Robust deep kernel-based fuzzy clustering with spatial information for image segmentation

Clustering algorithms with deep neural network has attracted wide attention to scholars. A deep fuzzy K-means clustering algorithm model on adaptive loss function and entropy regularization (DFKM) is proposed by combining automatic encoder and clustering algorithm. Although it introduces adaptive loss function and entropy regularization to improve the robustness of the model, its segmentation effect is not ideal for high noise. The research purpose of this paper is to focus on the anti-noise performance of image segmentation. Therefore, on the basis of DFKM, this paper focus on image segmentation, combine neighborhood median and mean information of current pixel, introduce neighborhood information of membership degree, and extend Euclidean distance to kernel space by using kernel function, propose a dual-neighborhood information constrained deep fuzzy clustering based on kernel function (KDFKMS). A large number of experimental results show that compared with DFKM and classical image segmentation algorithms, this algorithm has stronger anti-noise robustness.


Introduction
In real life, side scan sonar image segmentation, biomedical CT image segmentation, satellite remote sensing image and a real image segmentation for automatic driving, etc., they all suffer from the influence of noise, which is not desirable in practice. In order to solve this problem, many scholars have derived many denoting algorithms based on the traditional fuzzy clustering algorithm. At the same time, there are many deep learning algorithms for removing rain, fog and noise, but there are few literatures about deep learning applied to noisy image segmentation. In recent years, clustering methods with multilayered neural networks have attracted great attention [13]. Some of these Lujia Lei leilujia@stu.xupt.edu.cn Chengmao Wu wuchengmao123@sohu.com Xiaoping Tian xptian@xupt.edu.cn 1 School of Electronics and Engineering, Xi'an University of Posts and Telecommunications, Xi'an 710121, China models separate the clustering process from the deep network training process, that is, apply the clustering method to the feature representation of the hidden layer of the training deep network. For example, Mojtaba Yeganejou et al. (2018) [1] proposed the deep clustering model, which combines the convolutional neural network (CNN) and FCM algorithm, has a good performance on the MNIST data set. Zhao Ziyuan et al. (2020) [2] combined deep belief network (DBN) and fuzzy C-means (FCM), and proposed an unsupervised deep fuzzy C-means clustering network (UDFCMN) to process lung CT images. In the network, the pre-processed image is first encoded as a multi-layer hidden layer vector to extract deep features, then the FCM algorithm is used to cluster to generate initial cluster labels, and the network parameters are fine-tuned by using the labels through back propagation. Caron et al. (2018) [3] proposed a clustering method called Deepcluster, which also directly uses the features obtained by the convolutional neural network (CNN) for clustering, and utilizes the clustering results as pseudo-labels to update the convolutional network. So that deep feature learning and clustering be achieved in the network at the same time. Secondly, some feature learning processes and clustering processes is combined in a unified framework. For example, Xie et al. (2016) [4] first proposed Deep Embedded Cluster (DEC), which embeds the conventional K-means algorithm into the auto-encoder network model for feature mining and sample clustering, and used Kullback-Leibler divergence loss function to fine-tune the parameters of the model. Because DEC model uses stacked auto-encoder instead of convolutional auto-encoder, it is not appropriate for high-dimensional images, so Li and Qiao et al. (2018) [5] proposed a deep embedded clustering model (DBC) based on convolutional auto-encoder. WANG et al. (2020) [6] proposed that the parallel multi-space optimizer (PMVO) was used to solve the complex multi-layer image segmentation problem based on the minimum cross entropy threshold. Yang et al. (2016) [7] proposed an unsupervised learning recursive framework (JULE) which combines deep representation and image clustering.In this framework,the image clustering process and the deep representation are carried out in the forward propagation and backward propagation of convolution network, respectively. At the same time, the clustering results can provide supervision information for the deep representation. However, these methods all use stochastic gradient descent (SGD), which leads to relatively slow convergence and falls into local optimum, the MFO proposed by T. -T. Nguyen et al. [8] may solve this problem. In contrast, Maziarmoradi Fard et al. (2020) [9] proposed a deep K-mean algorithm, which embeds the Kmean algorithm into the autoencoder. In the model, the deep representation and clustering parameters are jointly learned by the gradient update. Zhang and Li et al. (2020) [10] combined autoencoder and fuzzy k-means algorithm to form a deep fuzzy k-means clustering algorithm (DFKM) with adaptive loss and entropy regularization, which proposed a new gradient update to execute simultaneously deep feature extraction and clustering has good performance in data set and image segmentation. However, most of these deep clustering algorithm models do not consider the anti-noise robustness of high-dimensional data. Therefore, we try to use the spatial information of image pixels to deal with noise pixels and achieve the effect of denoising. The contribution of this paper are three-fold: 1) The local spatial information restriction term is introduced into the objective function of DFKM. However, due to the high complexity of each iteration of the algorithm,mean filtering and median filtering image pixels are used to replace local spatial information pixels, and similar effects can be achieved. 2) At the same time, the kernel function is used to extend the Euclidean space to the kernel space. 3) Because the deep robust fuzzy clustering algorithm embedded in a single neighborhood spatial information has certain limitations, we combine the pixel median and mean neighborhood information, and introduce neighborhood membership information, propose the final algorithm (KDFKMS).
A large number of image segmentation experiments show that the proposed algorithm not only improves the segmentation performance of the algorithm, but also enhances the anti-noise robustness.

Kernel-based fuzzy clustering
At present, kernel method is widely used in nonlinear classification in pattern recognition. In order to understand kernel method accurately, we need to understand kernel function first. Kernel function is defined as follows.

Definition 1 (Kernel function):
Suppose that H is a feature space and a mapping x H. If K , K is called kernel function. Kernel function between and is expressed as: where, represents inner product operation of x,y mapping on the feature space, then 2

2
(2) At present, Gaussian kernel is one of the most widely used kernel function, which is defined as follows.
Where and are the and samples in the entire data set, respectively; 2 is the Gaussian kernel function scale parameter, which can be used to indicate the difference between all samples from the entire data set, and is defined as follows.
Where, represents the gray distance difference between the sample point and the center 1 of the sample data set X 1 2 ,and is the average of the distances of all the sample points ,that is, 1 . From (3), we know that 1 1. Therefore, for Gaussian kernel function, (2) can be updated as follows. 2 2 2 (5) Yang et al. (2008) [11] introduced Gaussian kernel function into fuzzy clustering and proposed KFCM, which is to map the original space samples to high-dimensional feature space through Gaussian kernel function.Based on the above description, the fuzzy clustering algorithm with Gaussian kernel function is used to optimize the objective function as follows.
represents membership matrix,and is the clustering center matrix.
The membership function and cluster center updating formula corresponding to the above algorithm are as follows.
Here, represents the fuzzy index; is the total number of samples; is the number of clusters; represents the sample.The kernel fuzzy C-means clustering algorithm makes the linearly inseparable samples linearly separable in the high-dimensional feature space, which overcomes the lack of FCM for non-linear separable samples to a certain extent.
To test the effectiveness of introducing kernel function to DFKM, comparing the clustering results of DFKM and improved algorithm(DFKM) on image datasets, the results are shown in Fig. 1.
As can be seen from Fig. 1, compared with DFKM, KDFKM has more outstanding ability to retain details of images, because the introduction of kernel function overcomes the shortage of DFKM for nonlinear separable samples.   [12] proposed a fuzzy C-means clustering algorithm (FCM-S) combining spatial information by introducing local spatial constraints. This algorithm enhances the segmentation performance of noisy images to a certain extent, which is defined as follows.

Robust fuzzy clustering with spatial information
Where, represents the set of pixels in the local window centered on ; represents the number of pixels in the neighborhood window; represents membership matrix,and is the clustering center matrix; is the coefficient of the control neighborhood item,which is used to adjust the influence of neighborhood information on the center point.The algorithm has achieved good results in reducing the influence of noise on the image. Later, scholars   [13] proposed FCM-S1 and FCM-S2 that use mean filtering and median filtering image pixels instead of FCM-S neighborhood pixels to reduce the complexity of the FCM-S. To further enhance the robustness of FCM-S algorithm to image noise,   [14] used the kernel induced distance instead of Euclidean distance, and proposed a kernel fuzzy C-means clustering algorithm (KFCM-S) based on the constraints of the neighborhood space.The optimization model of KFCM algorithm can be expressed as follows.
At the same time, the in KFCM-S is replaced with mean filter pixels and median filter pixels to get KFCM-S1 and KFCM-S2 algorithms.
To test the anti-noise effectiveness of introducing the spatial information to DFKM, the improved algorithm is tested with mixed noise image with Gaussian noise with normalized variance of 0.05 and salt-pepper noise with intensity of 10%, compared with DFKM at the same time. As can be seen from Fig. 2, after introducing the neighborhood spatial information, the improved algorithm can significantly improve the noise suppression ability, and can eliminate most noisy pixel blocks. Lchihashi et al. (2001) [15] proposed a fuzzy C-means clustering algorithm (KLFCM) based on KL divergence. This algorithm does not require a fuzzy factor of membership, and the regularization term about KL divergence is added to the objective function of FCM. The objective function is defined as follows.

Robust fuzzy clustering with local membership
In which represents the trade-off factor for controlling the fuzziness of the regularization term and satisfies 0; is used to control the clustering scale, which represents the prior probability of class and satisfies 1 1, can be expressed as: 1 (12) KLFCM improves the segmentation performance of the original algorithm. In order to improve the anti-noise robustness of the algorithm, Gharieb et al. (2014) [16] proposed a robust fuzzy clustering algorithm based on KL divergence constraint with local membership degree (LMKLFCM), and its objective function is defined as follows.
Among them, is the scale parameter that controls the degree of fuzzy clustering; is the membership degree of neighborhood mean, which is calculated by averaging the membership degrees in neighborhood windows around the membership degree of the -th pixel, and can be expressed as: 1 (14) In which is a group of pixels in the neighborhood window of the pixel, and is the cardinal number of . Obviously, all the pixels in the window are linearly weighted and averaged according to the weight 1 , which makes the membership of the pixel in the center of the neighborhood constrained by the local membership ,so that the boundary area of the image changes more smoothly, thus achieving the anti-noise effect. Since the improved algorithm is added to the neighborhood mean membership degree, the image segmentation effect has been significantly improved, but it cannot handle more complex nonlinear neighborhood membership data well. Zhao and Quanhua (2019) [17] et al. The replacement in (13) is expressed as follows: (15) Where is the neighborhood pixel set of pixel ,and is defined as follows: Where is a constant in the formula.
To verify the effectiveness of introducing neighborhood mixed membership degree to DFKM, we use image to test the improved algorithm and compare it with DFKM. Figure 3 show that after the introduction of mixed membership degree, the improved algorithm has the effect of suppressing some noise points when dealing with image boundary pixels, which can smooth the image boundary and make the image result more delicate.

Proposed deep fuzzy clustering algorithm model
In this article, we use the framework of a combination of convolutional autoencoders and fuzzy clustering algorithms. Auto-encoder is a special instance of deep neural network. After training the network, the data can be reduced to a low-dimensional vector, and then try to reconstruct the input according to this vector. The feature of the auto-encoder is that it can learn deep feature representations in a completely unsupervised manner. The basic framework of this article is shown in Fig. 4, where Conv is the convolution operation, Tanh is the activation function, and DeConv represents the deconvolution operation.It can be seen from the figure that firstly, after the original image is preprocessed, rgb values of the image pixel neighborhood information is introduced into the convolutional auto-encoder network. By training the network, the hidden layer features (HL feature) are extracted in the 2-th layer, then introduce them into the objective function of the fuzzy clustering algorithm.
Hidden layer features are added simultaneously to gradient and weight update iterations as regularization items to optimize the network. Finally, the segmented image and the real image are subjected to performance analysis to draw conclusions.

Deep kernel fuzzy clustering with dual-neighborhood information
Because the adjacent pixels of an image have the characteristics of correlation with each other, the neighborhood information of pixels can improve the effective segmentation of noisy images. Firstly, the neighborhood information of pixel membership degree is introduced, and then the Euclidean distance space measure is extended to the kernel space distance measure, a robust image segmentation algorithm based on the neighborhood information of pixels based on kernel function is proposed.
Combining kernel function and neighborhood pixel spatial information restriction term, the objective function and constraints are defined as follows.
Where, is the number of cluster samples; is the number of clusters; is the center of cluster of low-dimensional feature space sample data, R , 2 ; represents membership matrix,and is the clustering center matrix; is the control parameter of the spatial information penalty item, is the trade-off parameter that controls the distribution of ; 1 and 2 are the trade-off parameters; represents the input of the first layer network, and 2 represents the characteristic data output by the 2-th layer network.
is described as: Where, 1 2 ; and is the number of nodes of the -th layer of neural network; is the activation function of the network layer, and are the weight and bias matrix of the corresponding layer.Because the neighborhood information penalty items in the objective function of the algorithm in this paper are all filtered by the mean or median of the features trained by the network,the updating iteration of the weight and bias matrix of the corresponding layer is still the method proposed by Zhang and Li et al. (2020) [10], so it is iterated by the random gradient descent method, and the iteration formula is as follows: where, and are the descent gradients of the weight and bias, namely:  Among them, the -th element of is . In (17), is described as follows: Where, is a trade-off parameter used to control the degree of outliers of various types. means filtering or median filtering for . Where, 2 represents the neighborhood mean or median value of the feature data output from the 2-th layer network.From (5), it can be seen that the objective function can be rewritten as follows. Where, is the introduced Lagrange multiplier, find partial derivatives of the above formulas about and respectively, and make its value 0, which can be obtained: . According to (28), the partial derivative of the cluster center can be obtained as follows.
Making the value of it 0, the cluster center can be obtained as: . Because the deep fuzzy clustering algorithm embedded with single neighborhood spatial information has some limitations. So, in order to meet the segmentation requirements of different noisy images, mean filtering neighborhood information and median filtering neighborhood information are simultaneously embedded into the objective function of the fuzzy clustering algorithm to improve the algorithm's segmentation performance.Although the segmentation effect has been improved, it is sensitive to strong noise and outliers. Therefore, on the basis of the improved objective function, the neighborhood information of membership degree is introduced, which makes the membership degree within classes tend to be consistent, thus achieving the purpose of improving the noise resistance and segmentation accuracy of the algorithm. The improved objective function and constraints are defined as follows. where, and β are the control parameters of spatial information penalty, and are the mean and median neighborhood filtering information of ,respectively. is defined as follows.
Where, is a constant and is defined as: Using Lagrange multiplier method to solve the optimization model corresponding to the above objective function,we have (37) and (38). . The outline of the proposed KDFKMS is shown in Algorithm 1.

Experimental results and discussion
In this paper, many clustering algorithms studied in the literature mentioned above are compared and analyzed comprehensively. We mainly introduces kernel function and neighborhood information into deep clustering algorithm. Therefore, we compare the KDFKMS algorithm with several classic fuzzy clustering algorithms that integrate image local spatial information,which include KFCM S1, KFCM S2, FGFCM, RPFCM, ILKFCM, FLICM, KFLICM and KWFLICM. In addition, we also compared the proposed algorithm with DFKM to analyze the effectiveness and robustness of the improved algorithm for segmentation of noisy images.

Evaluation index of performance
To objectively and quantitatively evaluate the effectiveness and robustness performance of algorithm. In this paper, accuracy ( ), precision ( ), specificity ( ), sensitivity ( ), misclassification error ( ) and modified peak signal-to-noise ratio ( ) are used as quantitative evaluation indexes.
is defined as: In which, is the number of pixels belonging to -th class in the image obtained by algorithm,and represents the number of pixels belonging to -th classes in the ground truth image. is the total number of image segmentation categories, and is the size of the image. The smaller is, the segmentation result is closer to the ground truth image, and better the performance of clustering algorithm. On the contrary, the poorer performance of fuzzy clustering segmentation algorithm.
The peak signal-to-noise ratio is an important indicator for evaluating the quality of an image or signal. In the research work, the modified peak signal-to-noise ratio proposed by Guo et al. (2012) [18] is more suitable for evaluating the anti-noise ability of image segmentation algorithms, and is defined as follows: 10 log 255 2 (40) Where, 1 1 1 2 , A is the mean square error between the ideal segmentation result of noiseless image and the actual segmentation result of noiseless image, is the pixel value of the noise less image at is the pixel value at obtained by segmentation algorithm for image corrupted by noise, and is the size of the image.The larger the value of ,the better the segmentation performance and robustness of the algorithm; on the contrary, the worse the segmentation performance and robustness of the algorithm.
For , , and , the larger the value, the better the segmentation performance of the algorithm. See [19,20] for its definition.

Test results and analysis
In the proposed algorithm, after a large number of experimental tests, the parameters are set to 1 0.1 2 0.01 0.5 β 0.5 0.1 0.01 3.8 3. 8 10 4 . In the experiment, the maximum iteration times and the algorithm end threshold of all algorithms are set to 100 and 10 3 , respectively. Fuzzy weighted index of all algorithms 2, The restriction parameters of KFCM S1 and KFCM S2 are both set to 3.8 as suggested by Ahmed et al., setting 3 6 in FGFCM as suggested by Cai et al. It should be pointed out that for the local spatial information of pixels, the neighborhood window cannot be too large, otherwise, the segmentation results will become blurred, so the size of neighbor window is setting as 3 3.

Segmentation performance on synthetic image
To begin with, we evaluate these algorithms with a backgroundfree synthetic four-cluster image shown in Fig. 5a. Different levels of Gaussian noise, Salt and pepper noise and Rician noise are added to this image. Figure 5 shows the segmentation results of different algorithms in Gaussian noisy synthetic image with normalized variance of 0.1, and segmentation results of the noisy synthetic image with Salt-and-pepper noise with intensity of 15% and Rician noisy image with standard deviation of 72 are given in Figa. 6 and 7 , , , , and indexes of the noisy images at different levels of each algorithm.
For the convenience later, DFKM, FGFCM, ILKFCM, RPFCM, KFCM S1, KFCM S2, FLICM, KFLICM, As can be seen from Fig. 5c-k, the DFKM, FGFCM, ILK-FCM, RPFCM, KFCM S1, KFCM S2, FLICM, KFLICM and KWFLICM are affected by Gaussian noise to different extent.Among them, DFKM and RPFCM have the worst effects under the influence of Gaussian noise, because the DFKM algorithm does not use any of the images spatial information, so a large amount of noise remains in the segmentation result.The FGFCM, ILKFCM, KFCM S1, KFCM S2, FLICM, KFLICM and KWFLICM is second,moreover, it is found from Fig. 5l that the proposed KDFKMS algorithm can eliminate almost all the noise.For the Salt and pepper noisy image, it can be seen from Fig. 6f that the RPFCM has the worst effect and divides the four clusters of errors into three clusters;the DFKM, FGFCM, KFCM S1 and FLICM algorithms segmentation results have more noise; although ILKFCM, KFCM S2, KFLICM and KWFLICM can remove most of the noise, but the visual segmentation effect is slightly worse than the KDFKMS algorithm.With respect to Rician noisy images, it can be seen from Fig. 7d and g that FGFCM and KFCM S1 algorithms divide four types of errors into three types, Fig. 7f shows that the segmentation results of RPFCM have more noise points, while ILKFCM, KFCM S2, FLICM and KFLICM have a few noise points, among which KWFLICM and KDFKMS have better segmentation performance, while KDFKMS has a little less error points.
Furthermore, from Table 1 that KDFKMS can obtain a smaller value of , and a higher value of , , , and , which indicates that the improved algorithm can achieve better segmentation accuracy and anti-noise performance.
To test the robustness performance of each algorithm, different levels of Gaussian noise,Salt and pepper noise and Rician noise are added to Fig. 5a, the varying curves of , , and of different segmentation algorithms are shown as follows.  Fig. 8, with the increase of Gaussian noise, the and of each algorithm decline, and the index increases.Form Fig. 9, when the intensity of Salt and pepper noise is in the range of 5%-20%, as the noise increases, the of the does not decrease but increases, and the does not increase but decreases. The reason is that FGFCM is too sensitive to Salt and pepper noise, which leads to the segmented image retaining a large number of amplified Salt and pepper noise points.And from Fig. 10, the and of KFCM S1, KFCM S2, FLICM and DFKM algorithms jump, which is due to the misclassification, that is, the four categories are divided into three categories, so the indexes become unstable. However, the algorithm in this paper always obtains the highest and and the highest . Each index changes steadily with the influence of noise. Therefore, the improved algorithm has stronger robustness and the ability to suppress noise interference.
As shown in Fig. 11a, we used the synthetic four-cluster image with background to quantify the performance of the    Figure 11 shows the segmentation results of the algorithm on Gaussian noisy images with normalized variance of 0.05. Figure 12 shows the segmentation result of the algorithm on the noisy image of Salt and pepper with intensity of 10%. Figure 13 shows the segmentation result of the algorithm on the Rician noisy image with standard deviation of 42. The testing indexes are shown in Table 2.
It can be seen from Fig. 11k that the segmentation effect of KWFLICM algorithm is better than FGFCM, ILKFCM, RPFCM, KFCM S1, KFCM S2, FLICM and KFLICM, but there are still a large number of noisy pixel blocks, and DFKM has relatively few noise points; It can be seen from Fig. 11l that the segmentation result of KDFKMS has almost no noise points, and retains the details and contour information of the image, which is closer to the original segmentation image and achieves better segmentation performance.
For Salt and pepper noise images in Fig. 12, the segmentation results of DFKM, FGFCM, ILKFCM, RPFCM, KFCM S1, FLICM, KFLICM and KWFLICM all contain more noise points. KFCM S2 introduces pixel neighborhood median information, which enhances the algorithm's ability to suppress Salt and pepper noise to a certain extent, but its ability is limited, and the segmentation result still has a small amount of noise; However, the algorithm in this paper makes full use of the spatial neighborhood information of the image, which can suppress the noise and retain the details and texture information in the image.
From Fig. 13, the segmentation results of FGFCM, ILKFCM, RPFCM, KFCM S1 and KFCM S2 still contain more Rician noise, which can not effectively segment image objects. Although DFKM, FLICM, KFLICM and KWFLICM can segment the target well, there are still many noise points in the background area; But proposed algorithm can effectively segment the target and suppress the background noise, so this algorithm has certain advantages for Rician noisy image segmentation.
Test indexes of different algorithms are given in Table 2. The data show that KDFKMS has smaller ME, and the test indexes of , , , and are larger than those of other algorithms. From various test indexes, we can further objectively judge the superior segmentation performance and robustness of the improved algorithm.
Similarly,different levels of Gaussian noise, Salt and pepper noise and Rician noise are added to Fig. 11a, the varying curves of , and of different segmentation algorithms are shown as follows.
It can be seen from Fig. 14 that compared with other algorithms, the proposed algorithm obtains the highest , , and . Each index decreases steadily with the change of noise intensity. From Fig. 15, the algorithm in this paper and the KFCM S2 have stronger anti-noise robustness.It can be seen from Fig. 16 that the , and of the algorithm in this paper and the DFKM are higher than other algorithms, these two algorithms have stronger antinoise robustness against Rician noise, but with the increase of noise, each index of DFKM drops faster, which shows that DFKM is more sensitive to strong noise than proposed algorithm.
Next, in order to further evaluate the performance of the algorithm, mixed noises include Gaussian noise with normalized variance of 0.05 and Salt and pepper noise with intensity of 10%; Gaussian noise with normalized variance of 0.1 and Rician noise with standard deviation of 32, Speckle noise with normalized variance of 0.3 and Salt and pepper noise with intensity of 10% are added to three-cluster synthetic image in Fig. 17a. The segmentation results of the algorithm in different mixed noise images are shown in Figs. 17, 18 and 19, respectively. Table 3 shows testing indexes. It can be seen from Fig. 17 that KWFLICM and the proposed algorithm perform well in segmentation results, and the segmentation results contain fewer noise points; DFKM and RPFCM have the worst segmentation results; FGFCM, ILKFCM, KFCM S1, KFCM S2, FLICM and KFLICM suppress the influence of noise to a certain extent, but the segmentation results still have more noise points. Figure 18 shows that DFKM and RPFCM algorithms are very poor in segmentation results, followed by FGFCM, ILKFCM, KFCM S1, KFCM S2 and KFLICM. The FLICM and KWFLICM have good segmentation results, and the KDFKMS algorithm has fewer noise points. It can be seen from Fig. 19 that the proposed algorithm is slightly inferior to KWFLICM in visual effect, but superior to other algorithms in processing images with 0.3Speckle noise and 10% Salt and pepper noise. According to Table 3, the index of this algorithm is slightly inferior to KWFLICM algorithm except ME and SE index, and most indexes of noisy images are better than other algorithms. Therefore, from the objective point of view of the test data, it further   shows that KDFKMS algorithm has better segmentation performance and anti-noise interference ability.

Segmentation Performance on real-world images
Next, we use real-world images from the Berkeley segmentation dataset and benchmark (BSDS500) (2011) [21] to further evaluate the segmentation effectiveness and antinoise robustness of the proposed algorithm and the others algorithm. Mixed noise, Gaussian noise, Salt and pepper noise, Rician noise and Speckle noise with different noise levels are added to the image, and the segmentation indexes of each algorithm are recorded and analyzed. Figures 20 and  21 show the segmentation results of mixed noise include Gaussian noise with normalized variance of 0.1 and Salt and pepper noise with intensity of 4% images, test indexes are given in Table 4. Figures 22 and 23 show the segmentation results of Gaussian noise images with normalized variance of 0.1 and 0.03 respectively, and the indexes are given in Table 5. The segmentation result of Salt and pepper noise with noise intensity of 30% images are given in Figs. 24 and 25, and the indexes are given in Table 6. The segmentation result of Rician noise with standard deviation of 82 images are given in Figs. 26 and 27, and the indexes are given in Table 7. Figures 28 and 29 show the segmentation results of Speckle noise images with normalized variance of 0.2 and 0.3 respectively, and the indexes are given in Table 8. Figures 20-29 show that the proposed algorithm in this paper has better segmentation performance than other algorithms on Gaussian noise images, Salt and pepper noise images, mixed noise images, Speckle noise images, and Rician noise images. It contains less misclassified pixels caused by noise, and can well preserve image details. Therefore, compared with other algorithms, the improved               1 2 β To test the influence of different parameters on the proposed algorithm, Gaussian noise with zero mean and normalized variance 0.1 is added into Fig. 5a Figure 30 shows the influence of parameter selection of and β on the proposed algorithm.

Parameters selection analysis
As can be seen from Tables 9 and 10, in the process of increasing parameters 1 and 2 , the PSNR value of image segmentation results basically increased first and then decreased, while the ME value decreased first and then increased. This indicates that parameter 1 is set between 0.1 and 1, and parameter 2 is set between 0.001 and 10, the segmentation effect can achieve satisfactory results. The parameter is sensitive to the algorithm in this paper. The shows PSNR and ME respectively value of PSNR reaches the peak value at 0.001 or 0.01, while the ME index reaches the trough value at 0.001 and 0.01. Therefore, the reference value range of parameter is better between 0.001-0.1. Besides, from Fig. 30, and β achieve satisfactory results near 10.

Complexity analysis and test of the proposed algorithm
The algorithm in this paper is mainly divided into two parts: network training and clustering algorithm.Because this algorithm takes the network layer data after training and then goes through clustering algorithm, so the computational complexity of the algorithm is mainly composed of clustering algorithm.The complexity of clustering algorithm mainly consists of two parts, one part comes from the calculation of local spatial information of each pixel, and the other part comes from the iteration of the algorithm.In order to compare the efficiency of the algorithm proposed in this paper with that of the algorithm, the computational complexity of different algorithms is analyzed as follows.
Suppose the image has pixels, is the the number of clusters, is the number of iteration steps, is the length of local window, then the computational complexity of KFCM S1 or KFCM S2 algorithm mainly consists of two parts, one part comes from the computational complexit 2 of the mean (or median) information of each pixel, and the other part is the algorithm iteration complexity ; the computational complexity of FGFCM is mainly composed of two parts, namely the computational complexit 2 of nonlinear weighting and filtered image and the complexity of algorithm iteration . It is well-known that the computational complexity of the FLICM is 2 ; for the KFLICM, have the same computational complexity 2 ; the computational complexity of the KWFLICM is 2 2 5 2 . The computational complexity of the DFKM is ; the computational complexity of the ILKFCM is 2 4 2 . The computational complexity of the algorithm in this paper is mainly composed of three parts, which are the computational complexity of the mean information of all pixels, the computational complexity of the median information of all pixels, and the computational complexity of the algorithm iteration, which is The main reason is that the algorithm contains the operation of pixel neighborhood information, which greatly affects the real-time performance. Combining the segmentation results with running time, it is not difficult to see that the improved algorithm can achieve better segmentation performance at the expense of real-time, which is the main deficiency of the improved algorithm.

Conclusion
Deep fuzzy clustering algorithm has become a research hotspot in the field of computer vision in recent years. Although deep fuzzy clustering algorithm has achieved good results in data set classification, but few people focus on image segmentation and consider its anti-noise robustness. In this paper, the neighborhood information of membership degree is introduced into the entropy regularization penalty of the deep fuzzy K-means algorithm, and the dual neighborhood information of pixels is introduced into the objective function of the deep fuzzy Kmeans algorithm, and its Euclidean distance is extended to the kernel space. The results of image segmentation test show that the proposed algorithm has better segmentation performance and anti-noise robustness. Although the research has achieved some results, there are still some challenges in many aspects. For example, after the introduction of neural networks, the time overhead is large. In addition, it needs further improvement in the trade-off algorithm's anti-noise performance and preservation of image details.