engineering and Technology; Volume: 14; Issue: 3 Linguagem: Inglês
10.1049/iet-ipr.2018.5949
ISSN1751-9667
AutoresJinyu Wen, Shibin Xuan, Yuqi Li, Qihui Peng, Qing Gao,
Tópico(s)Face and Expression Recognition
ResumoIET Image ProcessingVolume 14, Issue 3 p. 576-584 Research ArticleFree Access Image segmentation algorithm based on neutrosophic fuzzy clustering with non-local information Jinyu Wen, Jinyu Wen School of Information Science and Engineering, Guangxi University for Nationalities, Daxue East Road 188, Nanning, People's Republic of ChinaSearch for more papers by this authorShibin Xuan, Corresponding Author Shibin Xuan xshibin1997@126.com School of Information Science and Engineering, Guangxi University for Nationalities, Daxue East Road 188, Nanning, People's Republic of ChinaSearch for more papers by this authorYuqi Li, Yuqi Li School of Information Engineering, Nanchang Hangkong University, Fenghe South Avenue 696, Nanchang, People's Republic of ChinaSearch for more papers by this authorQihui Peng, Qihui Peng School of Information Science and Engineering, Guangxi University for Nationalities, Daxue East Road 188, Nanning, People's Republic of ChinaSearch for more papers by this authorQing Gao, Qing Gao School of Information Science and Engineering, Guangxi University for Nationalities, Daxue East Road 188, Nanning, People's Republic of ChinaSearch for more papers by this author Jinyu Wen, Jinyu Wen School of Information Science and Engineering, Guangxi University for Nationalities, Daxue East Road 188, Nanning, People's Republic of ChinaSearch for more papers by this authorShibin Xuan, Corresponding Author Shibin Xuan xshibin1997@126.com School of Information Science and Engineering, Guangxi University for Nationalities, Daxue East Road 188, Nanning, People's Republic of ChinaSearch for more papers by this authorYuqi Li, Yuqi Li School of Information Engineering, Nanchang Hangkong University, Fenghe South Avenue 696, Nanchang, People's Republic of ChinaSearch for more papers by this authorQihui Peng, Qihui Peng School of Information Science and Engineering, Guangxi University for Nationalities, Daxue East Road 188, Nanning, People's Republic of ChinaSearch for more papers by this authorQing Gao, Qing Gao School of Information Science and Engineering, Guangxi University for Nationalities, Daxue East Road 188, Nanning, People's Republic of ChinaSearch for more papers by this author First published: 04 February 2020 https://doi.org/10.1049/iet-ipr.2018.5949Citations: 3AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract To improve the boundary processing ability and anti-noise performance of image segmentation algorithm,a neutrosophic fuzzy clustering algorithm based on non-local information is proposed here. Initially, the proposed approach uses the data distribution of deterministic subset to determine the clustering centre of the fuzzy subset. Besides, the fuzzy non-local pixel correlation is introduced into the neutrosophic fuzzy mean clustering algorithm. The experimental results on synthetic images, medical images and natural images show that the proposed method is more robust and more accurate than the existing clustering segmentation methods. 1 Introduction Image segmentation is an important task in image processing and computer vision applications [1, 2]. The purpose of image segmentation is to recognise the boundary and divide the image into the corresponding regions. It is still a challenging problem in practical image applications. At present, people have done extensive research on image segmentation and developed many different segmentation methods [3-5]. In recent years, many scholars have focused on the research and application of images [6-9]. In addition, the image segmentation is widely used in image processing [10-12]. However, the segmented image has usually noise and shows uncertainty. In order to process the uncertain information, a number of mathematical methods such as fuzzy set theory have been put forward. Fuzzy c-means (FCM) is a typical fuzzy clustering algorithm, which can provide the well-modelling results but cannot specify the number of clusters itself [13]. FCM can gain better segmentation result to non-noise images, but it is hard to obtain the desired effect when the segmented image includes noise, outliers or other image artefacts. The main reason is that the spatial information is not taken into account in the objective functions. The new clustering algorithm based on the fuzzy model is proposed by Choy et al. [14]. In this algorithm, the Kullback–Leibler metric, which measures the spatial difference between generalised probability distributions, is introduced into the objective function of the clustering, and a fuzzy generalised Gauss density segmentation is obtained. The advantage of the algorithm is that it is insensitive to initial parameters and effectively segments different texture images. It has a good segmentation effect, but its noise immunity is weak. In order to solve the shortcomings of the FCM algorithm in image segmentation, many kernel methods and even improved kernel methods have been proposed, for example an improved kernel-based fuzzy c-means (KFCM) clustering algorithm [15]. At present, there are many methods to optimise clustering by using the meta-heuristic algorithm [16-18]. Since then, some new image segmentation algorithms based on neutrosophic theory have been proposed [19-21]. Guo and Sengur [22] proposed a novel neutrosophic c-means (NCM) clustering algorithm, which can effectively define neutrosophic sets (NSs) and get more accurate segmentation images. However, these methods are more sensitive to noise, and do not consider the spatial information. In order to improve the sensitivity of algorithm to noise, various improved algorithms have been proposed. In literature [23], a new segmentation method is proposed, which combines the NCM clustering and indeterminacy filtering (IFNCM). First, the image is converted into an NS domain. Next, the filter is defined according to the uncertainty in the neutrosophic image. Then, based on global strength, the neutrosophic membership is defined by means of neutrosophic mean clustering. In order to eliminate the uncertainty of intensity, the method reuses the indeterminacy filter. Finally, the image is segmented according to the membership of the determined subset. In the offset corrected bias corrected version of FCM (BCFCM) [24], the spatial information is embedded into its objective function. When the neighbourhood information is calculated in each iteration, its efficiency is reduced. In order to reduce the time complexity of BCFCM, Chen and Zhang proposed two variants of BCFCM, named FCMS1 and FCMS2, in which the neighbourhood information computation was replaced by the median filter and the mean filter [25]. Most clustering algorithms have been developed in various fields in recent years. Among them, many algorithms adopt non-euclidean distance to measure the similarity and the kernel distance is their typical representative. However, it is difficult to choose the appropriate distance measure for a given data set. A good solution is to assign different weights to different distance measurements, and it is difficult to set appropriate weights as well [26]. Pointing to the power of spatial information in BCFCM is restricted by a parameter α, a method based on fuzzy local information clustering (FLICM) was proposed, which is mainly used in the diagnosis of lung cancer [27]. By means of this fuzzy factor, the membership of adjacent pixels in a local window will converge to a similar value. However, in FLICM, the attenuation degree of adjacent pixels to the central pixel is based on the Euclidean distance between pixels, which weakens the ability of FLICM in processing complex images. In order to speed up the process of image segmentation, enhanced FCM algorithm (EnFCM, fast generalised fuzzy C-means clustering (FGFCM)) was proposed [28]. EnFCM previously calculates the linear weighted sum between the original image and the mean filtered image before reality segmentation, so that it enjoys higher efficiency than BCFCM. So, based on the idea of EnFCM and FGFCM other algorithms have also been proposed one after the other. Similarly, FGFCM segments the image by means of image histograms, but the main distinction is that it is based on non-linear weighting. In literature [29] the kernel method is introduced into FLICM; furthermore, a novel segmentation algorithm KWFLICM is presented. However, only the local information is used in FLICM and KWFLICM, and it is very difficult to find a balance point between protecting image details and removing noise when the noise level is higher. In order to obtain the satisfactory segmentation results, this paper proposes a neutrosophic fuzzy clustering image segmentation algorithm based on the non-local information (NLNFC). In the segmentation process, the non-local information is taken into consideration. While maintaining the details of the image, the proposed algorithm enjoys stronger robustness to noise. In our NLNFC method, the data distribution of deterministic subset is used to define the cluster centre of fuzzy sets. At the same time, the pixel correlation measure based on non-local information is combined with the objective function of NLNFC. The experimental results show that the proposed algorithm can get better and more effective segmentation than the previous method on both noisy and noiseless images. The main contributions of this paper include: (i) By using the idea of neutral set, we improve the clustering centre of fuzzy terms of neutral mean fuzzy clustering, which can improve the segmentation accuracy. (ii) By referring to the non-local spatial information, we use the pixel location information to define the pixel correlation, which can enhance the robustness against noise. The rest of the paper is structured as follows: Section 2 introduces the related algorithms. Section 3 introduces the relevant work of our method. Section 4 has carried out the analysis and comparison of the experiment. Section 5 gives the conclusion of this study. 2 Related backgrounds 2.1 NCM clustering NCM [22] extends the traditional fuzzy clustering objective function and initialises three subsets (includes determining, uncertainty and noise subset) making use of the intensity and gradient of pixels and the clustering centre of certain subset. The membership of the certain subset can be obtained by means of the clustering centre. The clustering centre of the uncertain subset is determined by the centre of determinate subclass with maximum and secondary membership, that is the clustering centre of the fuzzy domain is replaced by the average of the two class centres. The membership of each sample point is calculated iteratively. Finally, all sample points are divided into several classes by the membership matrix. Its objective function is defined as follows: (1) (2) (3) (4)where N is the number of samples, C denotes the number of classes, m is a constant, and represent the class numbers corresponding to the largest and the second large membership of the sample point i, respectively. is defined by and . , and denote the membership values of sample I about deterministic clustering, edge region and noisy data set, respectively. Also, , and . The three memberships subject to the following constraint condition: (5)According to the conditions, the Lagrange objective function is constructed. By using the Lagrange multiplier method to minimise the objective function, that is to compute the partial derivative of the objective function for each unknown element and to set those derivatives equal to 0, we can obtain the equations for computing , and as follows: (6) (7) (8) (9) (10) According to the above equations, the NCM algorithm given in Fig. 1. Fig. 1Open in figure viewerPowerPoint NCM algorithm 2.2 FCM clustering of local information In the process of image segmentation, it is generally difficult to balance the noise and the reservation details, especially for the images with complex background and texture. The FCM clustering of local information algorithm (FLICM) is proposed to solve this problem. It is a non-parametric algorithm and a merging spatial information fuzzy factor is introduced into its objective function. The fuzzy factor is used to balance the noise reduction and the reservation details[27], which is defined as follows: (11)where represents the Euclidean distance between pixels j and r. represents the neighbourhood of pixel j and this fuzzy factor can ensure that the membership of both the non-noise pixels and the noise pixels that falling in the local window will all converge to a similar value, so that the membership of the pixels located in the window can be balanced. The objective function of FLICM is defined as (12)The membership and the cluster centre are calculated iteratively by (13) and (14): (13) (14) 2.3 FCM clustering of non-local information In fuzzy mean clustering based on local information, the fuzzy factor of a pixel expresses its influence on the centre pixel in a neighbourhood by the spatial Euclidean distance. If the distance is small, the influence on the centre pixel is larger, else the effect is smaller. In order to balance the quality and efficiency of the segmentation, in the FCM clustering algorithm for non-local information [28], a search window centred on a given pixel is defined, and only the pixels in the window with similar configuration will play a positive role in guiding the procedure of image segmentation. Based on the pixel correlation, the algorithm takes into account the non-local information of the image to improve the robustness. Therefore, it is very important to accurately measure the similarity between the pixels. represents the search window with size for pixel i. At the same time, the correlation between the two pixel points i and j is represented by , which is defined as (15)In addition, the following conditions should be satisfied: and are two image blocks centred on pixels i and j. If is close to 1, it means that two image blocks are similar. The larger between the pixels i and j, the greater effect the pixel j has on the pixel i and vice versa. Therefore, the membership of the two pixels will converge to a similar value, so that the robustness is enhanced. h is a smoothing parameter which is calculated based on the statistical characteristics of its search window. Its function is to control the attenuation degree of . is a normalisation factor, which defined as (16) (17) is the set of pixels in the window and is the cardinal number. The fuzzy factor of fuzzy mean clustering algorithm for non-local information is defined as (18)The algorithm is based on the above equation and its specific steps are given in Fig. 2. Fig. 2Open in figure viewerPowerPoint Fuzzy factor of fuzzy mean clustering algorithm 3 Image segmentation algorithm based on neutrosophic fuzzy clustering with non-local information The original neutrosophic fuzzy clustering algorithm is not very satisfactory for dealing with the boundary with weak discrimination and the selection of clustering centres does not take local information into account and the anti-noise performance is poor. In order to improve the accuracy of segmentation and enhance the robustness to noise, we use the idea of NS to improve the clustering centre of the fuzzy items in the neutral mean fuzzy clustering. At the same time, we use the non-local spatial information to define the correlation of pixels by using the location information of the pixels, and then deal with the noise-sensitive situation. 3.1 Clustering based on pixel correlation In order to reduce the influence of noise interference on image segmentation, we propose a clustering segmentation algorithm based on the correlation between pixels. Its objective function is defined as (19) It is similar to some parameters in the NCM algorithm. N is the number of samples, C is the number of classes and m is a constant. represents the clustering centre of the determined subset and denotes the clustering centre of the fuzzy subset. , and indicate the three memberships of sample i belonging to deterministic clustering, edge region and noise data set, respectively, and , and . The Lagrange objective function is constructed as follows: (20) In order to minimise the Lagrange objective function, we calculate the partial derivatives of the objective function by each unknown variables: (21) (22) (23) (24)Let every derivative be equal to 0 and set , then we can obtain the computational formulas as follows: (25) (26) (27) (28)The relationship between the three membership degrees should be satisfied: (29)Namely (30)In this way, the Lagrange factor can be obtained as (31) 3.2 Selection of the centre of fuzzy class In the image segmentation based on fuzzy clustering, the most prone to erroneous points are pixels located in connected regions between different classes. In fact, NCM is proposed to solve such problems, but the centre point () of the second item in the objective function, which is used to control the edge area, is always fixed by two classes with the maximum membership of the sample point , but the membership of other class is not fully taken into account. For this reason, we apply the following methods to redefine clustering centres in fuzzy items: (32) (33) (34)where denotes the clustering centre of the fuzzy subset and tt denotes the global threshold of the membership degree of the point i belonging to different classes. and represent the class number corresponding to the largest and second large membership of the sample point i, respectively. We use to represent the clustering centre of fuzzy subset. The selection of the cluster centre can be divided into two cases according to the data distribution of membership. In the first case, when there is a small list of mountains, that is when the other membership values are less than the threshold, equals the centre of the largest membership; on the other hand, when the polarisation trend of the membership degree occurs, that is when more than two membership values exceed the global threshold, is equal to the mean value of the centre of the class of the maximum and secondary memberships. Assuming there are five categories, the membership values of the sample point i are 0.0162, 0.0438, 0.5820, 0.0595 and 0.2985, respectively. Its threshold tt can be set as 0.1765. It can be seen that the point i most likely falls into the third category and the degree of membership is 0.5820. Secondly, it might also belong to the fifth category and this is due to the next largest membership value is 0.2985. Then and are 3 and 5, respectively. By calculating, we can get the average value of the third and fifth categories of fuzzy clustering centres. In this case, the fuzzy item clustering centre we defined is consistent with the centre of the NCM algorithm, that is is equal to . If the five membership degrees are 0.6870, 0.0378, 0.0058, 0.0035 and 0.0125, respectively, the sample points belonging to the first category have the largest membership values and the second largest is the second category. However, the global threshold of membership degree of this group is 0.3608 and the sub-maximum membership does not exceed the threshold, and the maximum membership degree is much larger than the others. In this case, the clustering centre corresponding to the second largest membership should be ignored. When the probability of some sample belongs to a class is much larger than that of other classes, the second membership does not have much effect. It shows that the category of the point does not exist fuzziness, so that the value of does not reduce the degree of uncertainty, but increases the ambiguity degree. On the above data comparison analysis, we can see that our proposal is closer to the actual situation than NCM. The redefined more accurately represents the fuzzy centre compared to the NCM method and shortens the convergence time of the algorithm. 3.3 Concrete steps of the algorithm The proposed neutrosophic fuzzy clustering algorithm with non-local information (NLNFC) is given in Fig. 3. Fig. 3Open in figure viewerPowerPoint Neutrosophic fuzzy clustering algorithm with non-local information (NLNFC) 4 Experimental analyses In order to prove the performance of the proposed algorithm, we choose three kinds of images for segmentation experiment, such as synthetic image, medical image and natural image, and add two different kinds of noise into testing all the images before the segmentation, and these noises include the Gaussian noise and salt and pepper noise. FCM, BCFCM, FCMS1, FCMS2, ENFCM, FGFCM, NCM, FLICM, KWFLICM are used to compare with the proposed algorithm. The experiment set, i.e. , and the size of the search window, is set to 7*7. 4.1 Example 1: 'Women' image By comparing and analysing the segmentation effect under different intensities of salt and pepper and Gaussian noise background, it can be seen that our algorithm has relatively good anti-noise performance. In this experiment, the real image 'Woman' is used to test the performance of the proposed NLNFC. The first column in Fig. 4 indicates that the image added salt and pepper noise with different variance, and the noise level is 0, 10 and 20%. In Fig. 4, the b column is the segmentation result of NCM. This method can almost correctly segment the image when the noise variance is zero, but produces also some error classification for noisy image and the resulting images are slightly rough with the noise intensity enhancement, so that NCM is affected more by the noise. Column c is the result of IFNCM, which is an improvement of NCM, and an uncertain filter is added before and after NCM. The filter is mainly used to remove the noise. IFNCM creates uncertain Gaussian filters based on uncertainty sets, which can reduce the noise to a some extent, but the intrinsic uncertainty and fuzziness of the indeterminate subset cannot ensure to select complete correct parameters, and this may further causes improper segmentation. As shown in column c, IFNCM can effectively restrain noise in the process of segmentation, but some details are smoothed out. Column d denotes our method. Intuitively, it has good results not only in the segmentation but also in the noise immunity compared with the foregoing two methods. Fig. 4Open in figure viewerPowerPoint Comparison results on 'Woman' image (a) 'Woman' image corrupted by different salt and pepper noise of level: 0, 10 and 20%, (b) NCM results, (c) IFNCM results, (d) NLNFC results Furthermore, the partition coefficient [30] is used to estimate the anti-noise performance of the proposed algorithm, which is a clustering validity function. is the cluster validity of the fuzzy set. is a matrix associated with the membership matrix. It has a unique global minimum in a continuous convex domain. is called the partition coefficient of membership matrix . Since its value is inversely proportional to the (overall) average coupling between pairs of fuzzy subsets in , indicates the average relative amount of membership sharing done between pairs of fuzzy subsets in by combining into a single number the average contents of pairs of fuzzy algebraic products. Partition coefficients indicate that the data set corresponding to all sets of fuzzy classes is compact and has a unique classification. When the partition coefficients reach the maximum, the clustering effect is the best. Furthermore, the partition coefficient is used to estimate the anti-noise performance of the proposed algorithm, which is a clustering validity function. The greater its value, the better the partition effect, and it can be defined as follows: (35)NCM, IFNCM and NLNFC are calculated ten times on noisy image contaminated by different noises. The results are shown in Table 1. Table 1. Two women images' partition coefficient (Vpc%) in different noise Fig. 4 NCM IFNCM NLNFC Gaussian 0% 40.41 44.03 58.4 Gaussian 10% 29.64 32.89 58.53 Gaussian 20% 25.52 38.84 57.09 salt and pepper 0% 40.41 44.03 58.4 salt and pepper 10% 39.69 43.27 58.13 salt and pepper 20% 39.94 42.52 57.78 It can be seen from Table 1 that IFNCM is slightly better than NCM and their anti-noise performance with the salt and pepper noise is stronger than the Gaussian noise. However, irrespective of Gaussian noise or salt and pepper noise, our method (NLNFC) has larger partition coefficients than other two methods and also it shows that our method has better robustness to noise. When there is a noise, the main reason for reduction of other algorithms is that the ambiguity becomes larger and this ambiguity will change the membership degree. The partition coefficient is directly related to the membership degree. Our algorithm takes spatial information into account. By introducing the pixel correlation, we can get more appropriate membership degree, thus reducing the ambiguity. 4.2 Example 2: synthetic image By comparing and analysing the segmentation effect under different intensities of salt and pepper and Gaussian noise background, we can show that our algorithm has better anti-noise performance and higher segmentation accuracy. In this experiment, two synthetic images are used for testing the image to compare the segmentation results of several algorithms. Figs. 5 and 6 show the results of several segmentation methods on two synthetic images added by 20% Gaussian noise and 30% salt and pepper noise, respectively. The original image in Fig. 5 is composed of two parts of the same size, the left pixel intensity value is 20 and the right one is 120. The original image in Fig. 6 consists of four parts and their pixel intensity values are 0, 85, 170 and 255, respectively. Fig. 5Open in figure viewerPowerPoint Comparison of segmentation results of first synthetic image corrupted by Gaussian noise of 20% level (a1) Original image, (a2) Result of FCM, (a3) Result of BCFCM, (a4) Result of FCMS1, (a5) Result of FCMS2, (b1) Result of EnFCM, (b2) Result of FGFCM, (b3) Result of FLICM, (b4) Result of KWFLICM, (b5) Result of NLNFC Fig. 6Open in figure viewerPowerPoint Comparison of segmentation results of second synthetic image corrupted by salt and pepper noise of 30% level (a1) Original image, (a2) Result of FCM, (a3) Result of BCFCM, (a4) Result of FCMS1, (a5) Result of FCMS2, (b1) Result of EnFCM, (b2) Result of FGFCM, (b3) Result of FLICM, (b4) Result of KWFLICM, (b5) Result of NLNFC All experimental results are shown in Figs. 5 and 6. It needs to explain that the intensities of pixels in our experiments are assigned as the value of the corresponding cluster centres and the segmentation results may have different colours. From the segmentation results of two synthetic images, it can be seen that the anti-noise performance of the traditional FCM is relatively poor. BCFCM, FLICM, KWFLICM and NLFCM in Fig. 5 have better robustness to Gaussian noise, and the resulting image almost does not include any noise, but it is evidently seen that this four algorithms cannot effectively erase noise in the image when the image is added by 30% salt and pepper noises. From segmentation results, our NLNFC method can almost eliminate all noises in the image under two kinds of noise. Considering that there exist different types of noise in the synthetic image, in order to further compare the effectiveness of each algorithm, we use the partition entropy to evaluate the performance of the algorithm. It is also an evaluation function of the clustering validity. If the value is small, there will be a better effect of partition, which is defined as (36)The partition entropy of the segmentation methods in Figs. 5 and 6 is shown in Table 2 as follows. Table 2. Two synthetic images' partition entropy (Vpe%) in different noise Fig. 5 FCM BCFCM FCMS1 FCMS2 EnFCM FGFCM FLICM KWFLICM NLNFC Gaussian 15% 25.94 31.1 17 16.52 9.19 7.78 25.64 15.58 13.97 Gaussian 20% 26.41 31.52 17.42 17 9.56 8.33 25.91 15.73 16.32 Gaussian 30% 26.67 32.15 17.97 17.31 10.03 8.59 26.17 15.94 19.87 salt and pepper 15% 12.64 53.9 28.02 11.55 19.06 5.69 25.85 10.22 8.6 salt and pepper 20% 17.31 66.77 34.9 15.89 22.92 8.76 32.32 13.52 10.51 salt and pepper 30% 30.99 86.84 48.68 25.45 30.42 16.21 47.57 20.35 14.51 Fig. 6 FCM BCFCM FCMS1 FCMS2 EnFCM FGFCM FLICM KWFLICM NLNFC Gaussian 15% 30.23 33.01 55.63 54.74 13.97 41.2 48.33 18.77 18.66 Gaussian 20% 28.85 31.81 28.12 56.33 16.01 42.53 80.26 19.97 22.78 Gaussian 30% 26.97 30.83 36.21 33.63 22.57 20.41 68.36 28.1 29.74 salt and pepper 15% 35.66 51.78 54.82 33.75 33.77 16.9 81.57 48.64 12.01 salt and pepper 20% 35.57 56.83 68.73 45.69 41.32 25.02 101.638 63.26 15.61 salt and pepper 30% 35.37 67.68 95.21 72.78 53.75 41.81 140.81 94.82 22.21 From the 12 sets of data listed in Table 2, we can see that in the case of Gaussian noise, the minimum partition entropy of EnFCM and FGFCM is the lowest, both of which are around 10%, but the partition entropy of two methods presents the significant variation with the enhancement o
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