Robust intuitionistic fuzzy clustering with bias field estimation for noisy image segmentation

Abstract

The concept of intuitionistic fuzzy set has been found to be highly useful to handle vagueness in data. Based on intuitionistic fuzzy set theory, intuitionistic fuzzy clustering algorithms are proposed and play an important role in image segmentation. However, due to the influence of initialization and the presence of noise in the image, intuitionistic fuzzy clustering algorithm cannot acquire the satisfying performance when applied to segment images corrupted by noise. In order to solve above problems, a robust intuitionistic fuzzy clustering with bias field estimation (RIFCB) is proposed for noisy image segmentation in this paper. Firstly, a noise robust intuitionistic fuzzy set is constructed to represent the image by using the neighboring information of pixels. Then, initial cluster centers in RIFCB are adaptively determined by utilizing the frequency statistics of gray level in the image. In addition, in order to offset the information loss of the image when constructing the intuitionistic fuzzy set of the image, a new objective function incorporating a bias field is designed in RIFCB. Based on the new initialization strategy, the intuitionistic fuzzy set representation, and the incorporation of bias field, the proposed method preserves the image details and is insensitive to noise. Experimental results on some Berkeley images show that the proposed method achieves satisfactory segmentation results on images corrupted by different kinds of noise in contrast to conventional fuzzy clustering algorithms.

Keywords

Image segmentation intuitionistic fuzzy set fuzzy clustering initial cluster centers noise robustness

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1. Introduction

Image segmentation plays an important role in image analysis and understand. The purpose of image segmentation is dividing the image into several homogeneous segments according to some features such as intensity, color, tone, texture, and shape. In general, these features show high intra-similarity in the same group and low inter-similarity between two different group [1]. At present, there are many representative gray image segmentation techniques which include edge-based segmentation algorithms [2], thresholding-based segmentation algorithms [3], region-based segmentation algorithms [4], and clustering-based segmentation algorithms [5, 6]. Clustering-based segmentation methods cluster image pixels into different groups by using some cluster validation measures [7, 8, 9, 10, 11]. In 1965, Zadeh introduced the concept of fuzzy sets into the crisp clustering algorithms and proposed fuzzy c-means clustering algorithm (FCM) [12]. By optimizing the objective function of FCM, the pixels are clustered to different groups according to the membership degree [13]. The traditional FCM algorithm is very sensitive to noise because of not considering any spatial information of pixels in the image [14, 15, 16]. Moreover, the unsuitable initial cluster centers may lead to the poor segmentation result.

In order to decrease the noise sensitivity of FCM, many researchers have incorporated local spatial information into FCM. Ahmed et al. [17] introduced a spatial neighborhood term into the objective function of FCM and proposed FCM algorithm with spatial information (FCM_S). In order to avoid the computation of spatial neighborhood term in each iteration, Chen and Zhang [18] presented FCM with mean spatial information (FCM_S1) and FCM with median spatial information (FCM_S2) to reduce the computational complexity of FCM_S. The neighborhood terms in FCM_S1 and FCM_S2 are calculated based on the mean filtered and median filtered images, respectively. Ji et al. [19] proposed a modified possibilistic fuzzy c-means clustering algorithm for bias field estimation for fuzzy segmentation of magnetic resonance (MR) images. In order to improve the robustness of FCM to noise, Cai et al. [20] proposed a fast generalized fuzzy c-means clustering algorithm (FGFCM) incorporating both local spatial coordinates and gray information. Moreover, by integrating the guided filter, Guo et al. [21] designed a general framework to improve the fuzzy clustering based noisy image segmentation. In the iteration of this algorithm, the original noisy image is utilized to post-process the fuzzy memberships.

The concept of intuitionistic fuzzy set (IFS) is presented for fuzzy set generalizations by Atanassov [22] and has been successfully applied to many application areas. Compared with fuzzy sets, the intuitionistic fuzzy set (IFS) theory considers the membership and non-membership degree and can accurately and completely express the uncertain information of data. Moreover, intuitionistic fuzzy sets theory is frequently utilized in image processing, especially as a means to coping with noise [23]. Based on intuitionistic fuzzy set, Chaira [24] proposed a novel intuitionistic fuzzy c-means clustering algorithm (IFCM1). By utilizing the intuitionistic fuzzy distance to replace the distance, Xu and Wu [25] proposed another version of the intuitionistic fuzzy c-means clustering algorithm (IFCM2). In IFCM2, Yager operator [26] is utilized to construct the hesitation degree. Although the intuitionistic fuzzy c-means clustering algorithm can successfully handle the uncertainty in the data, these algorithms are very sensitive to image noise due to not considering any spatial information derived from the image. In order to overcome this shortcoming, Verma utilized the local spatial information in an intuitionistic fuzzy way and proposed an improved intuitionistic fuzzy c-means clustering algorithm (IIFCM) [27]. However, IIFCM algorithm only presents the superior performance on images with Salt&pepper or Poisson noise. Furthermore, existing intuitionistic fuzzy c-means clustering algorithms are still sensitive to the initial cluster centers.

In order to solve the above two problems of intuitionistic fuzzy c-means clustering algorithms, a noise robust intuitionistic fuzzy clustering c-means (RIFCB) clustering algorithm based on bias field estimation is presented in this paper. Firstly, a noise robust intuitionistic fuzzy set is firstly constructed by using the neighboring information of pixels in the image. This novel noise robust intuitionistic fuzzy set can overcome the influence of image noise. Secondly, the initial cluster centers of RIFCB are selected by utilizing the frequency statistics strategy of gray level. Finally, in order to avoid the loss of noise removal, a bias field is designed to estimate the original image and introduced into the objective function of RIFCB. Through minimizing this objective function, the updating functions of membership degree, cluster centers, and bias field are obtained. Then the final segmentation result can be achieved according to the membership degree of each pixel. The advantages of the proposed approach are given as follows: 1) it is not easy to fall into the local optimal solution because of initializing the cluster centers. 2) it has strong robustness to different types of noise when using the noise robust intuitionistic fuzzy representation. 3) based on the designed bias field, it can reserve more image details. In the experiments, FCM, FCM_S1, FCM_S2, FGFCM, IFCM1, and IIFCM algorithms are chosen as the comparison algorithms. Segmentation results of a synthetic image and some Berkeley segmentation images corrupted by noise show that the RIFCB algorithm can achieve the satisfactory segmentation performance.

The remainder of this paper is organized as follows: Section 2 presents a brief introduction of fuzzy c-means clustering and intuitionistic fuzzy c-means clustering algorithms. Section 3 describes robust Intuitionistic fuzzy clustering c-means algorithm for noisy image segmentation in detail. In Section 4, the performance of the proposed method is verified by the segmentation of Berkeley images. Some concluding remarks and discussions are given in Section 5.

2. Fuzzy c-means clustering and intuitionistic fuzzy c-means clustering algorithms

2.1 Fuzzy sets and intuitionistic fuzzy sets

Fuzzy sets theory was first proposed by Zadeh [12]. The fuzzy set $A$ on the pixels set $X=\{x_{1},\linebreak x_{2},\ldots,x_{n}\}$ of the image is defined as:

$\displaystyle A=\{\mu_{A}(x_{i})|0\leqslant\mu_{A}(x_{i})\leqslant 1,x_{i}\in X\}$ (1)

where $\mu_{A}(x_{i}):X\to[0,1]$ denotes the membership degree of the pixel $x_{i}$ in fuzzy set $A$ . The non-membership degree of the pixel $x_{i}$ in fuzzy set $A$ is defined as $v_{A}(x)=1-\mu_{A}(x)$ . In image segmentation, if $\mu_{A}$ ( $x_{i}$ ) close to 1, the pixel $x_{i}$ is more likely to belong to $A$ . If $\mu_{A}$ ( $x_{i}$ ) close to 0, this means that the pixel $x_{i}$ is less likely to belong to $A$ [28].

However, there may be some hesitation degree while determining the degree of membership. Thus, Atanassov introduced the concept of intuitionistic fuzzy sets (IFSs) theory [29] and defined the intuitionistic fuzzy set $B$ which is given as follows:

$\displaystyle B=\{(\mu_{B}(x_{i}),v_{B}(x_{i}))|0\leqslant\mu_{B}(x_{i}),v_{B}% (x_{i})\leqslant 1,\mu_{B}(x_{i})+v_{B}(x_{i})\leqslant 1,x_{i}\in X\}$ (2)

where $\mu_{B}(x_{i}):X\to[0,1]$ and $v_{B}(x_{i}):X\to[0,1]$ denote the membership degree and non-membership degree for each $x_{i}\in X$ in the set $B$ , respectively. The hesitation degree is defined by:

$\displaystyle\pi_{B}(x_{i})=1-\mu_{B}(x_{i})-v_{B}(x_{i})$ (3)

Therefore, the value of membership degree is no longer a single value and is expressed as the interval form $[\mu_{B}(x_{i}),\mu_{B}(x_{i})+\pi_{B}(x_{i})]$ .

2.2 Fuzzy c-means clustering algorithm

Fuzzy c-means clustering algorithm (FCM) [7] is a traditional clustering method in the field of image segmentation. Its objective function is defined as:

$\displaystyle J_{\textit{FCM}}(U,C)=\sum\limits_{i=1}^{n}\sum\limits_{k=1}^{K}% u_{ki}^{m}d^{2}(x_{i},c_{k})$ (4)

where $U={\{}u_{ki}{\}}$ is the membership matrix and $u_{ki}$ should satisfy the following formula [30]:

$\displaystyle\sum\limits_{k=1}^{K}u_{ki}=1,u_{ki}\in[0,1],\forall i=1,2,\cdots,n$ (5)

$C={\{}{c}_{1},{c}_{2},\ldots,{c_{K}}{\}}$ is the set of cluster centers, $K$ is the number of clusters, $m$ is the fuzzy exponent which controls the fuzziness of the image segmentation results, $d(x_{i},c_{k})$ denotes the Euclidean distance $\sqrt{\|x_{i}-c_{k}\|^{2}}$ between the sample $x_{i}$ and the cluster center $c_{k}$ .

The membership degree $u_{ki}$ and the cluster center $c_{k}$ can be derived by using Lagrange multipliers method to minimize the objective function. Their updating functions are given as follows:

$\displaystyle u_{ki}=\frac{{1}}{\sum\limits_{j=1}^{K}\left(\frac{d(x_{i},c_{k}% )}{d(x_{i},c_{j})}\right)^{\frac{2}{m-1}}}$ (6) $\displaystyle c_{k}=\frac{\sum\limits_{i=1}^{n}u_{ki}^{m}x_{i}}{\sum\limits_{i% =1}^{n}u_{ki}^{m}}$ (7)

2.3 Intuitionistic fuzzy c-means clustering algorithm

Intuitionistic fuzzy c-means clustering algorithm is based on the intuitionistic fuzzy set theory. Xu and Wu [25] utilized the Euclidean intuitionistic fuzzy distance to replace the Euclidean distance in FCM and proposed the intuitionistic fuzzy c-means clustering algorithm (IFCM2). The objective function of IFCM2 is given as follows:

$\displaystyle J_{\textit{IFCM}}=\sum\limits_{i=1}^{n}\sum\limits_{k=1}^{K}u_{% ki}^{m}d^{2}(x_{i}^{\textit{IFS}},c_{k}^{\textit{IFS}})$ (8)

where $d(x_{i}^{\textit{IFS}},c_{k}^{\textit{IFS}})$ denotes the Euclidean intuitionistic fuzzy distance and is defined by:

$\displaystyle d(x_{i}^{\textit{IFS}},c_{k}^{\textit{IFS}})=\sqrt{(\mu(x_{i})-% \mu(c_{k}))^{2}+(v(x_{i})-v(c_{k})){}^{2}+(\pi(x_{i})-\pi(c_{k}))^{2}}$ (9)

In order to construct a non-membership function, researchers have proposed a variety of intuitionistic fuzzy generation operators based on membership functions.

(1) Yager’s intuitionistic fuzzy complement [26]

$\displaystyle v(x_{i})=(1-\mu(x_{i})^{\alpha})^{1/\alpha}0<\alpha<\infty$ (10)

$v(x_{i})$ is the non-membership degree of $x_{i}$ , $\alpha$ is constant parameter, the hesitation degree $\pi(x_{i})$ is defined as:

$\displaystyle\pi(x_{i})=1-\mu(x_{i})-(1-\mu(x_{i})^{\alpha})^{1/\alpha}$ (11)

(2) Sugeno’s intuitionistic fuzzy complement [31]

$\displaystyle v(x_{i})=\frac{1-\mu(x_{i})}{1+\alpha\mu(x_{i})}\alpha>0$ (12)

where the meaning of parameters is same as Yager’s intuitionistic fuzzy complement. The hesitation degree is defined as:

$\displaystyle\pi(x_{i})=1-\mu(x_{i})-\frac{1-\mu(x_{i})}{1+\alpha\mu(x_{i})}$ (13)

Through minimizing the objective function of IFCM2 by using Lagrange multipliers, the updating equations of the membership degree $u_{ki}$ and the cluster center $c_{k}^{\textit{IFS}}$ are given as follows:

$\displaystyle u_{ki}=\frac{{1}}{\sum\limits_{j=1}^{K}\left(\frac{d(x_{i}^{% \textit{IFS}},c_{k}^{\textit{IFS}})}{d(x_{i}^{\textit{IFS}},c_{j}^{\textit{IFS% }})}\right)^{\frac{2}{m-1}}}$ (14) $\displaystyle c_{k}^{\textit{IFS}}=\left(\frac{\sum\limits_{i=1}^{n}u_{ki}^{m}% \mu(x_{i})}{\sum\limits_{i=1}^{n}u_{ki}^{m}}\frac{\sum\limits_{i=1}^{n}u_{ki}^% {m}v(x_{i})}{\sum\limits_{i=1}^{n}u_{ki}^{m}}\frac{\sum\limits_{i=1}^{n}u_{ki}% ^{m}\pi(x_{i})}{\sum\limits_{i=1}^{n}u_{ki}^{m}}\right)$ (15)

3. Robust intuitionistic fuzzy clustering c-means algorithm for noisy image segmentation

In order to solve the problems of the noise sensitiveness and the random initialization of cluster centers of IFCM2, a robust intuitionistic fuzzy clustering with bias field estimation (RIFCB) is proposed for noisy image segmentation in this paper.

3.1 Construction of the noise robust intuitionistic fuzzy set for the image

Let $X=\{x_{1},{x}_{2},\ldots,x_{n}\}$ represent the input image. To improve the robustness of IFCM1 against noise, a noise robust intuitionistic fuzzy set $I=\{(\delta_{i},\mu_{I}(\delta_{i}),\nu_{I}(\delta_{i}),\pi_{I}(\delta_{i}))\}$ is constructed for the image. The noise robust membership degree $\mu_{I}(\delta_{i})$ of $i$ th pixel is computed by

$\displaystyle\mu_{I}(\delta_{i})=\frac{\delta_{i}}{255}$ (16)

where $\delta_{i}$ is the weight mean gray value of the $i$ th pixel and computed by using its neighboring pixel. $\delta_{i}$ is defined as follows:

$\displaystyle\delta_{i}=\frac{\sum\limits_{p\in N_{i}}E_{ip}x_{p}}{\sum\limits% _{p\in N_{i}}E_{ip}}$ (17)

where $x_{p}$ is the gray value of the $p$ th pixel in the neighboring windows $N_{i}$ around the $i$ th pixel. The size of the neighboring windows is $\omega$ . $E_{ip}$ represents the local similarity weight between the centered pixel $x_{i}$ and the neighboring pixel $x_{p}$ . This similarity weight utilizes the spatial coordinates and gray values, and is defined as:

$\displaystyle E_{ip}=\exp\left(-\frac{\max(|a_{i}-a_{p}|,|b_{i}-b_{p}|)}{% \lambda_{s}}-\frac{\|x_{p}-\bar{x}\|^{2}}{\lambda_{p}\sigma_{i}^{2}}\right)$ (18)

where ( $a_{i},b_{i}$ ) is the spatial coordinates of $x_{i}$ . $\lambda_{s}$ denotes the spatial scale parameter of $E_{ip}$ and is generally fixed to 3 [20]. $\lambda_{g}$ denotes the adaptive grey weight parameter of $E_{ip}$ and is calculated by:

$\displaystyle\lambda_{g}=\frac{o_{p}-\bar{o}}{\sum\limits_{p\in N_{i}}(o_{p}-% \bar{o})}+1$ (19)

where $o_{p}=\|x_{p}-\bar{x}\|$ denotes the local gray statistic similarity measure of the pixel in neighborhood window. $\bar{x}$ is the average gray value of all pixels in the neighborhood window $N_{i}$ . $\bar{o}$ denotes the average value of all $o_{p}$ . $\sigma_{i}$ is calculated by:

$\displaystyle\sigma_{i}=\sqrt{{\sum\limits_{p\in N_{i}}\left\|x_{p}-\bar{x}% \right\|^{2}\mathord{\left/{\vphantom{\sum\limits_{p\in N_{i}}\left\|x_{p}-% \bar{x}\right\|^{2}\left|N_{i}\right|}}\right.\kern-1.2pt}\left|N_{i}\right|}}$ (20)

where $|N_{i}|$ is the number of pixels in the neighborhood window $N_{i}$ .

After constructing the membership degrees of the image pixel, the non-membership degrees can be computed by some fuzzy complement operators [5, 31]. Since, compared to Sugeno’s generator, Yager’s generator produces better results [24]. Therefore, we used Yager’s generator to construct the non-membership degree in this paper, which is defined as:

$\displaystyle v_{I}(\delta_{i})=(1-\mu_{I}(\delta_{i})^{\alpha})^{\frac{1}{% \alpha}}$ (21)

where $\alpha$ is a constant parameter and is generally set between 0.8 and 0.9 [24]. In our method, $\alpha$ is fixed to 0.85 according to the analysis in Ref. [24]. The hesitation degree is defined by:

$\displaystyle\pi_{I}(\delta_{i})=1-\mu_{I}(\delta_{i})-v_{I}(\delta_{i})$ (22)

3.2 Determination of initial cluster centers

The common initialization method is choosing random $K$ vectors as initial center from the data sample space. However, this random initialization may lead to the unstable cluster performance. Because there are some pixels owning the similarity feature in the image, we can use the statistic characters of pixels to initial cluster centers. Gray histogram can reflect the frequency distribution of the images grayscale well [32]. The optimal cluster centers are likely to be located in regions with larger diversity. However, the grayscale histogram may have burrs or small local fluctuations when the image is corrupted by noise [33]. Therefore, a Gaussian filter method is utilized to process the burrs in the histogram and an initialization method of cluster centers is proposed based on the smoothed histogram of the image $\delta$ in this paper.

Let $h$ be the histogram of the image $\delta$ obtained by Eq. (17) and $h({l})$ denote the frequency of gray level $l(0\leqslant l\leqslant 255)$ in $\delta$ . In order to remove the burrs near the peak of the gray histogram, the smoothed value $f(l)$ of gray level $l$ is calculated by:

$\displaystyle f(l)=[[h(l)]_{\textit{fft}}\times g(l)]_{\textit{ifft}}$ (23)

where $[\,]_{\textit{fft}}$ represents the fast Fourier transform and $[\,]_{\textit{ifft}}$ is the fast inverse Fourier transform. $g(l)$ is the Gaussian filter function which is calculated by:

$\displaystyle g(l)=\frac{1}{\sqrt{2\pi}}e^{\left(-\frac{l^{2}}{2\theta^{2}}% \right)}$ (24)

where $\theta$ denotes the standard deviation and is calculated by:

$\displaystyle\theta=\sqrt{\frac{\sum(l-\bar{l})^{{2}}}{N-1}}$ (25)

where $\bar{l}$ denotes the average value of all gray level $l$ and $N$ is the number of gray level. The purpose of the above steps is to filter the outliers of the signal $h({l})$ in the frequency domain.

After obtaining the smoothed histogram, we use $f^{\prime}(l)$ to denote the differential between two gray values in the $f(l)$ . If $f^{\prime}(l)$ is equal to 0, $f(l)$ is a minimum or maximum value of the local area. If $f(l)$ satisfies Eq. (26), $f(l)$ is selected as the peak point. The corresponding gray level $l$ is selected as one of the initialized cluster centers.

$\displaystyle\left\{\begin{array}[]{lll}{f^{\prime}(l)<0}&&{l=0}\\ {f^{\prime}(l-1)>0,f^{\prime}(l)=0,f^{\prime}(l+1)<0}&&{0<l<255}\\ {f^{\prime}(l)>0}&&{l=255}\\ \end{array}\right.$ (26)

Let $Z$ denote the number of peak points $\{l_{1},l_{2},\ldots,l_{Z}\}$ . If the cluster number is $K$ , the initial cluster centers are selected according to the following conditions:

(1)

If $K=Z$ , the initialized cluster centers are $\{l_{1},l_{2},\ldots,l_{Z}\}$ .

(2)

If $K<Z$ , that is too many peak points. According to the value of $f(l_{i})$ in descending order, the first $K$ gray values are selected as initialized cluster centers.

(3)

If $K>Z$ , the numbers of peak points are not enough. Therefore, $K-Z$ random gray values are selected from the set [0, 255] and $Z$ peak points $\{l_{1},l_{2},\ldots,l_{Z}\}$ are adopted as the initial cluster centers.

After obtaining the initial cluster centers, the intuitionistic fuzzy representation of cluster centers can be computed. Therefore, the membership degree $\mu_{I}(c_{k})$ of $k$ th cluster center is computed by:

$\displaystyle\mu_{I}(c_{k})=\frac{l_{k}}{255}$ (27)

The non-membership degree and the hesitation degree of initial cluster centers are computed according to Eqs (21) and (22).

Figure 1a shows the image #3096 selected from the Berkeley Image Segmentation Database and Fig. 1b shows the frequency distribution curve of the gray value of image #3096. After frequency domain filtering, the new frequency distribution curve is shown in Fig. 1c. The peak points in the red circle are selected as the peak points and their corresponding normalization gray values are the initialized cluster centers.

Figure 1.

The effect of frequency domain filtering. (a) Image #3096, (b) gray value frequency distribution curve of image #3096, (c) gray value frequency distribution curve after frequency domain filtering.

3.3 Objective function of RIFCB

The construction of noise robust intuitionistic fuzzy set utilizes the weight mean gray value of the pixel. Although this construction strategy can remove the noise in the image, the own information of the original image may be simultaneously removed. Therefore, in order to compensate for the loss of useful information, a bias field $\gamma_{i}^{\textit{IFS}}$ is designed in our method. Therefore, the intuitionistic fuzzy set $\chi_{i}^{\textit{IFS}}$ of the original image is modeled as the noise robust intuitionistic fuzzy set plus the intuitionistic fuzzy set of a bias field.

$\displaystyle\chi_{i}^{\textit{IFS}}=\delta_{i}^{\textit{IFS}}+\gamma_{i}^{% \textit{IFS}}$ (28)

Generally, the bias field $\gamma_{i}^{\textit{IFS}}$ is unknown. Through introducing $\chi_{i}^{\textit{IFS}}$ into the objective function of the proposed method, we can estimate $\gamma_{i}^{\textit{IFS}}$ by using an iterative algorithm based on fuzzy logic. The objective function of RIFCB is defined as:

$\displaystyle J=\sum\limits_{i=1}^{n}\sum\limits_{k=1}^{K}u_{ki}^{m}d^{2}(% \delta_{i}^{\textit{IFS}},c_{k}^{\textit{IFS}})+\beta\sum\limits_{i=1}^{n}\sum% \limits_{k=1}^{K}u_{ki}^{m}d^{2}(\gamma_{i}^{\textit{IFS}},c_{k}^{\textit{IFS}})$ (29)

where $\delta_{i}^{\textit{IFS}}$ and $c_{k}^{\textit{IFS}}$ are IFSs of the pixel and cluster centers, respectively. $K$ is the number of clusters. $u_{ki}$ is the membership degree value of $\delta_{i}^{\textit{IFS}}$ belonging to the $k$ th cluster. $m$ is the fuzzy exponent. $d(\delta_{i}^{\textit{IFS}},c_{k}^{\textit{IFS}})$ denotes the Euclidean intuitionistic fuzzy distance which can be computed by Eq. (9). In the second term, $\beta$ is the deviation information correction factor.

By using the Lagrange method to minimize the above objective function, the updating functions of the membership matrix $u_{ki}$ , cluster centers $c_{k}^{\textit{IFS}}$ and bias field $\gamma_{i}^{\textit{IFS}}$ are defined as follows:

$\displaystyle u_{ki}=\frac{1}{\sum\limits_{j=1}^{K}\left(\frac{d^{2}(\delta_{i% }^{\textit{IFS}},c_{k}^{\textit{IFS}})+\beta d^{2}(\delta_{i}^{\textit{IFS}}+% \gamma_{i}^{\textit{IFS}},c_{k}^{\textit{IFS}})}{d^{2}(\delta_{i}^{\textit{IFS% }},c_{j}^{\textit{IFS}})+\beta d^{2}(\delta_{i}^{\textit{IFS}}+\gamma_{i}^{% \textit{IFS}},c_{j}^{\textit{IFS}})}\right)^{\frac{1}{m-1}}}$ (30) $\displaystyle c_{k}^{\textit{IFS}}=(\mu(c_{k}),\nu(c_{k}),\pi(c_{k}))=\left(% \frac{\sum\limits_{i=1}^{n}u_{ki}^{m}((1+\beta)\mu(\delta_{i}^{\textit{IFS}})+% \beta\mu(\gamma_{i}^{\textit{IFS}}))}{(1+\beta)\sum\limits_{i=1}^{n}u_{ki}^{m}% },\right.\left.\frac{\sum\limits_{i=1}^{n}u_{ki}^{m}((1+\beta)v(\delta_{i}^{% \textit{IFS}})+\beta v(\gamma_{i}^{\textit{IFS}}))}{(1+\beta)\sum\limits_{i=1}% ^{n}u_{ki}^{m}},\frac{\sum\limits_{i=1}^{n}u_{ki}^{m}((1+\beta)\pi(\delta_{i}^% {\textit{IFS}})+\beta\pi(\gamma_{i}^{\textit{IFS}}))}{(1+\beta)\sum\limits_{i=% 1}^{n}u_{ki}^{m}}\right)$ (31) $\displaystyle\gamma_{i}^{\textit{IFS}}=(\mu(\gamma_{i}),\nu(\gamma_{i}),\pi(% \gamma_{i}))=\left(\frac{\sum\limits_{k=1}^{K}u_{ki}^{m}\mu(c_{k})}{\sum% \limits_{k=1}^{K}u_{ki}^{m}}-\mu(x_{i}),\frac{\sum\limits_{k=1}^{K}u_{ki}^{m}v% (c_{k})}{\sum\limits_{k=1}^{K}u_{ki}^{m}}-v(x_{i}),\frac{\sum\limits_{k=1}^{K}% u_{ki}^{m}\pi(c_{k})}{\sum\limits_{k=1}^{K}u_{ki}^{m}}-\pi(x_{i})\right)$ (32)

The detail of the derivation process of these three updating functions is presented in the Appendix A.

3.4 Framework of RIFCB algorithm

In this section, the procedure of RIFCB is shown in Algorithm 1.

Algorithm 1: Robust Intuitionistic fuzzy c-means clustering based on bias field estimation for noisy image segmentation
Input: Image $X=\{{x}_{1},{x}_{2},\ldots,{x}_{n}\}$ , the number of clusters $K$ , the fuzzy exponent $m$ , the size $\omega$ of neighborhood window, the parameters $\lambda_{s}$ and $\beta$ , the maximum number of iterations $T$ .
Output: Image segmentation result. Step 1: Construct a noise robust intuitionistic fuzzy set. The membership degree $\mu_{I}(\delta_{i})$ , non-membership degree $v_{I}(\delta_{i})$ , and hesitation degree $\pi_{I}(\delta_{i})$ are calculated by Eqs (16), (21), and (22) respectively. Step 2: Determine initial cluster centers by using the proposed initialization strategy and compute the corresponding intuitionistic fuzzy set of cluster centers. Step 3: Set the number of iterations $\textit{iter}=1$ . Step 4: Computer the membership matrix $u_{ki}$ using Eq. (30). Step 5: Computer the cluster centers $c_{k}^{\textit{IFS}}$ using Eq. (31). Step 6: Computer the bias field $\gamma_{i}^{\textit{IFS}}$ using Eq. (32). Step 7: $\textit{iter}=\textit{iter}+1$ . If iter is larger than $T$ , goto step 8. Otherwise, go to step 4. Step 8: The maximum membership procedure is used as the defuzzification process. The $i$ th pixel is assigned to the class $k$ with the highest membership $\displaystyle k=\arg_{k}\{\max(u_{ki})\}(2\leqslant k\leqslant K)$ (33) Then, the final segmentation result is obtained.

Algorithm 1: Robust Intuitionistic fuzzy c-means clustering based on bias field estimation for noisy image segmentation

Input: Image

X=\{{x}_{1},{x}_{2},\ldots,{x}_{n}\}

, the number of clusters

K

, the fuzzy exponent

m

, the size

\omega

of neighborhood window, the parameters

\lambda_{s}

and

\beta

, the maximum number of iterations

T

Output: Image segmentation result.

Step 1:
Construct a noise robust intuitionistic fuzzy set. The membership degree $\mu_{I}(\delta_{i})$ , non-membership degree $v_{I}(\delta_{i})$ , and hesitation degree $\pi_{I}(\delta_{i})$ are calculated by Eqs (16), (21), and (22) respectively. Step 2:
Determine initial cluster centers by using the proposed initialization strategy and compute the corresponding intuitionistic fuzzy set of cluster centers.
Step 3:
Set the number of iterations $\textit{iter}=1$ .
Step 4:
Computer the membership matrix $u_{ki}$ using Eq. (30).
Step 5:
Computer the cluster centers $c_{k}^{\textit{IFS}}$ using Eq. (31).
Step 6:
Computer the bias field $\gamma_{i}^{\textit{IFS}}$ using Eq. (32).
Step 7:
$\textit{iter}=\textit{iter}+1$ . If iter is larger than $T$ , goto step 8. Otherwise, go to step 4.
Step 8:
The maximum membership procedure is used as the defuzzification process. The $i$ th pixel is assigned to the class $k$ with the highest membership

$\displaystyle k=\arg_{k}\{\max(u_{ki})\}(2\leqslant k\leqslant K)$ (33)

Then, the final segmentation result is obtained.

4. Results and discussion

In order to verify the effectiveness of RIFCB, FCM [17], FCM_S1 [18], FCM_S2 [18], FGFCM [20], IFCM1 [24] and IIFCM [27] are adopted as comparative methods in this paper. In these methods, FCM_S1, FCM_S2, IIFCM, and RIFCB are all utilize the spatial information of the image. All the methods are carried out on one synthetic image and ten natural images (#3096, #15088, #24063, #135069, #238011, #42044, #42049, #100007, #118035, and #241004) selected from Berkeley Image Database [2]. For all the methods, the fuzziness exponent $m$ is set to 2 [17], the maximal iteration number $T$ is set to 300 and the termination threshold $\varepsilon$ is set to 10 ${}^{{-5}}$ . For FCM_S1, FCM_S2, FGFCM, and RIFCB, the neighborhood window size $\omega$ is 3. For FGFCM and RIFCB, $\lambda_{s}$ is set to 3 (Cai, Chen, & Zhang, 2007). For FGFCM, $\lambda_{g}$ is set to 3 (Cai, Chen, & Zhang, 2007).

In this paper, three evaluation criteria are adopted to compare the performance of all the algorithms.

(1) Segmentation accuracy (SA) [5] denotes the percentage of correctly classified pixels in the total pixels of the image and is computed by:

$\displaystyle{\rm SA}=\sum\limits_{k=1}^{K}\frac{|A_{k}\bigcap D_{k}|}{n}% \times 100\%$ (34)

where | | is the cardinality of a set. $A_{k}$ denotes the set of pixels belonging to the $k$ th cluster. $D_{k}$ denotes the set of pixels belonging to the $k$ th group in the ground truth.

(2) Partition coefficient ( $V_{pc}$ ) [34] is a validity function to evaluate the clustering algorithms, defined as

$\displaystyle V_{pc}=\left.\sum\limits_{i=1}^{n}\sum\limits_{k=1}^{K}u_{ki}^{2% }\right/n$ (35)

where $u_{ki}$ is the membership degree and $n$ is the number of pixels in the input image.

(3) Partition entropy ( $V_{pe}$ ) [34] is another validity function to evaluate the clustering algorithms, defined as

$\displaystyle V_{pe}=\left.-\sum\limits_{i=1}^{n}\sum\limits_{k=1}^{K}(u_{ki}% \log u_{ki})\right/n$ (36)

4.1 Parameter analysis of the deviation information correction factor

\beta

In this section, we focus on the discussion about the effect of the deviation information correction factor $\beta$ on the performance of RIFCB. One synthetic image and five Berkeley images (#3096, #15088, #24063, #135069, and #238011) with different noise are utilized to test the influence of $\beta$ in this section. According to the definition of this bias field and the designation of the objective function, the noise robust intuitionistic fuzzy set $\delta_{i}^{\textit{IFS}}$ should play more important part in the proposed objective function. Therefore, the deviation information correction factor $\beta$ should not be assigned to a large value. In the experiment, $\beta$ is tested on the set $\{$ 0, 0.001, 0.002, 0.005, 0.01, 0.02, 0.05, 0.1 $\}$ . Gaussian white noise with different noise variance (NV) and Salt&pepper noise with different noise density (ND) are added to these images, respectively. Figures 2 and 3 show the SA curves of RIFCB varying with the parameter $\beta$ on the noisy images. From the results of #3096 and #135069, we can see that a decreasing trend of SA against $\beta$ is demonstrated when $\beta$ is greater than 0.05. For other images, the changing trend of SA against $\beta$ assumes relative smoothness. It can be found that $\beta$ has a certain impact on the segmentation performance of the proposed method and should be set different value according to different images. By comprehensive consideration of noise type and noise level, $\beta=$ 0.01 is the preferable choice for our proposed method.

Figure 2.

Segmentation accuracy (SA) against the parameter $\beta$ on images corrupted by Gaussian white noise. (a) Synthetic image, (b) #3096, (c) #15088, (d) #24063, (e) #135069 and (f) #238011.

Figure 3.

Segmentation accuracy (SA) against the parameter $\beta$ on images corrupted by Salt&pepper noise. (a) Synthetic image, (b) #3096, (c) #15088, (d) #24063, (e) #135069 and (f) #238011.

4.2 Effectiveness verification of cluster centers initialization

In order to investigate the effectiveness of the strategy for initializing cluster centers, we apply FCM and RIFCB with different initialization strategies on the image #238011 corrupted by Gaussian white noise (NV $=$ 0.002) (Shown in Fig. 4a). Figure 4b shows the segmentation benchmark image of the noisy image. Figure 4c–f show the segmentation results of these methods. We can conclude from the results that the moon can be accurately segmented in Fig. 4d and f because of the effective cluster centers. According to analysis of the obtained results, the proposed initialization strategy is effective.

Figure 4.

Segmentation results with different initialization. (a) #238011, (b) benchmark image, (c) FCM algorithm with random initialization of cluster centers, (d) FCM algorithm that initializing the cluster center method by this paper, (e) RIFCB algorithm that randomly initialize the cluster center, (f) RIFCB algorithm that initializing the cluster center method by this paper.

Table 1

Segmentation results of all comparison methods on synthetic image with different noise

Noise type	Criteria	FCM		FCM_S1		FCM_S2		FGFCM		IFCM1		IIFCM		RIFCB
Gaussian (NV $=$ 0.01)	SA (%)	89.	79	99.	30	99.	89	99.	11	89.	61	99.	79	99.	92
	Vpc	0.	8184	0.	9178	0.	9061	0.	9358	0.	8115	0.	9568	0.	9693
	Vpe	0.	3561	0.	1919	0.	2154	0.	1492	0.	3693	0.	0759	0.	0629
Salt&pepper (ND $=$ 0.05)	SA (%)	96.	35	96.	55	99.	76	97.	70	96.	30	96.	24	99.	87
	Vpc	0.	9797	0.	9149	0.	9701	0.	9771	0.	9772	0.	9795	0.	9924
	Vpe	0.	0615	0.	1766	0.	0679	0.	0513	0.	0677	0.	0620	0.	0224

Figure 5.

Original square image and noisy image. (a) synthetic image, (b) benchmark image, (c) noisy image with Gaussian white noise, (d) noisy image with Salt&pepper noise.

Figure 6.

Segmentation results on the synthetic image corrupted by the Gaussian white noise (NV $=$ 0.01). (a) FCM, (b) FCM_S1, (c) FCM_S2, (d) FGFCM, (e) IFCM1, (f) IIFCM, and (g) RIFCB.

Figure 7.

Segmentation results on the synthetic image corrupted by Salt&pepper noise (ND $=$ 0.05). (a) FCM, (b) FCM_S1, (c) FCM_S2, (d) FGFCM, (e) IFCM1, (f) IIFCM, and (g) RIFCB.

Figure 8.

Images #3096, #24063, #135069, and #238011. (a) Image corrupted by Gaussian white noise (NV $=$ 0.006), (b) Image corrupted by Salt&pepper noise (ND $=$ 0.1), and (c) Benchmark image.

Figure 9.

Segmentation results on the Image #3096 corrupted by Gaussian white noise. (a) FCM, (b) FCM_S1, (c) FCM_S2, (d) FGFCM, (e) IFCM1, (f) IIFCM and (g) RIFCB.

Figure 10.

Segmentation results on the Image #3096 corrupted by Salt&pepper noise. (a) FCM, (b) FCM_S1, (c) FCM_S2, (d) FGFCM, (e) IFCM1, (f) IIFCM and (g) RIFCB.

Figure 11.

Segmentation results on the Image #24063 corrupted by Gaussian white noise. (a) FCM, (b) FCM_S1, (c) FCM_S2, (d) FGFCM, (e) IFCM1, (f) IIFCM and (g) RIFCB.

Figure 12.

Segmentation results on the Image #24063 corrupted by Salt&pepper noise. (a) FCM, (b) FCM_S1, (c) FCM_S2, (d) FGFCM, (e) IFCM1, (f) IIFCM and (g) RIFCB.

Figure 13.

Segmentation results on the Image #135069 corrupted by Gaussian white noise. (a) FCM, (b) FCM_S1, (c) FCM_S2, (d) FGFCM, (e) IFCM1, (f) IIFCM and (g) RIFCB.

Figure 14.

Segmentation results on the Image #135069 corrupted by Salt&pepper noise. (a) FCM, (b) FCM_S1, (c) FCM_S2, (d) FGFCM, (e) IFCM1, (f) IIFCM and (g) RIFCB.

Figure 15.

Segmentation results on the Image #238011 corrupted by Gaussian white noise. (a) FCM, (b) FCM_S1, (c) FCM_S2, (d) FGFCM, (e) IFCM1, (f) IIFCM and (g) RIFCB.

Figure 16.

Segmentation results on the Image #238011 corrupted by Salt&pepper noise. (a) FCM, (b) FCM_S1, (c) FCM_S2, (d) FGFCM, (e) IFCM1, (f) IIFCM and (g) RIFCB.

4.3 Segmentation experiments on synthetic image

To compare the segmentation performances of the comparative algorithms, we apply these algorithms to a synthetic image presented in Fig. 5a (with 256 $\times$ 256 pixels, four classes with grayscale values taken from the interval [0, 30], [70, 110], [150, 180], and [220, 250]) corrupted by Gaussian white noise (NV $=$ 0.01) and Salt&pepper noise (ND $=$ 0.05), respectively. The corresponding benchmark image and noisy images are presented in Fig. 5b–d.

We applied these algorithms to the synthetic image corrupted by the Gaussian white noise and Salt&pepper noise. The SA, $V_{pc}$ and $V_{pe}$ values of all comparison methods are shown in Table 1. Figures 6 and 7 show the segmentation results by FCM, FCM_S1, FCM_S2, FGFCM, IFCM1, IIFCM, and RIFCB algorithms. Table 1 shows that RIFCB algorithm obtains the highest SA, $V_{pc}$ and $V_{pe}$ values among all algorithms. Due to not considering any spatial information, FCM, IFCM1, and IIFCM cannot obtain satisfying segmentation results of noisy images. For the image corrupted by Gaussian white noise, FCM_S1, FCM_S2, FGFCM, and RIFCB algorithm can overcome the influence of the noise in the image. For the image corrupted by Salt&pepper noise, the visual segmentation result of RIFCB is the best among all the comparison methods. Therefore, RIFCB algorithm is robust to Gaussian white noise and Salt&pepper noise and obtain the better performance.

Table 2
Segmentation accuracy (%) of different segmentation algorithms on natural images with different noise

Image	Noise type	FCM	FCM_S1	FCM_S2	FGFCM	IFCM1	IIFCM	RIFCB
#3096	Gaussian (0,0.002)	90.35	96.99	97.00	96.88	95.93	97.89	97.71
	Gaussian (0,0.006)	70.84	94.72	94.82	95.40	74.48	89.55	97.03
	Gaussian (0,0.01)	61.14	88.87	86.59	89.24	68.89	71.57	96.12
	Salt&pepper 2%	96.94	95.30	97.18	97.03	97.29	73.40	97.98
	Salt&pepper 5%	95.83	87.55	96.04	95.86	95.95	95.96	97.97
	Salt&pepper 10%	93.49	77.06	92.45	92.25	93.56	93.55	97.74
#15088	Gaussian (0,0.002)	91.02	93.05	92.90	92.76	91.67	92.05	93.17
	Gaussian (0,0.006)	87.24	92.99	92.81	92.62	89.37	90.32	93.02
	Gaussian (0,0.01)	82.24	92.73	92.51	92.33	85.98	87.61	93.04
	Salt&pepper 2%	91.28	92.87	92.91	92.75	91.63	91.75	93.17
	Salt&pepper 5%	90.12	92.37	92.79	92.55	90.56	90.67	93.12
	Salt&pepper 10%	88.15	90.38	92.53	91.70	88.54	88.65	93.09
#24063	Gaussian (0,0.002)	93.28	94.93	94.99	94.74	94.11	94.37	95.98
	Gaussian (0,0.006)	87.15	94.51	94.15	94.24	89.09	90.45	95.55
	Gaussian (0,0.01)	79.41	94.05	93.39	93.54	82.96	85.16	95.07
	Salt&pepper 2%	93.83	94.20	95.01	94.57	94.27	94.25	96.08
	Salt&pepper 5%	91.57	92.58	94.81	93.84	92.04	92.06	96.04
	Salt&pepper 10%	88.14	89.09	94.41	92.37	88.30	88.49	95.79
#42044	Gaussian (0,0.002)	75.17	76.54	76.05	76.05	75.12	76.58	77.61
	Gaussian (0,0.006)	75.19	76.24	76.01	75.70	72.72	76.81	77.67
	Gaussian (0,0.01)	74.96	75.89	76.06	75.52	68.30	76.17	77.41
	Salt&pepper 2%	73.42	76.27	76.35	76.04	76.64	75.05	77.61
	Salt&pepper 5%	70.20	75.99	76.54	75.95	76.82	72.52	77.58
	Salt&pepper 10%	65.66	75.36	76.95	75.40	76.06	68.22	77.56
#42049	Gaussian (0,0.006)	95.11	95.94	96.00	96.03	95.20	95.84	96.09
	Gaussian (0,0.01)	94.12	95.90	95.91	95.97	94.46	95.33	95.99
	Gaussian (0,0.015)	92.03	95.76	95.75	95.77	93.10	94.78	95.87
	Salt&pepper 5%	93.84	95.48	96.22	95.92	93.55	95.11	96.23
	Salt&pepper 10%	91.38	94.68	96.06	95.16	91.09	93.55	96.19
	Salt&pepper 15%	89.08	93.02	95.95	93.88	88.78	91.14	96.01
#100007	Gaussian (0,0.002)	81.25	81.66	81.77	82.03	82.19	82.12	82.50
	Gaussian (0,0.006)	79.81	81.44	81.49	81.90	80.90	81.35	82.21
	Gaussian (0,0.01)	77.40	81.42	81.50	81.58	79.51	80.00	82.15
	Salt&pepper 2%	81.54	81.18	81.71	81.93	82.23	82.27	82.58
	Salt&pepper 5%	81.24	80.54	81.60	81.38	82.09	82.05	82.56
	Salt&pepper 10%	80.41	78.63	81.31	80.19	81.50	81.40	82.47
#118035	Gaussian (0,0.001)	88.71	89.13	89.82	89.61	88.10	84.72	90.07
	Gaussian (0,0.002)	85.13	89.25	89.78	89.55	83.29	74.18	90.04
	Gaussian (0,0.006)	75.62	89.26	88.84	88.71	74.29	71.32	89.36
	Salt&pepper 2%	89.33	88.13	89.68	88.94	89.36	89.44	90.05
	Salt&pepper 5%	87.73	86.39	89.46	88.18	85.80	85.83	90.03
	Salt&pepper 10%	78.78	83.17	88.94	86.53	79.06	79.08	89.81
#135069	Gaussian (0,0.002)	66.92	69.54	70.94	69.89	91.16	72.05	99.14
	Gaussian (0,0.006)	62.88	68.78	68.88	68.64	64.49	65.92	74.08
	Gaussian (0,0.01)	59.42	67.85	67.01	67.72	62.28	63.23	72.72
	Salt&pepper 2%	69.94	67.65	69.88	70.14	74.54	74.56	99.18
	Salt&pepper 5%	66.95	65.61	67.40	67.25	96.72	92.72	99.16
	Salt&pepper 10%	68.40	61.51	66.78	66.43	94.22	65.98	99.07
#238011	Gaussian (0,0.002)	60.07	66.00	63.31	63.05	58.97	59.70	95.91
	Gaussian (0,0.006)	56.05	61.11	60.38	60.35	52.77	53.19	95.87
	Gaussian (0,0.01)	55.07	59.63	59.06	60.15	51.82	51.36	95.61
	Salt&pepper 2%	94.00	71.51	78.68	79.06	93.99	93.96	95.01
	Salt&pepper 5%	91.42	80.62	74.86	73.52	91.45	91.47	95.91
	Salt&pepper 10%	87.49	69.93	69.17	90.05	87.44	87.24	95.88

Table 2, continued
Image	Noise type	FCM	FCM_S1	FCM_S2	FGFCM	IFCM1	IIFCM	RIFCB
#241004	Gaussian (0,0.01)	64.67	89.36	89.23	89.33	57.74	88.25	89.42
	Gaussian (0,0.015)	60.05	87.84	86.82	86.70	53.72	66.57	88.02
	Gaussian (0,0.02)	57.25	85.53	83.89	83.17	51.64	58.42	85.82
	Salt&pepper 5%	88.76	87.86	91.46	90.68	88.36	90.39	91.73
	Salt&pepper 10%	85.56	82.61	90.64	88.34	85.09	88.43	91.53
	Salt&pepper 15%	69.96	77.58	90.10	85.43	81.32	73.75	90.89

Table 3

Comparison of partition coefficient ( $V_{pc}$ ) of different segmentation algorithms on natural images with different noise

Image	Noise type	FCM	FCM_S1	FCM_S2	FGFCM	IFCM1	IIFCM	RIFCB
#3096	Gaussian (0,0.002)	0.8638	0.9141	0.9192	0.9220	0.9038	0.9859	0.9357
	Gaussian (0,0.006)	0.8142	0.8815	0.8850	0.8990	0.8191	0.8966	0.9235
	Gaussian (0,0.01)	0.8134	0.8281	0.8150	0.8553	0.8154	0.8394	0.9094
	Salt&pepper 2%	0.9346	0.9346	0.8970	0.9227	0.9363	0.9351	0.9408
	Salt&pepper 5%	0.9325	0.9325	0.8396	0.9027	0.9338	0.9334	0.9400
	Salt&pepper 10%	0.9255	0.9255	0.7892	0.8566	0.9296	0.9276	0.9375
#15088	Gaussian (0,0.002)	0.9086	0.9344	0.9331	0.9363	0.9188	0.9256	0.9449
	Gaussian (0,0.006)	0.8603	0.9208	0.9194	0.9302	0.8771	0.9001	0.9407
	Gaussian (0,0.01)	0.8237	0.8974	0.8934	0.9242	0.8335	0.8768	0.9373
	Salt&pepper 2%	0.9251	0.9284	0.9310	0.9366	0.9328	0.9350	0.9453
	Salt&pepper 5%	0.9237	0.8994	0.9187	0.9325	0.9324	0.9335	0.9453
	Salt&pepper 10%	0.9186	0.8340	0.8927	0.9241	0.9304	0.9312	0.9446
#24063	Gaussian (0,0.002)	0.8543	0.8742	0.8745	0.8799	0.8607	0.8688	0.8865
	Gaussian (0,0.006)	0.8114	0.8591	0.8543	0.8698	0.8202	0.8307	0.8780
	Gaussian (0,0.01)	0.7906	0.8445	0.8357	0.8585	0.7981	0.8055	0.8702
	Salt&pepper 2%	0.8856	0.8661	0.8787	0.8822	0.8898	0.8898	0.8900
	Salt&pepper 5%	0.8853	0.8418	0.8679	0.8768	0.8881	0.8873	0.8896
	Salt&pepper 10%	0.8847	0.7982	0.8499	0.8669	0.8856	0.8837	0.8882
#42044	Gaussian (0,0.002)	0.9228	0.9258	0.9274	0.9286	0.9148	0.9191	0.9296
	Gaussian (0,0.006)	0.8935	0.9170	0.9138	0.9263	0.8711	0.8903	0.9284
	Gaussian (0,0.01)	0.8662	0.9044	0.8971	0.9230	0.8444	0.8697	0.9234
	Salt&pepper 2%	0.9324	0.9178	0.9291	0.9307	0.9320	0.9321	0.9328
	Salt&pepper 5%	0.9286	0.8885	0.9185	0.9264	0.9272	0.9279	0.9324
	Salt&pepper 10%	0.9284	0.8368	0.8979	0.9202	0.9249	0.9247	0.9308
#42049	Gaussian (0,0.006)	0.9372	0.9402	0.9440	0.9452	0.9396	0.9422	0.9458
	Gaussian (0,0.01)	0.9069	0.9320	0.9334	0.9413	0.9138	0.9268	0.9429
	Gaussian (0,0.015)	0.8677	0.9194	0.9174	0.9374	0.8783	0.9126	0.9403
	Salt&pepper 5%	0.9458	0.9352	0.9436	0.9448	0.9463	0.9464	0.9470
	Salt&pepper 10%	0.9426	0.9146	0.9341	0.9412	0.9442	0.9436	0.9466
	Salt&pepper 15%	0.9390	0.8719	0.9154	0.9337	0.9410	0.9402	0.9457
#100007	Gaussian (0,0.002)	0.9074	0.9177	0.9176	0.9196	0.9114	0.9158	0.9227
	Gaussian (0,0.006)	0.8751	0.9085	0.9054	0.9154	0.8819	0.8981	0.9194
	Gaussian (0,0.01)	0.8434	0.8944	0.8869	0.9118	0.8514	0.8824	0.9170
	Salt&pepper 2%	0.9182	0.9122	0.9175	0.9198	0.9203	0.9212	0.9240
	Salt&pepper 5%	0.9153	0.8897	0.9068	0.9161	0.9181	0.9188	0.9237
	Salt&pepper 10%	0.9124	0.8446	0.8858	0.9103	0.9149	0.9147	0.9229
#118035	Gaussian (0,0.001)	0.8665	0.9047	0.9041	0.9010	0.8597	0.8668	0.9076
	Gaussian (0,0.002)	0.8447	0.8782	0.8691	0.8824	0.8422	0.8427	0.8924
	Gaussian (0,0.006)	0.8347	0.8512	0.8404	0.8723	0.8355	0.8373	0.8773
	Salt&pepper 2%	0.9110	0.8960	0.9140	0.9122	0.9033	0.9034	0.9158
	Salt&pepper 5%	0.8886	0.8652	0.9027	0.9115	0.9099	0.8666	0.9154
	Salt&pepper 10%	0.9406	0.8226	0.8860	0.9025	0.8945	0.9025	0.9129

Table 3, continued
Image	Noise type	FCM	FCM_S1	FCM_S2	FGFCM	IFCM1	IIFCM	RIFCB
#135069	Gaussian (0,0.002)	0.8111	0.7968	0.7947	0.8153	0.8131	0.8156	0.8169
	Gaussian (0,0.006)	0.8135	0.7560	0.7578	0.8124	0.8140	0.8121	0.8157
	Gaussian (0,0.01)	0.8168	0.7293	0.7371	0.8115	0.8164	0.8123	0.8128
	Salt&pepper 2%	0.8166	0.8101	0.8193	0.8216	0.9492	0.8116	0.9501
	Salt&pepper 5%	0.7940	0.7815	0.8061	0.8186	0.9481	0.9450	0.9495
	Salt&pepper 10%	0.9112	0.7449	0.7786	0.8122	0.9409	0.7984	0.9464
#238011	Gaussian (0,0.002)	0.7990	0.7511	0.7587	0.8275	0.7965	0.7972	0.9557
	Gaussian (0,0.006)	0.7747	0.7127	0.7194	0.8140	0.7729	0.7714	0.9306
	Gaussian (0,0.01)	0.7707	0.6891	0.6980	0.8033	0.7702	0.7676	0.9097
	Salt&pepper 2%	0.9403	0.7941	0.8591	0.8621	0.9702	0.9694	0.9697
	Salt&pepper 5%	0.9383	0.8218	0.8382	0.8323	0.9671	0.9669	0.9685
	Salt&pepper 10%	0.9343	0.7580	0.8061	0.9237	0.9607	0.9599	0.9635
#241004	Gaussian (0,0.01)	0.7755	0.8216	0.8122	0.8461	0.7705	0.8415	0.8502
	Gaussian (0,0.015)	0.7740	0.7924	0.7827	0.8274	0.7692	0.7773	0.8342
	Gaussian (0,0.02)	0.7728	0.7673	0.7580	0.8100	0.7683	0.7727	0.8183
	Salt&pepper 5%	0.8881	0.8669	0.8860	0.8906	0.8806	0.8840	0.8893
	Salt&pepper 10%	0.8829	0.8298	0.8729	0.8846	0.8739	0.8763	0.8886
	Salt&pepper 15%	0.8742	0.7752	0.8524	0.8737	0.8601	0.8425	0.8850

Table 4

Comparison of partition entropy ( $V_{pe}$ ) of different segmentation algorithms on natural images with different noise

Image	Noise type	FCM	FCM_S1	FCM_S2	FGFCM	IFCM1	IIFCM	RIFCB
#3096	Gaussian (0,0.002)	0.2306	0.1572	0.1491	0.1423	0.1716	0.0234	0.1203
	Gaussian (0,0.006)	0.3010	0.2106	0.2049	0.1787	0.2941	0.1719	0.1403
	Gaussian (0,0.01)	0.3024	0.2891	0.3065	0.2428	0.2995	0.2635	0.1627
	Salt&pepper 2%	0.1205	0.1803	0.1398	0.1371	0.1216	0.1136	0.1114
	Salt&pepper 5%	0.1228	0.2646	0.1694	0.1592	0.1249	0.1157	0.1128
	Salt&pepper 10%	0.1332	0.3365	0.2357	0.2063	0.1317	0.1251	0.1168
#15088	Gaussian (0,0.002)	0.1642	0.1272	0.1292	0.1222	0.1486	0.1380	0.1083
	Gaussian (0,0.006)	0.2364	0.1508	0.1529	0.1319	0.2119	0.1776	0.1148
	Gaussian (0,0.01)	0.2888	0.1889	0.1946	0.1413	0.2747	0.2124	0.1203
	Salt&pepper 2%	0.1382	0.1355	0.1310	0.1218	0.1264	0.1231	0.1077
	Salt&pepper 5%	0.1401	0.1800	0.1493	0.1280	0.1270	0.1254	0.1077
	Salt&pepper 10%	0.1483	0.2760	0.1880	0.1409	0.1303	0.1295	0.1087
#24063	Gaussian (0,0.002)	0.2798	0.2496	0.2492	0.2370	0.2698	0.2560	0.2265
	Gaussian (0,0.006)	0.3487	0.2781	0.2857	0.2540	0.3349	0.3176	0.2410
	Gaussian (0,0.01)	0.3817	0.3039	0.3180	0.2728	0.3700	0.3575	0.2538
	Salt&pepper 2%	0.2282	0.2603	0.2388	0.2329	0.2219	0.2218	0.2210
	Salt&pepper 5%	0.2295	0.3007	0.2561	0.2415	0.2251	0.2267	0.2216
	Salt&pepper 10%	0.2305	0.3725	0.2851	0.2572	0.2287	0.2327	0.2238
#42044	Gaussian (0,0.002)	0.1370	0.1284	0.1262	0.1289	0.1522	0.1450	0.1215
	Gaussian (0,0.006)	0.1866	0.1480	0.1542	0.1388	0.2217	0.1932	0.1342
	Gaussian (0,0.01)	0.2269	0.1722	0.1844	0.1461	0.2601	0.2239	0.1442
	Salt&pepper 2%	0.1125	0.1773	0.1182	0.1161	0.1219	0.1142	0.1135
	Salt&pepper 5%	0.1190	0.2978	0.1336	0.1231	0.1235	0.1199	0.1143
	Salt&pepper 10%	0.1358	0.3780	0.1634	0.1337	0.1812	0.1260	0.1172
#42049	Gaussian (0,0.006)	0.1153	0.1081	0.1022	0.0988	0.1106	0.1060	0.0970
	Gaussian (0,0.01)	0.1670	0.1259	0.1245	0.1069	0.1560	0.1345	0.1033
	Gaussian (0,0.015)	0.2261	0.1504	0.1543	0.1150	0.2106	0.1582	0.1084
	Salt&pepper 5%	0.0962	0.1155	0.0999	0.0983	0.0948	0.0947	0.0944
	Salt&pepper 10%	0.1009	0.1511	0.1148	0.1045	0.0981	0.0989	0.0951
	Salt&pepper 15%	0.1089	0.2182	0.1438	0.1170	0.1039	0.1051	0.0968

Table 4, continued
Image	Noise type	FCM	FCM_S1	FCM_S2	FGFCM	IFCM1	IIFCM	RIFCB
#100007	Gaussian (0,0.002)	0.1647	0.1484	0.1487	0.1437	0.1582	0.1508	0.1385
	Gaussian (0,0.006)	0.2148	0.1663	0.1713	0.1512	0.2046	0.1799	0.1445
	Gaussian (0,0.01)	0.2611	0.1912	0.2024	0.1579	0.2495	0.2040	0.1488
	Salt&pepper 2%	0.1450	0.1557	0.1464	0.1429	0.1414	0.1397	0.1363
	Salt&pepper 5%	0.1487	0.1916	0.1623	0.1486	0.1439	0.1426	0.1367
	Salt&pepper 10%	0.1543	0.2595	0.1931	0.1577	0.1483	0.1484	0.1382
#118035	Gaussian (0,0.001)	0.2411	0.1849	0.1853	0.1801	0.2510	0.2398	0.1757
	Gaussian (0,0.002)	0.2800	0.2320	0.2445	0.2012	0.2843	0.2836	0.2011
	Gaussian (0,0.006)	0.2997	0.2756	0.2900	0.2329	0.2991	0.2950	0.2250
	Salt&pepper 2%	0.1667	0.1972	0.1631	0.1685	0.1791	0.1788	0.1613
	Salt&pepper 5%	0.1995	0.2508	0.1814	0.1683	0.1633	0.2313	0.1622
	Salt&pepper 10%	0.1221	0.3244	0.2083	0.1842	0.1836	0.1774	0.1667
#135069	Gaussian (0,0.002)	0.3046	0.3302	0.3324	0.2972	0.3017	0.2984	0.2962
	Gaussian (0,0.006)	0.3023	0.3878	0.3843	0.3024	0.3016	0.3046	0.2979
	Gaussian (0,0.01)	0.2982	0.4232	0.4117	0.3041	0.2988	0.3046	0.3023
	Salt&pepper 2%	0.2942	0.3055	0.2912	0.2887	0.1102	0.3018	0.1039
	Salt&pepper 5%	0.3232	0.3464	0.3076	0.2929	0.1048	0.1100	0.1047
	Salt&pepper 10%	0.1615	0.3978	0.3423	0.3018	0.1149	0.3128	0.1096
#238011	Gaussian (0,0.002)	0.3603	0.4212	0.4109	0.3014	0.3634	0.3613	0.0927
	Gaussian (0,0.006)	0.4048	0.4928	0.4855	0.3326	0.4069	0.4085	0.1402
	Gaussian (0,0.01)	0.4126	0.5367	0.5251	0.3536	0.4129	0.4166	0.1763
	Salt&pepper 2%	0.0634	0.3484	0.2459	0.2399	0.0608	0.0632	0.0624
	Salt&pepper 5%	0.0668	0.3322	0.2788	0.2852	0.0663	0.0664	0.0649
	Salt&pepper 10%	0.0762	0.4375	0.3298	0.1540	0.0804	0.0815	0.0742
#241004	Gaussian (0,0.01)	0.4295	0.3628	0.3774	0.3086	0.4391	0.3164	0.3018
	Gaussian (0,0.015)	0.4328	0.4147	0.4291	0.3412	0.4423	0.4237	0.3298
	Gaussian (0,0.02)	0.4354	0.4582	0.4714	0.3707	0.4447	0.4331	0.3571
	Salt&pepper 5%	0.2291	0.2694	0.2344	0.2254	0.2440	0.2374	0.2292
	Salt&pepper 10%	0.2383	0.3380	0.2577	0.2363	0.2557	0.2513	0.2306
	Salt&pepper 15%	0.2542	0.4397	0.2945	0.2561	0.2801	0.3076	0.2370

4.4 Segmentation experiments on real images

In the following experiments, seven segmentation algorithms are performed on ten real images contaminated by different noise types with different noise level to investigate the robustness of the algorithms. The quantitative comparison of these algorithms are presented in Tables 2–4. The noise types and the corresponding noise level have given in second column of this table. From Tables 2–4, we can see that the proposed method (RIFCB) outperforms the other methods with different noise types and levels. In order to visually compare the performance, the segmentation results of Images #3096, #24063, #135069, and #238011 (shown in Fig. 8) are shown in Figs 9–16. For Images #3069 and #24063, FCM_S1, FCM_S2, FGFCM and RIFCB algorithms can overcome the influence of Gaussian white noise (shown in Figs 9 and 11). For Images #135069 and #238011, only RIFCB algorithm obtains the satisfying segmentation results because of utilizing the effective initialization, spatial information and intuitionistic fuzzy theory. For all images corrupted by Salt&pepper noise, RIFCB presents better segmentation results than other comparison algorithms. It is evident that RIFCB outperforms all the comparative methods in quantitative and visual results, can well overcome the influence of noise and obtain the satisfying performance.

5. Conclusions

In this paper, a robust intuitionistic fuzzy clustering c-means algorithm for noisy image segmentation (RIFCB) is proposed to overcome the disadvantages of the IFCM1 algorithm. Firstly, the gray histogram of image is utilized to initialize the cluster centers in the proposed method. In addition, a noise robust intuitionistic fuzzy set is constructed by utilizing the local grayscale and spatial information which can overcome the influence of the noise in the image. Finally, the information guidance factor and deviation information correction factor are introduced into the objective function of RIFCB to improve the algorithm performance. Numerical results obtained in the paper show that the proposed RIFCB algorithm outperforms FCM, FCM_S1, FCM_S2, FGFCM, and IFCM1 in noise robustness and segmentation performance.

The local grayscale and spatial information are effective for the image corrupted with low noise level. However, when the image is heavily corrupted by noise, only utilizing the pixels in the neighboring windows cannot overcome the influence of noise. Therefore, the large neighboring windows and neighboring structure of pixel will be utilized to construct the more robust intuitionistic fuzzy set in our future work. Furthermore, further studies should investigate the suitable selection method of parameters in our method.

Footnotes

Acknowledgments

This study was funded by the National Natural Science Foundation of China (Grant Nos. (62071379 and 61571361), the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2020JM-299), the Fundamental Research Funds for the Central Universities (Grant No. GK202103085), the New Star Team of Xi’an University of Posts and Telecommunications (Grant No. xyt016-01).

Appendix A

In this section, we give the derivation processes of the updating equations of membership degree and cluster centers of RIFCB. In order to minimize the objective function (Shown in Eq. (29)), the Lagrange multiplier method is utilized and the following formula can be obtained:

(A.1) $\displaystyle L=\sum\limits_{i=1}^{n}\sum\limits_{k=1}^{K}u_{ki}^{m}d^{2}(% \delta_{i}^{\textit{IFS}},c_{k}^{\textit{IFS}})+\beta\sum\limits_{i=1}^{n}\sum% \limits_{k=1}^{K}u_{ki}^{m}d^{2}((\delta_{i}^{\textit{IFS}}+\gamma_{i}^{% \textit{IFS}}),c_{k}^{\textit{IFS}})+\sum\limits_{i=1}^{n}\lambda_{i}\left(1-% \sum\limits_{k=1}^{K}u_{ki}\right)$

Taking the derivative of $L$ with respect to membership matrix $u_{ki}$ and setting the result to zero,

(A.2) $\displaystyle\frac{\partial L}{\partial u_{ki}}=mu_{ki}^{m-1}d^{2}(\delta_{i}^% {\textit{IFS}},c_{k}^{\textit{IFS}})+\beta mu_{ki}^{m-1}d^{2}((\delta_{i}^{% \textit{IFS}}+\gamma_{i}^{\textit{IFS}}),c_{k}^{\textit{IFS}})-\lambda_{i}=0$

Then, $u_{ki}$ , can be computed by

(A.3) $\displaystyle u_{ki}=\left(\frac{\lambda}{m(d^{2}(\delta_{i}^{\textit{IFS}},c_% {k}^{\textit{IFS}}))+\beta md^{2}((\delta_{i}^{\textit{IFS}}+\gamma_{i}^{% \textit{IFS}}),c_{k}^{\textit{IFS}})}\right)^{\frac{1}{m-1}}$

since $\sum\limits_{j=1}^{K}u_{ki}=1$ , $\forall i$ , so

(A.4) $\displaystyle\sum\limits_{j=1}^{K}\left(\frac{\lambda}{m(d^{2}(\delta_{i}^{% \textit{IFS}},c_{j}^{\textit{IFS}}))+\beta md^{2}((\delta_{i}^{\textit{IFS}}+% \gamma_{i}^{\textit{IFS}}),c_{j}^{\textit{IFS}})}\right)^{\frac{1}{m-1}}=1$

Then $\lambda$ can be computed by

(A.5) $\displaystyle\lambda=\frac{m}{\left(\sum\limits_{j=1}^{K}\left(\frac{1}{m(d^{2% }(\delta_{i}^{\textit{IFS}},c_{j}^{\textit{IFS}}))+\beta md^{2}((\delta_{i}^{% \textit{IFS}}+\gamma_{i}^{\textit{IFS}}),c_{j}^{\textit{IFS}})}\right)^{\frac{% {1}}{m-1}}\right)^{m-1}}$

We use Eq. (A.5) to substitute $\lambda$ in Eq. (A.3), the membership degree $u_{ki}$ can be rewritten as

(A.6) $\displaystyle u_{ki}=\frac{1}{\sum\limits_{j=1}^{K}\left(\frac{d^{2}(\delta_{i% }^{\textit{IFS}},c_{k}^{\textit{IFS}})+\beta d^{2}(\delta_{i}^{\textit{IFS}}+% \gamma_{i}^{\textit{IFS}},c_{k}^{\textit{IFS}})}{d^{2}(\delta_{i}^{\textit{IFS% }},c_{j}^{\textit{IFS}})+\beta d^{2}(\delta_{i}^{\textit{IFS}}+\gamma_{i}^{% \textit{IFS}},c_{j}^{\textit{IFS}})}\right)^{\frac{1}{m-1}}}$

In order to obtain the updating function of cluster centers, we take the derivative of $L$ with respect to $c_{k}^{\textit{IFS}}$ and setting the result to zero. Because $c_{k}^{\textit{IFS}}$ contains membership degree, non-membership degree, and hesitation degree, we can obtain

(A.7) $\displaystyle\left\{\begin{array}[]{c}{\frac{\partial L}{\partial\mu(c_{k}^{% \textit{IFS}})}=\sum\limits_{i=1}^{n}u_{ki}^{m}(\mu(\delta_{i}^{\textit{IFS}})% -\mu(c_{k}^{\textit{IFS}}))+\beta\sum\limits_{i=1}^{n}u_{ki}^{m}(\mu(\delta_{i% }^{\textit{IFS}})+\mu(\gamma_{i}^{\textit{IFS}})-\mu(c_{k}^{\textit{IFS}}))=0}% \\ {\frac{\partial L}{\partial v(c_{k}^{\textit{IFS}})}=\sum\limits_{i=1}^{n}u_{% ki}^{m}(v(\delta_{i}^{\textit{IFS}})-v(c_{k}^{\textit{IFS}}))+\beta\sum\limits% _{i=1}^{n}u_{ki}^{m}(v(\delta_{i}^{\textit{IFS}})+v(\gamma_{i}^{\textit{IFS}})% -v(c_{k}^{\textit{IFS}}))=0}\\ {\frac{\partial L}{\partial\pi(c_{k}^{\textit{IFS}})}=\sum\limits_{i=1}^{n}u_{% ki}^{m}(\pi(\delta_{i}^{\textit{IFS}})-\pi(c_{k}^{\textit{IFS}}))+\beta\sum% \limits_{i=1}^{n}u_{ki}^{m}(\pi(\delta_{i}^{\textit{IFS}})+\pi(\gamma_{i}^{% \textit{IFS}})-\pi(c_{k}^{\textit{IFS}}))=0}\end{array}\right.$

Then

(A.8) $\displaystyle\left\{\begin{array}[]{c}{\mu(c_{k}^{\textit{IFS}})=\frac{\sum% \limits_{i=1}^{n}u_{ki}^{m}((1+\beta)\mu(\delta_{i}^{\textit{IFS}})+\beta\mu(% \gamma_{i}^{\textit{IFS}}))}{(1+\beta)\sum\limits_{i=1}^{n}u_{ki}^{m}}}\\ {v(c_{k}^{\textit{IFS}})=\frac{\sum\limits_{i=1}^{n}u_{ki}^{m}((1+\beta)v(% \delta_{i}^{\textit{IFS}})+\beta v(\gamma_{i}^{\textit{IFS}}))}{(1+\beta)\sum% \limits_{i=1}^{n}u_{ki}^{m}}}\\ {\pi(c_{k}^{\textit{IFS}})=\frac{\sum\limits_{i=1}^{n}u_{ki}^{m}((1+\beta)\pi(% \delta_{i}^{\textit{IFS}})+\beta\pi(\gamma_{i}^{\textit{IFS}}))}{(1+\beta)\sum% \limits_{i=1}^{n}u_{ki}^{m}}}\end{array}\right.$

Therefore, the updating function of the cluster centers $c_{k}^{\textit{IFS}}$ is:

(A.9) $\displaystyle c_{k}^{\textit{IFS}}=(\mu(c_{k}),\nu(c_{k}),\pi(c_{k}))=\left(% \frac{\sum\limits_{i=1}^{n}u_{ki}^{m}((1+\beta)\mu(\delta_{i}^{\textit{IFS}})+% \beta\mu(\gamma_{i}^{\textit{IFS}}))}{(1+\beta)\sum\limits_{i=1}^{n}u_{ki}^{m}% },\right.\left.\frac{\sum\limits_{i=1}^{n}u_{ki}^{m}((1+\beta)v(\delta_{i}^{% \textit{IFS}})+\beta v(\gamma_{i}^{\textit{IFS}}))}{(1+\beta)\sum\limits_{i=1}% ^{n}u_{ki}^{m}},\frac{\sum\limits_{i=1}^{n}u_{ki}^{m}((1+\beta)\pi(\delta_{i}^% {\textit{IFS}})+\beta\pi(\gamma_{i}^{\textit{IFS}}))}{(1+\beta)\sum\limits_{i=% 1}^{n}u_{ki}^{m}}\right)$

The last one is about the bias field updating. Taking the derivative of $L$ with respect to $\gamma_{i}^{\textit{IFS}}$ and setting the result to zero, that is

(A.10) $\displaystyle\left\{\begin{array}[]{c}{\frac{\partial L}{\partial\mu(\gamma_{i% }^{\textit{IFS}})}=\beta\sum\limits_{k=1}^{K}u_{ki}^{m}(\mu(\delta_{i}^{% \textit{IFS}})+\mu(\gamma_{i}^{\textit{IFS}})-\mu(c_{k}^{\textit{IFS}}))=0}\\ {\frac{\partial L}{\partial v(\gamma_{i}^{\textit{IFS}})}=\beta\sum\limits_{k=% 1}^{K}u_{ki}^{m}(v(\delta_{i}^{\textit{IFS}})+v(\gamma_{i}^{\textit{IFS}})-v(c% _{k}^{\textit{IFS}}))=0}\\ {\frac{\partial L}{\partial\pi(\gamma_{i}^{\textit{IFS}})}=\beta\sum\limits_{k% =1}^{K}u_{ki}^{m}(\pi(\delta_{i}^{\textit{IFS}})+\pi(\gamma_{i}^{\textit{IFS}}% )-\pi(c_{k}^{\textit{IFS}}))=0}\end{array}\right.$

Solving the above formula, we have

(A.11) $\displaystyle\left\{\begin{array}[]{c}{\mu(\gamma_{i}^{\textit{IFS}})=\frac{% \sum\limits_{k=1}^{K}u_{ki}^{m}\mu(c_{k}^{\textit{IFS}})}{\sum\limits_{k=1}^{K% }u_{ki}^{m}}-\mu(\delta_{i}^{\textit{IFS}})}\\ {v(\gamma_{i}^{\textit{IFS}})=\frac{\sum\limits_{k=1}^{K}u_{ki}^{m}v(c_{k}^{% \textit{IFS}})}{\sum\limits_{k=1}^{K}u_{ki}^{m}}-v(\delta_{i}^{\textit{IFS}})}% \\ {\pi(\gamma_{i}^{\textit{IFS}})=\frac{\sum\limits_{k=1}^{K}u_{ki}^{m}\pi(c_{k}% ^{\textit{IFS}})}{\sum\limits_{k=1}^{K}u_{ki}^{m}}-\pi(\delta_{i}^{\textit{IFS% }})}\end{array}\right.$

so, the updating function of bias field $\gamma_{i}^{\textit{IFS}}$ is:

(A.12) $\displaystyle\gamma_{i}^{\textit{IFS}}=(\mu(\gamma_{i}^{\textit{IFS}}),\nu(% \gamma_{i}^{\textit{IFS}}),\pi(\gamma_{i}^{\textit{IFS}}))=\left(\frac{\sum% \limits_{k=1}^{K}u_{ki}^{m}\mu(c_{k}^{\textit{IFS}})}{\sum\limits_{k=1}^{K}u_{% ki}^{m}}-\mu(\delta_{i}^{\textit{IFS}}),\frac{\sum\limits_{k=1}^{K}u_{ki}^{m}v% (c_{k}^{\textit{IFS}})}{\sum\limits_{k=1}^{K}u_{ki}^{m}}-v(\delta_{i}^{\textit% {IFS}}),\frac{\sum\limits_{k=1}^{K}u_{ki}^{m}\pi(c_{k}^{\textit{IFS}})}{\sum% \limits_{k=1}^{K}u_{ki}^{m}}-\pi(\delta_{i}^{\textit{IFS}})\right)$

References

Gonzales

R.C.

and Woods

R.E.

, Digital image processing, third ed., Pearson, 2002.

Arbeláez

Maire

Fowlkes

and Malik

, Contour detection and hierarchical image segmentation, IEEE Trans. Pattern Analysis and Machine Intelligence 33 (2011), 898–916. https://www2eecs.berkeley.edu/Research/Projects/CS/vision/grouping/resources.html.

Bhandari

A.K.

, A novel beta differential evolution algorithm-based fast multilevel thresholding for color image segmentation, Neural Computing and Applications 32(9) (2020), 4583–4613. doi: 10.1007/s00521-018-3771-z.

H.P.

F.Z.

and Pan

Y.T.

, A scalable region-based level set method using adaptive bilateral filter for noisy image segmentation, Multimedia Tools and Applications 79(9–10) (2020), 5743–5765. doi: 10.1016/j.patrec.2004.11.023.

Krinidis

and Chatzis

, A robust fuzzy local information c-means clustering algorithm, IEEE Trans. on Image Processing 19 (2010), 1328–1337. doi: 10.1109/TIP.2010.2040763.

Zhang

Tang

and Wei

, Medical image segmentation using improved FCM, Science China Information Sciences 55 (2012), 1052–1061. doi: 10.1007/s11432-012-4556-0.

Bezdek

J.C.

, Pattern recognition with fuzzy objective function algorithms, Boston, MA: Springer US; 1981. doi: 10.1007/978-1-4757-0450-1.

Dhanachandra

Manglem

and Chanu

Y.J.

, Image segmentation using k-means clustering algorithm and subtractive clustering algorithm, Procedia Computer Science 54 (2015), 764–771. doi: 10.1016/j.procs.2015.06.090.

Gharieb

R.R.

Gendy

and Abdelfattah

, Image segmentation using fuzzy c-means algorithm incorporating weighted local complement membership and local data distances, in: 2016 World Symposium on Computer Applications & Research (WSCAR), Cairo, Egypt: IEEE, 2016. pp. 6–11. doi: 10.1109/WSCAR.2016.18.

10.

Ferraro

M.B.

and Giordani

, A review and proposal of (fuzzy) clustering for nonlinearly separable data, International Journal of Approximate Reasoning 115 (2019), 13–31. doi: 10.1016/j.ijar.2019.09.004.

11.

Zhou

and Yang

, Fuzziness parameter selection in fuzzy c-means: The perspective of cluster validation, Science China Information Sciences 57 (2014), 1–8. doi: 10.1007/s11432-014-5146-0.

12.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8 (1965), 338–353. doi: 10.1016/S0019-995(65)90241-X.

13.

Ghosh

and Kumar

, Comparative analysis of k-means and fuzzy c-means algorithms, International Journal of Advanced Computer Science and Applications 4 (2013), 35–39. doi: 10.14569/IJACSA.2013.040406.

14.

Biswal

Behera

H.S.

Bisoi

and Dash

P.K.

, Classification of power quality data using decision tree and chemotactic differential evolution based fuzzy clustering, Swarm and Evolutionary Computation 4 (2012), 12–24. doi: 10.1016/j.swevo.2011.12.003.

15.

Zhao

Liu

H.Q.

Fan

J.L.

Chen

C.W.

Lan

and Li

, Intuitionistic fuzzy set approach to multi-objective evolutionary clustering with multiple spatial information for image segmentation, Neurocomputing 312 (2018), 296–309. doi: 10.1016/j.neucom.2018.05.116.

16.

Zhao

Fan

J.L.

Liu

H.Q.

Lan

and Chen

C.W.

, Noise robust multi-objective evolutionary clustering image segmentation motivated by the intuitionistic fuzzy information, IEEE Transactions on Fuzzy Systems 27 (2019), 387–401. doi: 10.1109/TFUZZ.2018.2852289.

17.

Ahmed

M.N.

Yamany

S.M.

Mohamed

Farag

A.A.

and Moriarty

, A modified fuzzy c-means algorithm for bias field estimation and segmentation of MRI data, IEEE Transactions on Medical Imaging 21 (2002), 193–199. doi: 10.1109/42.996338.

18.

Chen

and Zhang

, Robust image segmentation using FCM with spatial constraints based on new kernel-induced distance measure, IEEE Transactions on Systems, Man, Cybern B. 34 (2004), 1907–1916. doi: 10.1109/TSMCB.2004.831165.

19.

Z.X.

Sun

Q.S.

and Xia

D.S.

, A modified possibilistic fuzzy c-means clustering algorithm for bias field estimation and segmentation of brain MR image, Computerized Medical Imaging and Graphics 35(5) (2011), 383–397. doi: 10.1016/j.compmedimag.2010.12.001.

20.

Cai

Chen

and Zhang

, Fast and robust fuzzy c-means clustering algorithms in corporating local information for image segmentation, Pattern Recognition (40) (2007), 825–838. doi: 10.1016/j.patcog.2006.07.011.

21.

Guo

Chen

C.L.P.

and Zhou

, Integrating guided filter into fuzzy clustering for noisy image segmentation, Digital Signal Processing 83 (2018), 235–248. doi: 10.1016/j.dsp.2018.08.022.

22.

Atanassov

K.T.

, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986), 87–96. doi: 10.1016/S0165-0114(86)80034-3.

23.

Iakovidis

D.K.

Pelekis

Kotsifakos

and Kopanakis

, Intuitionistic fuzzy clustering with applications in computer vision, in: Blanc-Talon

Bourennane

Philips

Popescu

and Scheunders

, eds., Advanced Concepts for Intelligent Vision Systems, Berlin, Heidelberg: Springer Berlin Heidelberg, 2008. pp. 764–774. doi: 10.1007/978-3-540-88458-3_69.

24.

Chaira

, A novel intuitionistic fuzzy C means clustering algorithm and its application to medical images, Applied Soft Computing 11 (2011), 1711–1717. doi: 10.1016/j.asoc.2010.05.005.

25.

and Wu

, Intuitionistic fuzzy c-means clustering algorithms, Journal of Systems Engineering and Electronics 21 (2010), 580–590. doi: 10.3969/j.issn.1004-4132.2010.04.009.

26.

Yager

R.R.

, On the measure of fuzziness and negation Part I: Membership in the unit interval, International Journal of General Systems 5 (1979), 221–229. doi: 10.1080/03081077908547452.

27.

Verma

Agrawal

R.K.

and Sharan

, An improved intuitionistic fuzzy c-means clustering algorithm incorporating local information for brain image segmentation, Applied Soft Computing 46 (2016), 543–557. doi: 10.1016/j.asoc.2015.12.022.

28.

Chaira

and Ray

A.K.

, Fuzzy measures for color image retrieval, Fuzzy Sets and Systems 150 (2005), 545–560. doi: 10.1016/j.fss.2004.09.003.

29.

Atanassov

and Gargov

, Interval valued intuitionistic fuzzy sets, Fuzzy Sets and Systems 31 (1989), 343–349. doi: 10.1016/0165-0114(89)90205-4.

30.

Kaur

Soni

A.K.

and Gosain

, Robust intuitionistic fuzzy c-means clustering for linearly and nonlinearly separable data, in: 2011 International Conference on Image Information Processing, Shimla, Himachal Pradesh, India: IEEE, 2011. pp. 1–6. doi: 10.1109/ICIIP.2011.6108908.

31.

Sugeno

, Fuzzy measures and fuzzy integrals-a survey, Readings in Fuzzy Sets for Intelligent Systems, Elsevier, 1993; 251–257. doi: 10.1016/B978-1-4832-1450-4.50027-4.

32.

Kim

G.R.

Kim

E.K.

Kim

S.J.

et al., Evaluation of underlying lymphocytic thyroiditis with histogram analysis using grayscale ultrasound images, Journal of Ultrasound in Medicine 35 (2016), 519–526. doi: 10.7863/ultra.15.04014.

33.

and Fan

J.L.

, A novel segmentation method for uneven lighting image with noise injection based on non-local spatial information and intuitionistic fuzzy entropy, EURASIP Journal on Advances in Signal Processing 2017 (2017), 74. doi: 10.1186/s13634-017-0509-5.

34.

Zhang

Guo

Sun

et al., Patch-based fuzzy clustering for image segmentation, Soft Computing 23 (2019), 3081–3093. doi: 10.1007/s00500-017-2955-2.

Robust intuitionistic fuzzy clustering with bias field estimation for noisy image segmentation

Abstract

Keywords

1. Introduction

2. Fuzzy c-means clustering and intuitionistic fuzzy c-means clustering algorithms

2.1 Fuzzy sets and intuitionistic fuzzy sets

3.1 Construction of the noise robust intuitionistic fuzzy set for the image

Table 2 Segmentation accuracy (%) of different segmentation algorithms on natural images with different noise

5. Conclusions

Footnotes

Acknowledgments

Appendix A

References

Table 2
Segmentation accuracy (%) of different segmentation algorithms on natural images with different noise