Robust Harmonic Fuzzy Partition Local Information C-Means Clustering for Image Segmentation

Abstract

Considering the shortcomings of Ruspini partition-based fuzzy clustering in revealing the intrinsic correlation between different classes, a series of harmonic fuzzy local information C-means clustering for noisy image segmentation are proposed. Firstly, aiming at the shortage of Zadeh’s fuzzy sets, a new concept of generalized harmonic fuzzy sets is originally introduced and the corresponding harmonic fuzzy partition is further defined. Then, based on the concept of symmetric harmonic partition, a new harmonic fuzzy local information C-means clustering (HLICM) is proposed and the local convergence of the algorithm is rigorously proved using Zangwill’s theorem. Finally, inspired by the improved fuzzy local information C-means clustering (IFLICM) and kernel-based weighted fuzzy local information C-means clustering (KWFLICM), two enhanced robust HLICM algorithms are constructed to further improve the ability of the algorithm to suppress noise. Compared with existing state-of-the-art robust fuzzy clustering-related algorithms, it has been confirmed that the two proposed algorithms have significant competitiveness and superiority.

1. Introduction

Clustering analysis [1] is the process of dividing a set of abstract objects or physical targets into multiple classes containing similar features for analysis and understanding. The main purpose of cluster analysis [2] is to group data samples based on their similarities, so that samples in the same class are similar to each other and samples in different classes are different from each other. Clustering [3] is an unsupervised partition method that does not rely on priori knowledge and does not require human intervention. The origin of cluster analysis [4] can be traced back to too many different fields, such as mathematics, statistics, economics, biology, etc. It is widely used in image recognition, data mining [5], and other fields to determine the similarity between different data and achieve data partitioning. Common clustering analysis methods include hierarchical-based clustering [6], partition-based clustering [7], and density-based clustering [8]. Among them, partition-based clustering methods have been widely applied in recent years, including hard partition and fuzzy partition [9]. Hard C-mean (HCM) clustering [10] is one of the classical algorithms for implementing hard C-partition of datasets. It uses a hard clustering algorithm [11] to assign data points to a fixed number of clusters and each example can only belong to one cluster. Therefore, considering the obvious drawbacks of the HCM algorithm, Dunn [12] extended the HCM algorithm to a fuzzy form based on the concept of fuzzy partition given by Ruspini [13]. Subsequently, Bezdek [14] further generalized Dunn’s objective function to a more general form, enabling the FCM algorithm to preserve more information in the original data. However, the biggest drawback of the FCM algorithm is that it does not utilize any spatial information in the image [15], which makes the algorithm sensitive to noise and outliers and prevents it from obtaining satisfactory segmentation results in noise-corrupted images.

In view of this, researchers have proposed many robust fuzzy clustering algorithms that utilize the spatial information of images [16]. Firstly, Ahmed [17] et al. introduced spatial neighborhood terms into the objective function of the FCM algorithm and proposed a spatial fuzzy C-mean clustering algorithm (FCM-S) that combines spatial information, which was applied to the segmentation of MRI images. To reduce the computational complexity of the FCM-S algorithm, Chen and Zhang [18] proposed three improved algorithms, FCM-S, FCM-S1, and FCM-S2. In addition, Chen and Zhang [18] proposed a kernel version of an improved algorithm that uses kernel-induced distance metric instead of the squared Euclidean distance. As we have clearly pointed out, the number of gray levels in an image is much smaller than the number of pixels in the image. Based on this fact, Szilagyi et al. [19] proposed the enhanced fuzzy C-mean clustering algorithm (EnFCM) in an attempt to further improve the efficiency of image segmentation. It should be noted that the EnFCM algorithm achieves segmentation performance comparable to the FCM-S algorithm while significantly reducing computational complexity. Subsequently, Cai et al. [20] proposed the fast generalized fuzzy C-means clustering algorithm (FGFCM). The FGFCM algorithm first defines a novel linearly weighted sum image using the gray values and spatial location information within the pixel neighborhood window of the original image and then performs clustering on the gray histogram of the image. Similar to the EnFCM algorithm, the FGFCM algorithm consumes less running time while achieving satisfactory segmentation results.

In recent years, Krinidis and Chatzis [21] proposed the fuzzy local information C-means (FLICM) clustering algorithm, which suppresses image noise and preserves image details by constructing fuzzy local factors incorporating local spatial information, achieving better segmentation performance in noisy image segmentation. To make FCM and FGFCM clustering more robust to noise, Mithra and Emmanuel [22] proposed an improved fuzzy-based clustering algorithm for sputum images called IFLICM. It combines local spatial and grayscale information in a fuzzy way to maintain robustness and noise insensitivity. A new improved kernel factor is introduced into the traditional FCM algorithm to improve the segmentation performance of sputum images. It manages the effect of neighboring pixels based on their distance from the center pixel. The FLICM algorithm balances the membership values corresponding to each pixel point in the local neighborhood window by fuzzy local factors so that the membership values of non-noise pixels and noise pixels in the neighborhood window converge to similar values and therefore improves the robustness of the algorithm. The KWFLICM algorithm proposed by Gong et al. [23] improved the fuzzy local factor. By introducing the spatial weight influencing factor composed of local variance coefficients into the algorithm more information about spatial interactions can be obtained, resulting in more satisfactory segmentation results.

Problem statement: Due to the limitation of the sum of membership degrees, existing fuzzy clustering algorithms have not fully considered the intrinsic correlation between classes. Therefore, GLCA, as a special unsupervised clustering algorithm [24], uses a non-additive Sugeno fuzzy measure to represent the intrinsic correlation between classes. This enables GLCA to maintain the simplicity and robustness of the Bezdek-type clustering method while effectively extracting expected outliers. However, the fuzzy membership in GLCA may have negative values and the algorithm will not converge locally [25], so it has not attracted more attention from scholars. After studying the characteristics of the GLCA algorithm, Leski [26] proposed the generalized weighted conditional fuzzy clustering (GWCFCM) algorithm. The fuzzy membership in GWCFCM may be greater than 1, which conflicts with the existing definition of membership in Zadeh’s fuzzy set theory. Therefore, Zadeh’s fuzzy set theory cannot support these fuzzy clustering algorithms, which poses a great challenge to the development of these algorithms.

Motivation: Based on the above considerations, this paper first introduces a new concept of generalized harmonic fuzzy sets. Then, inspired by the theoretical research of the GWCFCM algorithm, it is emphasized that clustering algorithms need to fully consider the intrinsic correlation between different categories. Therefore, this paper modifies the constraints of the GWCFCM algorithm by using the idea of non-linear constraints in GLCA to define non-additive harmonic fuzzy partition, which are supported by the new harmonic fuzzy sets. Finally, based on the concept of harmonic fuzzy partition, the related theories and algorithms of harmonic fuzzy C-means clustering and harmonic fuzzy local information C-means clustering are studied.

The main contributions of this paper are summarized below:

• The new concept of generalized harmonic fuzzy sets is defined for the first time, and on this basis, the concept of symmetric harmonic fuzzy partition is proposed.
• On the basis of existing robust fuzzy local information clustering for noisy image segmentation, a harmonic fuzzy local information C-means clustering algorithm is proposed through extension. The convergence of the proposed clustering algorithm is rigorously analyzed using Zangwill’s theorem and the bordered Hessian matrix theorem.
• For images with high noise, two improved algorithms, IHLICM and KWHLICM, are proposed to improve the HLICM algorithm by using the idea of the KWFLICM algorithm.
• The experimental results indicate that the proposed harmonic fuzzy clustering correlation algorithms are superior to the existing state-of-the-art robust fuzzy clustering correlation algorithms.

The rest of the paper is organized as follows: Section 2 introduces the theory of fuzzy sets, fuzzy partition, and fuzzy clustering related algorithms. Section 3 presents the basic concepts of generalized harmonic fuzzy sets and harmonic fuzzy partition and constructs the harmonic fuzzy C-mean clustering algorithm and its improved robust segmentation algorithms. Section 4 compares and analyzes the clustering results of the proposed HLICM and its improved algorithms with other state-of-the-art robust fuzzy clustering algorithms. Finally, Section 5 provides a summary of the entire paper and offers prospects.

2. Related Work

In this section, we focus on introducing the basic theories involved in this paper, including fuzzy sets, fuzzy partition, and fuzzy C-means clustering and its robust algorithms based on local information.

2.1. Fuzzy Sets and Fuzzy Partition

Basic fuzzy sets, also known as a Zadeh fuzzy sets or a type-1 fuzzy sets, were proposed by Zadeh in 1965 [27]. The membership function, decomposition theorem, and extension principle constitute the three pillars of the classical fuzzy set theory.

Definition 1.

[27]. Let $X=\{x_i,i=1,\cdots ,n\}$ be a nonempty finite set, where $i$ is the index of the element of the set, $n$ is the number of elements, and $x_i$ is the value taken by the element $i$. For any element $x_i$ in this set, the fuzzy set $A$ built on $X$ can be defined as $A=\{(x_i,{\mu }_A(x_i)),x_i\in X\}$, where ${\mu }_A(x_i)\in [0,1]$ is the fuzzy membership degree of $x_i$ belonging to the finite set $X$.

Zadeh generalized various common concepts of union sets, intersection sets, etc. He defined some basic operation on fuzzy sets, equality, containment, complementation, intersection, and union related to classical Zadeh’s fuzzy sets $A$, $B$ in any universe of discourse $X$ as follows.

(a)

$A=B$ if and only if ${\mu }_A(x)={\mu }_B(x)$, $\forall x\in X$;

(b)

$A\subseteq B$ if and only if ${\mu }_A(x)\le {\mu }_B(x)$, $\forall x\in X$;

(c)

$A^c$ is the complement of $A$, if and only if ${\mu }_{A^c}(x)=1-{\mu }_A(x)$, $\forall x\in X$;

(d)

$A\cup B$ if and only if ${\mu }_{A\cup B}(x)=\max ({\mu }_A(x),{\mu }_B(x))$, $\forall x\in X$;

(e)

$A\cap B$ if and only if ${\mu }_{A\cap B}(x)=\min ({\mu }_A(x),{\mu }_B(x))$, $\forall x\in X$.

So far, fuzzy sets have been widely used in pattern recognition, image processing, and intelligent decision making [28].

With the deeper study of fuzzy set theory, Ruspini [29,30] defined the concept of fuzzy partition as a generalization of hard partition. The definition of fuzzy partition is as follows.

Definition 2.

[29]. Let $\tilde{X}=\left\{A_k\right|k=1,2,\cdots ,c\}$, where $A_j$ is a fuzzy set on domain $X=\{x_1,x_2,\cdots ,x_n\}$. $\tilde{X}$ is a fuzzy partition if and only if

(a) 

$\forall k\in \{1,2,\cdots ,c\}$, $A_k\ne \varphi$;

(b) 

$\forall x\in X$, ${\displaystyle {\sum }_{k=1}^c{\mu }_{A_k}}(x)=1$.

Later, Pan and Xu [31] also gave a generalized axiomatization definition of fuzzy partition as follows.

Definition 3.

[31]. Let $X=[a,b]$. A fuzzy partition of $X$ is an objective having the following form $\tilde{X}=\left\{{\mu }_{A_1}(x),{\mu }_{A_2}(x),\cdots ,{\mu }_{A_c}(x)\right\},c\in {\mathbb{N}}^{+}$, where the fuzzy membership function ${\mu }_{A_k}(x):X\to [0,1](k=1,2,\cdots ,c)$ defines the degrees of memberships of the element $x\in X$ belonging to the class $A_k$, respectively, and satisfy the following conditions:

(a) 

For any  $x\in X$, there is at least one  $k\in \{1,2,\cdots ,c\}$  such that  ${\mu }_{A_k}(x)>0$;

(b) 

For any  $k\in \{1,2,\cdots ,c\}$,${\mu }_{A_k}(x)$  is continuous on  $X$;

(c) 

For any  $k\in \{1,2,\cdots ,c\}$, there is at least one  $x_0\in X$  such that  ${\mu }_{A_k}(x_0)=1$;

(d) 

For any  $k\in \{1,2,\cdots ,c\}$, if  ${\mu }_{A_k}(x_0)=1$  for  $x_0\in X$, then  ${\mu }_{A_k}(x)$  is non-decreasing on  $[a,x_0]$, and is non-increasing on  $[x_0,b]$;

(e) 

$0<{\displaystyle {\sum }_{k=1}^c{\mu }_{A_k}}(x)\le 1$ holds for $x\in X$.

If ${\displaystyle {\sum }_{k=1}^c{\mu }_{A_k}}(x)=1,\forall x\in X$, then $\tilde{X}$ is said to be a regular fuzzy partition, namely Ruspini’s fuzzy partition.

In addition, Mesiar and Rybátrik [32] used the assignment of an exemplar generating function to define the following g-fuzzy partition.

Definition 4.

[32]. Let $\tilde{X}=\left\{A_k\right|k=1,2,\cdots ,c\}$, where $A_k$ is a fuzzy set on domain $X=\{x_1,x_2,\cdots ,x_n\}$. $\tilde{X}$ is a generalized fuzzy partition if and only if $\forall x\in X$, ${\displaystyle {\sum }_{k=1}^{c}g({\mu }_{A_k}}(x))=1$, where $g(\cdot )$ is a normed generator; i.e., it is a continuous strictly increasing function $g: [0,1]\to [0,1]$, $g(0)=0$, $g(1)=1$. The typical normed generator is $g(x)=x^r(r>0)$, $g(x)=\ln (1+x)/\ln 2$, and $g(x)=(\exp (ax)-1)/(\exp (a)-1)(a>0)$.

Overall, fuzzy partition is the theoretical basis of fuzzy clustering [33]. For a long time, the fuzzy partition problem in fuzzy clustering theory has attracted extensive attention from scholars [34,35].

2.2. Fuzzy Partition-Related Clustering

The FCM algorithm is an unsupervised fuzzy clustering method proposed by Dunn [12] and developed by Bezdek [14]. The optimization model for FCM is as follows:

$$\min J_{FCM}(U,V)={\displaystyle {\sum }_{k=1}^c{\displaystyle {\sum }_{i=1}^n{u}_{ki}^m{d}^2(x_i,v_k)}}$$

(1)

s.t. (a) $u_{ki}\in [0,1],\forall i,k$; (b) ${\displaystyle {\sum }_{k=1}^c{u}_{ki}}=1,\forall i$; (c) $0<{\displaystyle {\sum }_{i=1}^n{u}_{ki}}<n$, $\forall k$.

Fuzzy membership $u_{ik}$ and the clustering center $v_k$ corresponding to the optimization model are obtained as follows:

$$u_{ki}={\displaystyle \frac{1}{{\displaystyle {\sum }_{j=1}^c{\left(d^2(x_i,v_k)/d^2(x_i,v_j)\right)}^{1/(m-1)}}}}$$

(2)

$$v_k={\displaystyle \sum _{i=1}^n{u}_{ki}^m{x}_i}/{\displaystyle \sum _{i=1}^n{u}_{ki}^m}$$

(3)

The FCM algorithm calculates fuzzy membership of each sample to different clusters in the range [0, 1], ensuring that the sum of membership degrees of each sample to all clusters is 1. Namely, the fuzzy partition space in the FCM algorithm is described as follows:

$${\overline{P}}_{fc}=\left\{U={[u_{ki}]}_{c\times n}\in {ℝ}^{c\times n}|u_{ki}\in [0,1],\forall i,k;{\displaystyle {\sum }_{k=1}^c{u}_{ki}}=1,\forall i\right\}$$

(4)

However, this clustering method fails to reveal the inherent correlations between different categories and has some difficulty in dealing with data with noise or outliers. Considering the shortcomings of the FCM algorithms, Wang et al. [36] and Zhu et al. [37] proposed a double-indices-induced fuzzy C-means clustering method, and the corresponding optimization model is established as follows:

$$\min J_{DIFCM}(U,V;X)={\displaystyle {\sum }_{k=1}^c{\displaystyle {\sum }_{i=1}^n{u}_{ki}^m}}{d}^2(x_i,v_k)$$

(5)

s.t. (a) $u_{ki}\in [0,1],\forall i,k$; (b) ${\displaystyle {\sum }_{k=1}^c{u}_{ki}^r}=1,\forall i$,$0<r$; (c) $0<{\displaystyle {\sum }_{i=1}^n{u}_{ki}}<n,\forall k$.

The iterated formulas of membership and clustering center are given by

$$u_{ki}={\displaystyle \frac{1}{{\left({\displaystyle {\sum }_{j=1}^c{\left(d^2(x_i,v_k)/d^2(x_i,v_j)\right)}^{\frac{2r}{m-r}}}\right)}^{1/r}}}$$

(6)

$$v_k={\displaystyle \frac{{\displaystyle \sum _{i=1}^n{u}_{ki}^m{x}_i}}{{\displaystyle \sum _{i=1}^n{u}_{ki}^m}}}$$

(7)

For double-indices-induced fuzzy C-means clustering, if $r=1$, this algorithm degenerates to the classical FCM algorithm. Therefore, the FCM algorithm is a specific case of the DIFCM algorithm. In addition, from the perspective of fuzzy partition, this DIFCM algorithm is a generalized fuzzy partition clustering method [32], and its g-fuzzy partition space may be defined as follows:

$${\overline{P}}_{gc}=\left\{U={[u_{ki}]}_{c\times n}\in {ℝ}^{c\times n}|u_{ki}\in [0,1],\forall i,k;{\displaystyle {\sum }_{k=1}^c{u}_{ki}^r}=1(0<r),\forall i\right\}$$

(8)

2.3. Robust Fuzzy Clustering with Local Information

The FCM algorithm introduces fuzziness to the attributes of each image pixel, allowing the FCM algorithm to preserve more detailed information in the original image and achieve more satisfactory segmentation results. However, the biggest drawback of the FCM algorithm in image segmentation is that it does not consider any spatial information in the image [15], which makes the algorithm sensitive to noise in the image and unable to obtain satisfactory segmentation results in noisy images. Krinidis and Chatzis [21] defined the concept of fuzzy local factor using the local spatial information of image pixels, called local spatial fuzzy factor [38] and introduced this factor into the FCM algorithm to propose the fuzzy local information C-means clustering algorithm (FLICM). Due to the local spatial fuzzy factor fusing the spatial distance and grayscale information between neighborhood pixels in a fuzzy manner [39], it can find a balance between suppressing noise and image detail preservation to a certain extent. The optimization model of this FLICM algorithm is as follows:

$$\min J_{FLICM}(U,V)={\displaystyle {\sum }_{k=1}^c{\displaystyle {\sum }_{i=1}^n{u}_{ki}^m\left[d^2(x_i,v_k)+G_{ki}\right]}}$$

(9)

• s.t. (a) $0\le u_{ki}\le 1,\forall i,k$; (b) ${\displaystyle {\sum }_{k=1}^c{u}_{ki}=1},\forall i$; (c) $0<{\displaystyle {\sum }_{i=1}^n{u}_{ki}}<n,\forall k$.
• with the fuzzy local information factor $G_{ki}={\displaystyle \sum _{j\in N_i,j\ne i}\frac{1}{d_{ji}+1}{(1-u_{kj})}^m{d}^2}(x_j,v_k)$.

Using the Lagrange multiplier method, the following fuzzy membership $u_{ki}$ and the clustering center $v_k$ can be obtained:

$$u_{ki}={\displaystyle \frac{1}{{\displaystyle {\sum }_{l=1}^c{\left(\frac{d^2(x_i,v_k)+G_{ki}}{d^2{(x_i,v_l)}^2+G_{li}}\right)}^{\frac{1}{m-1}}}}}$$

(10)

$$v_k={\displaystyle {\sum }_{i=1}^n{u}_{ki}^m{x}_i}/{\displaystyle {\sum }_{i=1}^n{u}_{ki}^m}$$

(11)

To make FCM clustering more robust to noise, Li and Zheng [22] also proposed an improved fuzzy-based clustering algorithm IFLICM for MR image segmentation. The optimization model for the IFLICM algorithm is as follows:

$$\min J_{IFLICM}(U,V)={\displaystyle {\sum }_{k=1}^c{\displaystyle {\sum }_{i=1}^n{u}_{ki}^m\left[d^2(x_i,v_k)+G_{ki}^{\ast }\right]}}$$

(12)

• s.t. (a) $0\le u_{ki}\le 1,\forall i,k$; (b) ${\displaystyle {\sum }_{k=1}^c{u}_{ki}=1},\forall i$; (c) $0<{\displaystyle {\sum }_{i=1}^n{u}_{ki}}<n,\forall k$.
• where the improved fuzzy local information factor is defined as

$$G_{ki}^{\ast }={\displaystyle \sum _{j\in N_i,j\ne i}{s}_{ji}{(1-u_{kj})}^m{d}^2}(x_j,v_k)$$

(13)

in which, $s_{ji}={\displaystyle \frac{\exp \left(-|x_i-x_j|/255\right)+\exp \left(-|x_i-{\overline{x}}_i|/255\right)}{2}}$, ${\overline{x}}_i={\displaystyle \frac{1}{\left|N_i\right|}}{\displaystyle {\sum }_{j\in N_i}{x}_j}$.

Using the Lagrange multiplier method, the following membership $u_{ki}$ and the clustering center $v_k$ can be also obtained:

$$u_{ki}={\displaystyle \frac{1}{{\displaystyle {\sum }_{l=1}^c{\left(\frac{d^2(x_i,v_k)+G_{ki}^{\ast }}{d^2{(x_i,v_l)}^2+G_{li}^{\ast }}\right)}^{\frac{1}{m-1}}}}}$$

(14)

$$v_k={\displaystyle {\sum }_{i=1}^n{u}_{ki}^m{x}_i}/{\displaystyle {\sum }_{i=1}^n{u}_{ki}^m}$$

(15)

The FLICM algorithm introduces unsupervised parameters, but still lacks sufficient robustness to high noise. In view of this, Gong et al. [23] proposed a kernel metric-based fuzzy local information FCM algorithm (KWFLICM), which introduces a kernel function [40] and the weighted factor $w_{ij}$ [41] to further improve the noise resistance performance. The combination of the coefficient of variation and the weighted factor $w_{ij}$ better expresses the difference between the center pixel and its neighboring pixels and can obtain more information about spatial interactions [39], thereby achieving segmentation results with less noise. The objective function of the KWFLICM algorithm is as follows:

$$J_{KWFLICM}(U,V)={\displaystyle {\sum }_{i=1}^n{\displaystyle {\sum }_{k=1}^c{u}_{ki}^m\left[(1-K(x_i,v_k))+G_{ki}^{\prime }\right]}}$$

(16)

where $U={(u_{ki})}_{c\times n}$ represents the membership matrix of size $c\times n$, $n$ represents the total number of samples, $c$ represents the number of categories into which the samples have been classified, $u_{ki}$ represents the degree to which sample $x_i$ belongs to category $k$, and $m$ is the fuzzy weighting exponent, which is defaulted to $m=2$ in this paper.

The weighted fuzzy local information factor with kernel metric ${G^{\prime }}_{ki}$ is defined as follows:

$${G^{\prime }}_{ki}={\displaystyle \sum _{j\in N_i,j\ne i}{w}_{ji}{(1-u_{kj})}^m(1-K(x_j,v_k))}$$

(17)

where, the weighted factor $w_{ji}$ consists of spatial information $w_{sc}$ and neighborhood gray scale information $w_{gc}$. Using the least square method to optimize its objective function, the update formulas for the degree of membership and clustering center are obtained as follows:

$$u_{ki}={\displaystyle \frac{1}{{\displaystyle \sum _{l=1}^c{\left(\frac{(1-K(x_i,v_k))+G_{ki}^{\prime }}{(1-K(x_i,v_l))+G_{kl}^{\prime }}\right)}^{1/(m-1)}}}}$$

(18)

$$v_k={\displaystyle \sum _{i=1}^n{u}_{ki}^{m}K(x_i,v_k)x_i}/{\displaystyle \sum _{i=1}^n{u}_{ki}^{m}K(x_i,v_k)}$$

(19)

Overall, the KWFLICM algorithm combines neighborhood spatial information and grayscale features [39] to establish a new fuzzy local information factor, which greatly improves the robustness of the algorithm to noise [42].

3. Proposed Clustering Algorithm

To address the shortcomings of existing fuzzy clustering algorithms, a harmonic fuzzy set is first defined; secondly, a harmonic fuzzy partition concept is proposed; and finally, a series of harmonic fuzzy local information C-means algorithms for noisy image segmentation are constructed and its convergence is rigorously proved.

3.1. New Harmonic Fuzzy Set Theory

Zadeh [43] proposed a fuzzy set which uses the degree of membership to indicate the positivity level of elements in the set. However, the fuzzy membership values in fuzzy sets are limited to the range of [0, 1] and their shortcomings are gradually exposed in practical applications. Therefore, this paper will propose a new concept of generalized harmonic fuzzy sets, providing theoretical support for the research of harmonic fuzzy partition and harmonic fuzzy C-means clustering algorithms.

Definition 5.

Let $X=\{x_1,x_2,\cdots ,x_n\}$, and its power set is denoted as $\wp (X)$. The generalized harmonic fuzzy set $\stackrel{⌢}{A}$ built on $X$ is defined as $\stackrel{⌢}{A}=\{(x_i,h_{\stackrel{⌢}{A}}(x_i))|x_i\in X\}$, where $h_{\stackrel{⌢}{A}}(x):X\to (1,+\infty )$ is the generalized harmonic fuzzy membership function, and $h_{\stackrel{⌢}{A}}(x_i)\in (1,+\infty )$ is the harmonic fuzzy membership degree of $x_i$ belonging to the finite set $X$.

We generalize various common concepts of union sets, intersection sets, etc. and define some basic operation on the generalized fuzzy sets, equality, containment, complementation, intersection, and union related to the generalized harmonic fuzzy sets $\stackrel{⌢}{A}$, $\stackrel{⌢}{B}$ in any universe of discourse $X$ as follows.

(a)

$\stackrel{⌢}{A}=\stackrel{⌢}{B}$ if and only if $h_{\stackrel{⌢}{A}}(x)=h_{\stackrel{⌢}{B}}(x)$, $\forall x\in X$;

(b)

$\stackrel{⌢}{A}\subseteq \stackrel{⌢}{B}$ if and only if $h_{\stackrel{⌢}{A}}(x)\le h_{\stackrel{⌢}{B}}(x)$, $\forall x\in X$;

(c)

${\stackrel{⌢}{A}}^c$ is the standard complement of $\stackrel{⌢}{A}$, if and only if $h_{{\stackrel{⌢}{A}}^c}(x)={\displaystyle \frac{h_{\stackrel{⌢}{A}}(x)}{h_{\stackrel{⌢}{A}}(x)-1}}$, $\forall x\in X$;

(d)

$\stackrel{⌢}{A}\cup \stackrel{⌢}{B}$ if and only if $h_{\stackrel{⌢}{A}\cup \stackrel{⌢}{B}}(x)=\max (h_{\stackrel{⌢}{A}}(x),h_{\stackrel{⌢}{B}}(x))$, $\forall x\in X$;

(e)

$\stackrel{⌢}{A}\cap \stackrel{⌢}{B}$ if and only if $h_{\stackrel{⌢}{A}\cap \stackrel{⌢}{B}}(x)=\min (h_{\stackrel{⌢}{A}}(x),h_{\stackrel{⌢}{B}}(x))$, $\forall x\in X$.

Definition 6.

Harmonic fuzzy complement operator is defined an operation on harmonic fuzzy set $\stackrel{⌢}{A}$, namely  $N^{(h)}(h_{\stackrel{⌢}{A}}(x)):(1,+\infty )\to (1,+\infty )$  that assigns a value to each harmonic fuzzy membership degree  $h_{\stackrel{⌢}{A}}(x)$  of any given harmonic fuzzy set  $\stackrel{⌢}{A}$. Some common harmonic fuzzy complement operators are given as follows:

(a) 

Standard harmonic fuzzy complement operator  $N_S^{(h)}(h_{\stackrel{⌢}{A}}(x))={\displaystyle \frac{h_{\stackrel{⌢}{A}}(x)}{h_{\stackrel{⌢}{A}}(x)-1}}$,  $\forall x\in X$;

(b) 

Yager’s harmonic fuzzy complement operator is first constructed as  $N_{Yager}^{(h)}(h_{\stackrel{⌢}{A}}(x))={\displaystyle \frac{h_{\stackrel{⌢}{A}}(x)}{{({(h_{\stackrel{⌢}{A}}(x))}^{\alpha }-1)}^{1/\alpha }}}(\alpha >0)$,  $\forall x\in X$;

(c) 

Sugeno’s harmonic fuzzy complement operator $N_{\mathrm{Sugeno}}^{(h)}(h_{\stackrel{⌢}{A}}(x))={\displaystyle \frac{h_{\stackrel{⌢}{A}}(x)+\lambda }{h_{\stackrel{⌢}{A}}(x)-1}}(\lambda >1)$, $\forall x\in X$.

For standard harmonic fuzzy complement operator $N_S^{(h)}(\cdot )$, if $h_{\stackrel{⌢}{A}}(x)=1.5$, $N_S(h_{\stackrel{⌢}{A}}(x))=3.0$; if $h_{\stackrel{⌢}{A}}(x)=5.0$, $N_S^{(h)}(h_{\stackrel{⌢}{A}}(x))=1.25$; if $h_{\stackrel{⌢}{A}}(x)=2.0$, $N_S^{(h)}(h_{\stackrel{⌢}{A}}(x))=h_{\stackrel{⌢}{A}}(x)=2.0$. Therefore, 2.0 is a fixed point of the standard harmonic complement operator, namely 2.0 is a fixed point of the standard harmonic complement operator; namely 2.0 is an intermediary element of the generalized harmonic fuzzy logistic and reasoning.

To measure the fuzzy uncertainty of generalized harmonic fuzzy sets, we give the axiomatic definition of fuzzy entropy of generalized harmonic fuzzy sets according to existing fuzzy entropy theory of Zadeh’s fuzzy sets [44].

Definition 7.

Let $E^{(h)}$ denotes a mapping $E^{(h)}:$ $\wp (X)\to [0,1]$, where $\wp (X)$ is the set composed of all generalized harmonic fuzzy sets on the finite set $X$. Then $E$ is called the fuzzy entropy of generalized harmonic fuzzy sets, if it satisfies the following conditions:

(a) 

$E^{(h)}(\stackrel{⌢}{A})\to 0$ if and only if $h_{\stackrel{⌢}{A}}(x)\to 1^{+}$ or $+\infty$, $\forall x\in X$;

(b) 

$E^{(h)}(\stackrel{⌢}{A})=1$  if and only if  $h_{\stackrel{⌢}{A}}(x)=2.0$,  $\forall x\in X$;

(c) 

$E^{(h)}(\stackrel{⌢}{A})\le E^{(h)}(\stackrel{⌢}{B})$ if and only if the generalized harmonic fuzzy set $\stackrel{⌢}{A}$ is having lesser overall fuzziness than the generalized harmonic fuzzy set $\stackrel{⌢}{B}$.

In practical applications, some formulas of fuzzy entropy for the generalized harmonic fuzzy sets are given as follows:

(a)

$E^{(h)}(\stackrel{⌢}{A})={\displaystyle \frac{2}{n}}{\displaystyle {\sum }_{i=1}^n\min \left(\frac{1}{h_{\stackrel{⌢}{A}}(x_i)},\frac{h_{\stackrel{⌢}{A}}(x_i)-1}{h_{\stackrel{⌢}{A}}(x_i)}\right)}$;

(b)

$E^{(h)}(\stackrel{⌢}{A})={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n\min \left(\frac{1}{h_{\stackrel{⌢}{A}}(x_i)},\frac{h_{\stackrel{⌢}{A}}(x_i)-1}{h_{\stackrel{⌢}{A}}(x_i)}\right)}}{{\displaystyle {\sum }_{i=1}^n\max \left(\frac{1}{h_{\stackrel{⌢}{A}}(x_i)},\frac{h_{\stackrel{⌢}{A}}(x_i)-1}{h_{\stackrel{⌢}{A}}(x_i)}\right)}}}$

In order to measure the similarity between generalized harmonic fuzzy sets, we give an axiomatic definition of the fuzzy similarity measure based on the existing Zadeh fuzzy similarity theory of fuzzy sets [45].

Definition 8.

A mapping $S^{(h)}:\wp (X)\times \wp (X)\to [0,1]$ is a similarity measure between generalized harmonic fuzzy sets if $S$ satisfies the following conditions:

(a) 

$0\le S^{(h)}(\stackrel{⌢}{A},\stackrel{⌢}{B})\le 1,\forall \stackrel{⌢}{A},\stackrel{⌢}{B}\in \wp (X)$;

(b) 

$S^{(h)}(\stackrel{⌢}{A},\stackrel{⌢}{B})=S^{(h)}(\stackrel{⌢}{B},\stackrel{⌢}{A}),\forall \stackrel{⌢}{A},\stackrel{⌢}{B}\in \wp (X)$;

(c) 

If $\stackrel{⌢}{A}=\stackrel{⌢}{B}$, then $S^{(h)}(\stackrel{⌢}{A},\stackrel{⌢}{B})=1$;

(d) 

If $\stackrel{⌢}{A}\subseteq \stackrel{⌢}{B}\subseteq \stackrel{⌢}{C}$, $\forall \stackrel{⌢}{A},\stackrel{⌢}{B},\stackrel{⌢}{C}\in \wp (X)$, then $S^{(h)}(\stackrel{⌢}{A},\stackrel{⌢}{C})\le S^{(h)}(\stackrel{⌢}{A},\stackrel{⌢}{B})$ and $S^{(h)}(\stackrel{⌢}{A},\stackrel{⌢}{C})\le S^{(h)}(\stackrel{⌢}{B},\stackrel{⌢}{C})$.

Some common formulas for the fuzzy similarity measure are given as follows:

(a)

$S^{(h)}(\stackrel{⌢}{A},\stackrel{⌢}{B})={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n\min (h_{\stackrel{⌢}{A}}(x_i),h_{\stackrel{⌢}{B}}(x_i))}}{{\displaystyle {\sum }_{i=1}^n\max (h_{\stackrel{⌢}{A}}(x_i),h_{\stackrel{⌢}{B}}(x_i))}}}$;

(b)

$S^{(h)}(\stackrel{⌢}{A},\stackrel{⌢}{B})={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n\min (h_{\stackrel{⌢}{A}}^{-1}(x_i),h_{\stackrel{⌢}{B}}^{-1}(x_i))}}{{\displaystyle {\sum }_{i=1}^n\max (h_{\stackrel{⌢}{A}}^{-1}(x_i),h_{\stackrel{⌢}{B}}^{-1}(x_i))}}}$.

In addition, we use the similarity measure between the generalized harmonic fuzzy sets to induce a new formula of fuzzy entropy as follows [46].

$$E^{(h)}(\stackrel{⌢}{A})=S^{(h)}(\stackrel{⌢}{A}\cup {\stackrel{⌢}{A}}^c,\stackrel{⌢}{A}\cap {\stackrel{⌢}{A}}^c)={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n\min \left(\frac{1}{h_{\stackrel{⌢}{A}}(x_i)},\frac{h_{\stackrel{⌢}{A}}(x_i)-1}{h_{\stackrel{⌢}{A}}(x_i)}\right)}}{{\displaystyle {\sum }_{i=1}^n\max \left(\frac{1}{h_{\stackrel{⌢}{A}}(x_i)},\frac{h_{\stackrel{⌢}{A}}(x_i)-1}{h_{\stackrel{⌢}{A}}(x_i)}\right)}}}$$

(20)

In summary, the fuzzy entropy and similarity of generalized reconciled fuzzy sets can be widely used in pattern recognition, image processing, and intelligent decision making [47].

3.2. Harmonic Fuzzy Partition

In 2003, Leski [25] proposed a weighted generalized conditional fuzzy clustering algorithm that introduces additional information about the fuzzy environment as a constraint in the clustering process. The algorithm is effective for clustering abnormal data and constructing radial basis neural networks. The specific constraints are constructed below:

$${\displaystyle {\cup }_{k=1}^c{}_{(\alpha ,\beta )}u_{ki}}={({\displaystyle \sum _{k=1}^c{\beta }_{ki}{(u_{ki})}^{\alpha }})}^{1/\alpha }$$

(21)

when $\alpha \in (-\infty ,0)\cup (0,+\infty )$, and weight constraints ${\beta }_k\ge 0,{\displaystyle {\sum }_{k=1}^c{\beta }_k=1}$, we analyze the following three special cases:

(1)

If $\alpha =1.0$, ${\displaystyle {\cup }_{j=1}^c{}_{(\alpha ,\beta )}u_{ki}}={({\displaystyle \sum _{k=1}^c{\beta }_k{(u_{ki})}^{\alpha }})}^{1/\alpha }={\displaystyle {\sum }_{k=1}^c{\beta }_k{u}_{ki}}$;

(2)

If $\alpha \to 0$, $u_{ki}\ne 0$, $\underset{\alpha \to 0}{\lim }{\displaystyle {\cup }_{j=1}^c{}_{(\alpha ,\beta )}u_{ki}}=\underset{\alpha \to 0}{\lim }{({\displaystyle \sum _{k=1}^c{\beta }_k{(u_{ki})}^{\alpha }})}^{1/\alpha }={\displaystyle {\prod }_{k=1}^c{u}_{ki}^{{\beta }_k}}$;

(3)

If $\alpha =-1.0$, ${\displaystyle {\cup }_{j=1}^c{}_{(\alpha ,\beta )}u_{ki}}={({\displaystyle \sum _{k=1}^c{\beta }_k{(u_{ki})}^{\alpha }})}^{1/\alpha }={\displaystyle \frac{1}{{\displaystyle {\sum }_{k=1}^c{\beta }_k/u_{ki}}}}$.

The weighted generalized conditional fuzzy clustering optimization model is as follows:

$$\min J_{GWCFCM}(U,V;X)={\displaystyle {\sum }_{k=1}^c{\displaystyle {\sum }_{i=1}^n{\beta }_k{u}_{ki}^m{d}^2\left(x_i,v_k\right)}}$$

(22)

$\mathrm{s}.\mathrm{t}.$ (a) $u_{ij}\in [0,+\infty ),\forall i,k$; (b) $c\cdot {({\displaystyle {\sum }_{k=1}^c{\beta }_k{u}_{ki}^{\alpha }})}^{1/\alpha }=f_i,\forall i$; (c) $0<{\displaystyle {\sum }_{i=1}^n{u}_{ki}}<n,\forall k$.

The iterated formulas of fuzzy membership $u_{ij}$ and the clustering center $v_j$ are shown below:

$$u_{ki}={\displaystyle \frac{f_i}{c}}{\displaystyle \frac{1}{{\left({\displaystyle {\sum }_{j=1}^c{\beta }_j{\left(d^2(x_i,v_k)/d^2(x_i,v_j)\right)}^{\alpha /(m-\alpha )}}\right)}^{1/\alpha }}}$$

(23)

$$v_k={\displaystyle {\sum }_{i=1}^n{u}_{ki}^m{x}_i}/{\displaystyle {\sum }_{i=1}^n{u}_{ki}^m}$$

(24)

when $\alpha \to 0$,${\beta }_k=1/c(\forall k)$, and $f_i=1(\forall i)$, fuzzy membership is $u_{ki}={\displaystyle \frac{{\displaystyle {\prod }_{j=1}^c{(d^2(x_i,v_j))}^{1/(cm)}}}{{(d^2(x_i,v_k))}^{1/m}}}$, and this corresponds the generalized multiplicative fuzzy clustering based on non-additive multiplicative fuzzy partition, and can be also explained by the theory of the generalized multiplicative fuzzy sets [48]. If $\alpha =1.0$, ${\beta }_k=1/c(\forall k)$, and $f_i=1(\forall i)$, membership $u_{ki}={\displaystyle \frac{1}{{\displaystyle {\sum }_{j=1}^c{(d^2(x_i,v_k)/d^2(x_i,v_j))}^{1/(m-1)}}}}$, and this corresponds the classical fuzzy C-means clustering based on Ruspini’s additive fuzzy partition, and can be also explained by the theory of Zadeh’s fuzzy sets. However, if $\alpha =-1.0$, ${\beta }_k=1/c(\forall k)$, and $f_i=1(\forall i)$, membership $u_{ij}={\displaystyle {\sum }_{r=1}^c{(d^2(x_i,v_j)/d^2(x_i,v_r))}^{1/(m+1)}}$. This clustering has not attracted the attention of relevant scholars, but it can be explained by the theory of the generalized harmonic fuzzy sets proposed in this paper. Therefore, on the basis of harmonic fuzzy set theory proposed in this paper, we will explore a new partition theory and optimization model to support this clustering method to promote the widespread application in machine learning and machine intelligence.

Based on the generalized harmonic fuzzy sets proposed in this paper, a new concept of harmonic fuzzy partition is originally proposed as follows.

Definition 9.

Given a dataset $X=\{x_1,x_2,\cdots ,x_n\}$, these samples are then categorized into $c$ classes. $H={[h_{ki}]}_{c\times n}$ being a symmetric harmonic fuzzy partition matrix of size $c\times n$. $h_{ki}$ denotes the harmonic fuzzy membership of each sample belonging to different classes and the corresponding harmonic fuzzy partition space ${\tilde{H}}_c$ is defined as follows:

$${\tilde{P}}_{hc}=\left\{H={[h_{ij}]}_{n\times c}\in {ℝ}^{n\times c}|h_{ij}\in (1,+\infty ),\forall i,k;{\displaystyle \frac{1}{{\displaystyle {\sum }_{k=1}^{c}1/h_{ki}}}}=1,\forall i\right\}$$

(25)

The above concept of symmetric harmonic fuzzy partition provides theoretical support for the research of the generalized harmonic fuzzy partition C-means clustering.

3.3. Modeling of Harmonic Fuzzy Clustering

Based on the related theories of harmonic fuzzy sets and harmonic fuzzy partition, the harmonic fuzzy C-means clustering algorithm (HCM) is researched, which satisfies the constraint that the harmonic sum of membership degrees of each sample to all classes is 1. Therefore, it can reveal the intrinsic correlation between data and help further discover representation of data structures [49].

The HCM algorithm is an abstract process to find the minimum value of the objective function by dividing the dataset $X$ into $c$ classes. It can be modeled as follows:

$$\min J_{GHFCM}(H,V)={\displaystyle {\sum }_{k=1}^c{\displaystyle {\sum }_{i=1}^n{h}_{ki}^m{d}^2(x_i,v_k)}}$$

(26)

• s.t. (a) $h_{ki}\in (1,+\infty ),\forall i,k$; (b) $1/{\displaystyle {\sum }_{k=1}^{c}1/h_{ki}}=1,\forall i$; (c) $0<{\displaystyle {\sum }_{i=1}^{n}1/h_{ki}}<n,\forall k$.
• where $m$ is the fuzzy weighting exponential which affects the fuzziness of the clustering structure. $h_{ki}$ denotes the harmonic fuzzy membership degree of each sample belonging to different classes.

Using the Lagrange multiplier method, the following unconstrained objective function can be constructed:

$$L(H,V,\lambda )={\displaystyle {\sum }_{k=1}^c{\displaystyle {\sum }_{i=1}^n{h}_{ki}^m{d}^2(x_i,v_k)}}+{\displaystyle {\sum }_{i=1}^n{\lambda }_i(1-({\displaystyle {\sum }_{k=1}^c{h}_{ki}^{-1}}{)}^{-1})}$$

(27)

In addition, according to the constraint condition $1=({\displaystyle {\sum }_{k=1}^c{h}_{ki}^{-1}}{)}^{-1}$, we may obtain a new unconstrained objective function as follows:

$$L(H,V,\lambda )={\displaystyle {\sum }_{k=1}^c{\displaystyle {\sum }_{i=1}^n{h}_{ki}^m{d}^2(x_i,v_k)}}+{\displaystyle {\sum }_{i=1}^n{\lambda }_i({\displaystyle {\sum }_{k=1}^c{h}_{ki}^{-1}}-1)}$$

(28)

By taking the partial derivatives of $L(H,V,\lambda )$ with respect to the harmonic fuzzy membership $h_{ki}$ and the clustering center $v_k$ in Equation (28) and making them zero, we obtain the following:

$$h_{ki}={\displaystyle \frac{{\displaystyle {\sum }_{l=1}^c{(d^2(x_i,v_l))}^{1/(m+1)}}}{{(d^2(x_i,v_k))}^{1/(m+1)}}}$$

(29)

$$v_k={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n{h}_{ki}^m{x}_i}}{{\displaystyle {\sum }_{i=1}^n{h}_{ki}^m}}}$$

(30)

With harmonic fuzzy clustering, like the existing Bezdek fuzzy clustering, it is difficult to suppress the impact of noise in image segmentation when directly applied. To this end, the robust FLICM algorithm proposes a series of improvements to harmonic fuzzy clustering. Figure 1 shows the main framework of the proposed algorithm.

3.3.1. Robust Harmonic Fuzzy Clustering with Local Information

However, the above harmonic fuzzy C-means clustering algorithm has the same drawback as the classical FCM algorithm; i.e., it is very sensitive to noise and outliers. Inspired by the existing robust FLICM algorithm [21], this paper further proposes the harmonic fuzzy local information C-means (HLICM) algorithm to solve the above problem. The optimization model of the robust harmonic fuzzy algorithm for image segmentation is constructed as follows:

$$\min J_{\mathrm{HIICM} }(H,V)={\sum }_{k=1}^c{\sum }_{i=1}^n{h}_{ki}^m\left[d^2\left(x_i,v_k\right)+H_{kj}\right]$$

(31)

• s.t. (a) $h_{ki}\in (1,+\infty ),\forall i,k$; (b) $1/{\displaystyle {\sum }_{k=1}^{c}1/h_{ki}}=1,\forall i$; (c) $0<{\displaystyle {\sum }_{i=1}^{n}1/h_{ki}}<n,\forall k$.
• where $m$ is the fuzzy weighting exponent and $h_{ki}$ represents the harmonic fuzzy membership of each sample belonging to different categories.

$$H_{ki}={\displaystyle \sum _{j\in N_i,j\ne i}\frac{1}{d_{ji}+1}{\left(h_{kj}/(h_{kj}-1)\right)}^m{d}^2}(x_j,v_k)$$

(32)

where $h_{kj}/(h_{kj}-1)$ is the standard complement membership of harmonic fuzzy membership $h_{kj}$. Obviously, there is the equation ${\displaystyle \frac{1}{h_{kj}}}+{\displaystyle \frac{1}{h_{kj}/(1-h_{kj})}}=1$.

Compared with the FLICM algorithm, the proposed algorithm replaces the Zadeh fuzzy set in the model with a harmonic fuzzy set, which establishes a harmonic fuzzy local information C-means clustering algorithm, and therefore, the above model is reasonable.

Using the Lagrange multiplier method, the unconstrained objective function is constructed as follows:

$$J_{HLICM}(H,V,\lambda )={\displaystyle {\sum }_{k=1}^c{\displaystyle {\sum }_{i=1}^n{h}_{ki}^m[d^2(x_i,v_k)}+H_{ki}]}+{\displaystyle {\sum }_{i=1}^n{\lambda }_i(1-{\displaystyle {\sum }_{j=1}^{c}1/h_{ki}})}$$

(33)

We find the partial derivatives of the objective function with respect to $h_{ki}$, $v_k$, and ${\lambda }_i$ and make them equal to zero.

• Solution $h_{ki}$

  $${\displaystyle \frac{\partial J}{\partial h_{ki}}}=m{h}_{ki}^{m-1}[d^2(x_i,v_k)+H_{ki}]-{\lambda }_i{h}_{ki}^{-2}=0$$

  (34)

  $${\displaystyle \frac{\partial J}{\partial {\lambda }_i}}={\displaystyle {\sum }_{k=1}^c{h}_{ki}^{-1}-1}=0$$

  (35)

From Equation (34), it follows that

$$h_{ki}={\left({\displaystyle \frac{{\lambda }_i}{m[d^2(x_i,v_k)+H_{ki}]}}\right)}^{{\displaystyle \frac{1}{m+1}}}$$

(36)

Substituting Equation (36) into Equation (35), we can obtain

$${\displaystyle {\sum }_{k=1}^c{({\lambda }_i/m)}^{-1/(m+1)}\cdot {(d^2(x_i,v_k)+H_{ki})}^{1/(m+1)}=1}$$

(37)

In this way, we get

$${\left({\displaystyle \frac{{\lambda }_i}{m}}\right)}^{1/(m+1)}={\displaystyle {\sum }_{k=1}^c{\left(d^2(x_i,v_k)+H_{kj}\right)}^{1/(m+1)}}$$

(38)

Substituting Equation (38) into Equation (36), we obtain

$$h_{ki}={\displaystyle \frac{{\displaystyle {\sum }_{l=1}^c{(d^2(x_i,v_l)+H_{li})}^{\frac{1}{m+1}}}}{{(d^2(x_i,v_k)+H_{ki})}^{\frac{1}{m+1}}}}$$

(39)

2.

Solution $V$

According to ${\displaystyle \frac{\partial L}{\partial v_k}}=0$, we can obtain

$$v_k={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n{h}_{ki}^m(x_i+{\displaystyle {\sum }_{j\in N_i,j\ne i}\frac{1}{d_{ji}+1}{(h_{kj}/(h_{kj}-1))}^m{x}_j)}}}{{\displaystyle {\sum }_{i=1}^n{h}_{ki}^m(1+{\displaystyle {\sum }_{j\in N_i,j\ne i}\frac{1}{d_{ji}+1}{(h_{kj}/(h_{kj}-1))}^m)}}}}$$

(40)

To decrease the computational complexity, this clustering center is often approximated by the following equation:

$$v_k={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n{h}_{ki}^m{x}_i}}{{\displaystyle {\sum }_{i=1}^n{h}_{ki}^m}}}$$

(41)

Equations (41) and (39) are iterative formulas of the harmonic fuzzy membership and the clustering center, respectively, and they solve Equation (31) by alternately iterating between the harmonic fuzzy partition matrix and the clustering centers. The specific algorithmic algorithm 1 is described as follows:

Algorithm 1 Harmonic fuzzy local information C-means clustering

Input     Dataset X = {x1, x2,…, xn}, where $n$ is the number of samples.

Output    $\mathrm{Harmonic} \mathrm{fuzzy} \mathrm{partition} \mathrm{matrix} h={[h_{ij}]}_{n\times c}$ and the clustering center         matrix $V$.

Initialization Set algorithm stopping error $\epsilon$, set the maximum number of        iterations of the algorithm $T$, initialize the clustering center $v_k^{(0)}$, set        the number of iterations $t=0$, and set the size of neighborhood         window $w\times w$.

Repeat   $\mathrm{Calculate} \mathrm{the} \mathrm{degree} \mathrm{of} \mathrm{harmonic} \mathrm{fuzzy} \mathrm{membership} h_{ki}^{(t)}$ using Equation (39);        $\mathrm{Calculate} \mathrm{the} \mathrm{clustering} \mathrm{center} v_k$ using Equation (41);        $\mathrm{If} |V^{(t+1)}-V^{(t)}||<\epsilon$ or the number of iterations of the algorithm        $t>T$, the algorithm ends; Otherwise, $t=t+1$

End    Algorithm is over.

3.3.2. Harmonic Fuzzy Clustering with Weighted Local Information

However, due to the fact that the segmentation results and noise resistance of the HLICM algorithm are not as good as we expected, this paper was inspired by the KWFLICM algorithms [23], and a weighted local coefficient $w_{ji}$ was introduced into the HLICM algorithm to construct a new robust fuzzy local clustering algorithm IHLICM with good noise immunity and segmentation performance. The optimization model of the IHLICM algorithm is constructed below:

$$\min J_{IHLICM}(H,V)={\displaystyle {\sum }_{k=1}^c{\displaystyle {\sum }_{i=1}^n{h}_{ki}^m\left[d^2(x_i,v_k)+{H^{\prime }}_{ki}\right]}}$$

(42)

• s.t. (a) $h_{ki}\in (1,+\infty ),\forall i,k$; (b) $1/{\displaystyle {\sum }_{k=1}^{c}1/h_{ki}}=1,\forall i$; (c) $0<{\displaystyle {\sum }_{i=1}^{n}1/h_{ki}}<n,\forall k$.
• where $m$ is the fuzzy weighting exponent, which defaults to the range of [5, 10] and $h_{ki}$ denotes the harmonic fuzzy membership of each sample belonging to different categories.

The weighted harmonic fuzzy local information factor ${H^{\prime }}_{kj}$ is defined as follows:

$${H^{\prime }}_{ki}={\displaystyle \sum _{j\in N_i,j\ne i}{w}_{ji}(h_{kj}/(h_{kj}-1){)}^m}{d}^2(x_j,v_k)$$

(43)

Using the least squares method to solve this objective function, the updated formulas for harmonic fuzzy membership and the clustering center are obtained as follows:

$$h_{ik}={\displaystyle \frac{{\displaystyle \sum _{l=1}^c{(d^2(x_i,v_l)+H_{li}^{\prime })}^{1/(m+1)}}}{{(d^2(x_i,v_k)+H_{li}^{\prime })}^{1/(m+1)}}}$$

(44)

$$v_k={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n{h}_{ki}^m(x_i+{\displaystyle {\sum }_{j\in N_i,j\ne i}{w}_{ji}{(h_{kj}/(h_{kj}-1))}^m{x}_j)}}}{{\displaystyle {\sum }_{i=1}^n{h}_{ki}^m(1+{\displaystyle {\sum }_{j\in N_i,j\ne i}{w}_{ji}{(h_{kj}/(h_{kj}-1))}^m)}}}}$$

(45)

To reduce the computational complexity, this clustering center is often approximated by the following formula:

$$v_k={\displaystyle \sum _{i=1}^n{h}_{ki}^m{x}_i}/{\displaystyle \sum _{i=1}^n{h}_{ki}^m}$$

(46)

3.3.3. Robust Kernelized Harmonic Fuzzy Clustering with Local Information

Two robust HCM algorithms have significant improvement in terms of noise resistance and segmentation performance, but there are still shortcomings compared to the KWFLICM algorithm. Therefore, this paper was once again inspired by the KWFLICM algorithm and proposes a novel high performance KWHLICM algorithm. The optimization model of the KWHLICM algorithm is constructed as follows:

$$\min J_{KWHLICM}(H,V)={\displaystyle {\sum }_{k=1}^c{\displaystyle {\sum }_{i=1}^n{h}_{ki}^m\left[1-K(x_i,v_k)+{H^{″}}_{ki}\right]}}$$

(47)

• s.t. (a) $h_{ki}\in (1,+\infty ),\forall i,k$; (b) $1/{\displaystyle {\sum }_{k=1}^{c}1/h_{ki}}=1,\forall i$; (c) $0<{\displaystyle {\sum }_{i=1}^{n}1/h_{ki}}<n,\forall k$
• where $m$ is the fuzzy weighting exponent, which defaults to the range of [5, 10], $h_{ki}$ denotes the harmonic fuzzy membership of each sample belonging to different categories, and the kernel weighted harmonic fuzzy local information factor ${H^{″}}_{kj}$ is defined as follows:

$${H^{″}}_{ki}={\displaystyle \sum _{j\ne i,j\in N_i}{w}_{ij}({h_{kj}/(h_{kj}-1))}^m(1-K(x_j,v_k))}$$

(48)

where $K(x_j,v_k)=\exp \left(-{\displaystyle \frac{d^2(x_j,v_k)}{{\sigma }^2}}\right)$, $\sigma$ is the scale parameter of Gaussian function.

The weighted local coefficient $w_{ij}$ in [23] consists of spatial information $w_{sc}$ and neighborhood gray scale information $w_{gc}$. The least squares method is used to solve this objective function to obtain the updated formulas for harmonic membership and the clustering center, respectively, as follows:

$$h_{ki}={\displaystyle \frac{{\displaystyle \sum _{l=1}^c{((1-K(x_i,v_l))+H_{li}^{\prime \prime })}^{1/(m+1)}}}{{((1-K(x_i,v_k))+H_{ki}^{\prime \prime })}^{1/(m+1)}}}$$

(49)

$$v_k={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n{h}_{ki}^m(K(x_i,v_k)x_i+{\displaystyle {\sum }_{j\in N_i,j\ne i}{w}_{ji}{(h_{kj}/(h_{kj}-1))}^{m}K(x_j,v_k)x_j)}}}{{\displaystyle {\sum }_{i=1}^n{h}_{ki}^m(K(x_i,v_k)+{\displaystyle {\sum }_{j\in N_i,j\ne i}{w}_{ji}{(h_{kj}/(h_{kj}-1))}^{m}K(x_j,v_k))}}}}$$

(50)

To reduce the computational complexity, this clustering center is often approximated by the following formula:

$$v_k={\displaystyle \sum _{i=1}^n{h}_{ik}^{m}K(x_i,v_k)x_i}/{\displaystyle \sum _{i=1}^n{h}_{ik}^{m}K(x_i,v_k)}$$

(51)

Similar to the KWFLICM algorithm, the proposed KWHLICM algorithm also combines the neighborhood spatial information and grayscale features to establish a new fuzzy local information factor, which greatly improves the robustness of the algorithm to noise. The KWHLICM algorithm tests a noisy image, with the dynamic iteration process for image segmentation shown in Figure 2.

As shown in Figure 3, a 3 × 3 block of neighborhood pixels is selected from the noisy image. The iterative changes of fuzzy membership and the clustering centers intuitively reflect that the proposed algorithm is a two-level alternating iterative algorithm. In a neighborhood of the noisy image, we identify normal pixels with grayscale values of 118, 120, 124, and 140, and noisy pixels with grayscale values of 0, 26, 208, and 230. In the first iteration, the membership of the noise pixels is relatively low, and after several iterations using the KWHLICM algorithm, noisy pixels and surrounding pixels have similar membership. The iterated results show that the proposed KWHLICM algorithm can effectively suppress the effect of noise on this fuzzy clustering algorithm.

In summary, these two improved algorithms have a similar process to the HLICM algorithm; they are not repeated here due to the limited length of this paper.

3.4. Algorithm Convergence Analysis

Only the convergence of alternating iterative algorithms can ensure their effectiveness and rationality. So far, the related algorithms for fuzzy C-means clustering with constraints as additive expressions have been rigorously proved to be locally convergent and have been widely used in pattern analysis and data mining [50,51]. Saha and Das [52] provided a general convergence analysis method for iterative algorithms using Zangwill’s theorem. This paper uses this method to analyze the convergence of the proposed HLICM algorithm. If the objective function of the HLICM algorithm is decreasing, it indicates that the algorithm is convergent.

Considering the limited length of this paper, the proof of continuity and tightness constraints for the HLICM algorithm is the same as that for the FCM algorithm [53,54,55], and the IHLICM and KWHLICM algorithms are similar to the HLICM method. Therefore, in this article, we only prove that the objective function of the HLICM algorithm is a decreasing function, thus omitting the proof of the other two constraints in the theorem and the subsequent proof of the proposed algorithm.

Theorem 1.

[56] (Lagrange’s Theorem). Let functions $f:D_f\to ℝ$, $D_f\subseteq {\Re }^N$ and $g_i:D_{g_i}\to ℝ$, $D_{g_i}\to {\Re }^n$, $i=1,2,\cdots ,t,t<n$, be continuously partially differentiable and let $x^0=\left(x_1^0,\cdots ,x_N^0\right)\in D_f$ be a local extreme point of the function $f$ subject to the constraints $g_i(x_1,\cdots ,x_N)=0$, $i=1,2,\cdots ,t$. $L(x;\lambda )=f(x_1,\cdots ,x_N)+{\displaystyle {\sum }_{i=1}^t{\lambda }_i}{g}_i(x_1,\cdots ,x_N)$ and $\left|J\right|=\left[\begin{array}{cccc}{\displaystyle \frac{\partial g_1(x)}{\partial x_1}}& {\displaystyle \frac{\partial g_1(x)}{\partial x_2}}& \cdots & {\displaystyle \frac{\partial g_1(x)}{\partial x_t}}\\ {\displaystyle \frac{\partial g_2(x)}{\partial x_1}}& {\displaystyle \frac{\partial g_2(x)}{\partial x_2}}& \cdots & {\displaystyle \frac{\partial g_2(x)}{\partial x_t}}\\ ⋮& ⋮& \ddots & ⋮\\ {\displaystyle \frac{\partial g_t(x)}{\partial x_1}}& {\displaystyle \frac{\partial g_t(x)}{\partial x_2}}& \cdots & {\displaystyle \frac{\partial g_t(x)}{\partial x_t}}\end{array}\right]\ne 0$ at the point $x^0$. Then, we have that the gradient of $L(x;\lambda )$ at the point $(x^0,{\lambda }^0)$ is 0, i.e., $\nabla L(x^0;{\lambda }^0)=0$.

Theorem 2.

[56] (local sufficient conditions). Let functions $f:D_f\to ℝ$, $D_f\subseteq {\Re }^N$ and $g_i:D_{g_i}\to ℝ$, $D_{g_i}\to {\Re }^n$, $i=1,2,\cdots ,t,t<n$, be continuously partially differentiable and let $x^0=\left(x_1^0,\cdots ,x_N^0\right)\in D_f$ be a solution of the system $\nabla L(x^0;{\lambda }^0)=0$. Let $H_L(x,\lambda )=\left[\begin{array}{cccccc}0& \cdots & 0& {\displaystyle \frac{\partial L^2}{\partial {\lambda }_1\partial x_1}}& \cdots & {\displaystyle \frac{\partial L^2}{\partial {\lambda }_1\partial x_n}}\\ ⋮& \ddots & ⋮& ⋮& \ddots & \\ 0& \cdots & 0& {\displaystyle \frac{\partial L^2}{\partial {\lambda }_t\partial x_1}}& \cdots & {\displaystyle \frac{\partial L^2}{\partial {\lambda }_t\partial x_n}}\\ {\displaystyle \frac{\partial L^2}{\partial x_1\partial {\lambda }_1}}& \cdots & {\displaystyle \frac{\partial L^2}{\partial x_1\partial {\lambda }_t}}& {\displaystyle \frac{\partial L^2}{\partial x_1\partial x_1}}& \cdots & {\displaystyle \frac{\partial L^2}{\partial x_1\partial x_n}}\\ ⋮& \ddots & ⋮& ⋮& \ddots & ⋮\\ {\displaystyle \frac{\partial L^2}{\partial x_n\partial {\lambda }_1}}& \cdots & {\displaystyle \frac{\partial L^2}{\partial x_n\partial {\lambda }_t}}& {\displaystyle \frac{\partial L^2}{\partial x_n\partial x_1}}& \cdots & {\displaystyle \frac{\partial L^2}{\partial x_n\partial x_n}}\end{array}\right]$ be the bordered Hessian matrix and consider its leading principle minors $\left|{\overline{H}}_r(x^0;{\lambda }^0)\right|$ of the order $r=2t+1,2t+2,\cdots ,n+t$ at point $(x^0,{\lambda }^0)$. Therefore, the following expressions can be derived:

(1) 

If all leading principle minors $\left|{\overline{H}}_r(x^0;{\lambda }^0)\right|$, $2t+1\le r\le n+t$, have the sign ${(-1)}^t$, then $x^0=\left(x_1^0,\cdots ,x_N^0\right)$ is a local minimum point of function $f$ subject to the constraints $g_i(x)=0,i=1,2,\cdots ,t$.

(2) 

If the signs of all leading principle minors $\left|{\overline{H}}_r(x^0;{\lambda }^0)\right|$, $2t+1\le r\le n+t$, are alternated, the sign of $\left|{\overline{H}}_{n+1}(x^0;{\lambda }^0)\right|=\left|{\overline{H}}_L(x^0;{\lambda }^0)\right|$ being that of ${(-1)}^n$ then $x^0=\left(x_1^0,\cdots ,x_N^0\right)$ is a local minimum point of functions $f$ subject to the constraints $g_i(x)=0,i=1,2,\cdots ,t$.

(3) 

If neither the conditions of (1) nor those of (2) are satisfied, then  $x^0$  is not a local extreme point of function  $f$  subject to the constraints  $g_i(x)=0,i=1,2,\cdots ,t$. Here, the case in which one or several leading principal minors have a value of zero is not considered a violation of conditions (1) or (2).

For the optimization model of the HLICM algorithm, using constraint $1=1/{\displaystyle {\sum }_{j=1}^{c}1/h_{ij}}$, the final constructed unconstrained objective function is constructed as follows:

$$\begin{array}{ll}{J}_{HLICM}(H,V,\lambda )& ={\displaystyle {\sum }_{k=1}^c{\displaystyle {\sum }_{i=1}^n{h}_{ki}^m[d^2(x_i,v_k)}+{\displaystyle \sum _{j\in N_i,j\ne i}\frac{1}{d_{ji}+1}({h_{kj}/(h_{kj}-1))}^m{d}^2(x_j,v_k)}]}\\ & +{\displaystyle {\sum }_{i=1}^n{\lambda }_i(-1+{\displaystyle {\sum }_{k=1}^c\frac{1}{h_{ki}}})}\end{array}$$

(52)

where $H_{ki}={\displaystyle \sum _{j\in N_i,j\ne i}\frac{1}{d_{ji}+1}{\left(h_{kj}/(h_{kj}-1)\right)}^m{d}^2(x_j,v_k)}$.

If $J_{HLICM}(H,V,\lambda )$ is a decreasing function, then the following three propositions should hold true

Proposition 1.

Given a clustering center $V$, let $F(H)=J_{HLICM}(H,V,\lambda )$, if and only if Equation (39) is computed to obtain $h_{ki}^{\ast }$, $h_{ki}^{\ast }$ is a minima of function $F(H,\lambda )$.

Proof. 

Given $V$, if it is assumed that the minimum point of $F(H,\lambda )$ is $h^{\ast }$, then

$${\displaystyle \frac{\partial F(H^{\ast })}{\partial h_{ki}}}=m{h}_{ki}^{m-1}[d^2(x_i,v_k)+H_{ki}]-{\lambda }_i(1/h_{ki}^2)=0$$

(53)

$${\lambda }_i=m{h}_{ki}^{m+1}[d^2(x_i,v_k)+H_{ki}]$$

(54)

$$h_{ki}={\left({\lambda }_i/(m[d^2(x_i,v_k)+H_{ki}])\right)}^{{\displaystyle \frac{1}{m+1}}}$$

(55)

Finally, we can obtain

$$h_{ki}^{\ast }=h_{ki}={\displaystyle \frac{{\displaystyle {\sum }_{l=1}^c{(d^2(x_i,v_l)+H_{li})}^{\frac{1}{m+1}}}}{{(d^2(x_i,v_k)+H_{ki})}^{\frac{1}{m+1}}}}$$

(56)

Subsequently, we find the Hessian matrix of function $F(H)$ when $h_{ki}=h_{ki}^{\ast }$

$$\begin{array}{ll}{\displaystyle \frac{\partial }{\partial h_{ki}}}({\displaystyle \frac{\partial F(H^{\ast })}{\partial h_{ki}}})& =m(m-1)h_{ki}^{m-2}(d^2(x_i,v_k)+H_{ki})+2(m{h}_{ki}^{m+1}(d^2(x_i,v_k)+H_{ki}))h_{ki}^{-3}\\ & =m(m-1)h_{ki}^{m-2}(d^2(x_i,v_k)+H_{ki})+2m{h}_{ki}^{m-2}(d^2(x_i,v_k)+H_{ki})\\ & =m(m+1)h_{ki}^{m-2}(d^2(x_i,v_k)+H_{ki})\end{array}$$

(57)

As a result, we obtain

$${\displaystyle \frac{\partial }{\partial h_{ab}}}({\displaystyle \frac{\partial F(H^{\ast })}{\partial h_{ki}}})=\left\{\begin{array}{cc}0,\hfill & \hfill a\ne k \mathrm{or} b\ne i\\ m(m+1)h_{ki}^{m-2}(d^2(x_i,v_k)+H_{ki}),\hfill & \hfill a=k \mathrm{and} b=i\end{array}\right.$$

(58)

According to Equation (58), this Hessian matrix is positive definite when $m>-1$ However, for the HLICM algorithm, it only makes sense when $m>0$. Therefore, this Hessian matrix is positive definite when $m$ takes any real number greater than 0.

At the same time, we can know that the sufficient condition for determining the strict minima of the objective function is to analyze the Jacobi and Hessian matrices. Therefore, in addition to the above analysis, we also need to evaluate the bordered Hessian matrix [56]. Given the clustering center matrix $V$, the bordered Hessian matrix of $h_i=[h_{1i},h_{2i},\cdots ,h_{ci}]$ and ${\lambda }_i$ is as follows:

$$H_L(h_i,{\lambda }_i)=\left[\begin{array}{cccc}0& {\displaystyle \frac{{\partial }^{2}L}{\partial {\lambda }_i\partial h_{1i}}}& \cdots & {\displaystyle \frac{{\partial }^{2}L}{\partial {\lambda }_i\partial h_{ci}}}\\ {\displaystyle \frac{{\partial }^{2}L}{\partial {\lambda }_i\partial h_{1i}}}& {\displaystyle \frac{{\partial }^{2}L}{\partial h_{1i}\partial h_{1i}}}& \cdots & {\displaystyle \frac{{\partial }^{2}L}{\partial h_{1i}\partial h_{ci}}}\\ ⋮& ⋮& \ddots & ⋮\\ {\displaystyle \frac{{\partial }^{2}L}{\partial {\lambda }_i\partial h_{ci}}}& {\displaystyle \frac{{\partial }^{2}L}{\partial h_{ci}\partial h_{1i}}}& \cdots & {\displaystyle \frac{{\partial }^{2}L}{\partial h_{ci}\partial h_{ci}}}\end{array}\right]$$

(59)

For ${\displaystyle \frac{{\partial }^{2}L}{\partial {\lambda }_i\partial h_{ki}}}={\displaystyle \frac{1}{h_{ki}^2}},1\le k\le c$ the leading principle minors of matrix $H_L(h_i,{\lambda }_i)$ are as follows:

$$\begin{array}{ll}\left|{\overline{H}}_3(h_i^{\ast },{\lambda }_i^{\ast })\right|& =\det \left[\begin{array}{ccc}0& {\displaystyle \frac{{\partial }^{2}L}{\partial {\lambda }_i\partial h_{1i}}}& {\displaystyle \frac{{\partial }^{2}L}{\partial {\lambda }_i\partial h_{2i}}}\\ {\displaystyle \frac{{\partial }^{2}L}{\partial {\lambda }_i\partial h_{1i}}}& {\displaystyle \frac{{\partial }^{2}L}{\partial h_{1i}\partial h_{1i}}}& {\displaystyle \frac{{\partial }^{2}L}{\partial h_{1i}\partial h_{2i}}}\\ {\displaystyle \frac{{\partial }^{2}L}{\partial {\lambda }_i\partial h_{2i}}}& {\displaystyle \frac{{\partial }^{2}L}{\partial h_{2i}\partial h_{1i}}}& {\displaystyle \frac{{\partial }^{2}L}{\partial h_{2i}\partial h_{2i}}}\end{array}\right]\\ & =-{\left({\displaystyle \frac{{\partial }^{2}L}{\partial {\lambda }_i\partial h_{2i}}}\right)}^2{\displaystyle \frac{{\partial }^{2}L}{\partial h_{1i}\partial h_{1i}}}-{\left({\displaystyle \frac{{\partial }^{2}L}{\partial {\lambda }_i\partial h_{1i}}}\right)}^2{\displaystyle \frac{{\partial }^{2}L}{\partial h_{2i}\partial h_{2i}}}\\ & =(-h_{2i}^{-4}{m}^2{h}_{1i}^{m-2}(d^2(x_i,v_k)+H_{ki})-h_{1i}^{-4}{m}^2{h}_{2i}^{m-2}(d^2(x_i,v_k)+H_{ki}){)}_{h_i=h_i^{\ast },{\lambda }_i={\lambda }_i^{\ast }}<0\end{array}$$

(60)

$$\begin{array}{ll}\left|{\overline{H}}_4(h_i^{\ast },{\lambda }_i^{\ast })\right|& =\det \left[\begin{array}{cccc}0& {\displaystyle \frac{{\partial }^{2}L}{\partial {\lambda }_i\partial h_{1i}}}& {\displaystyle \frac{{\partial }^{2}L}{\partial {\lambda }_i\partial h_{2i}}}& {\displaystyle \frac{{\partial }^{2}L}{\partial {\lambda }_i\partial h_{3i}}}\\ {\displaystyle \frac{{\partial }^{2}L}{\partial {\lambda }_i\partial h_{1i}}}& {\displaystyle \frac{{\partial }^{2}L}{\partial h_{1i}\partial h_{1i}}}& {\displaystyle \frac{{\partial }^{2}L}{\partial h_{1i}\partial h_{2i}}}& {\displaystyle \frac{{\partial }^{2}L}{\partial h_{1i}\partial h_{3i}}}\\ {\displaystyle \frac{{\partial }^{2}L}{\partial {\lambda }_i\partial h_{2i}}}& {\displaystyle \frac{{\partial }^{2}L}{\partial h_{2i}\partial h_{1i}}}& {\displaystyle \frac{{\partial }^{2}L}{\partial h_{2i}\partial h_{2i}}}& {\displaystyle \frac{{\partial }^{2}L}{\partial h_{2i}\partial h_{3i}}}\\ {\displaystyle \frac{{\partial }^{2}L}{\partial {\lambda }_i\partial h_{3i}}}& {\displaystyle \frac{{\partial }^{2}L}{\partial h_{3i}\partial h_{1i}}}& {\displaystyle \frac{{\partial }^{2}L}{\partial h_{3i}\partial h_{2i}}}& {\displaystyle \frac{{\partial }^{2}L}{\partial h_{3i}\partial h_{3i}}}\end{array}\right]\\ & =-{{\displaystyle {\sum }_{k=1}^3\left(\frac{{\partial }^{2}L}{\partial {\lambda }_i\partial h_{ki}}\right)}}^2\cdot {\displaystyle {\prod }_{k_1=1,k_1\ne k}^3\frac{{\partial }^{2}L}{\partial h_{k_{1}i}\partial h_{k_{1}i}}}\\ & =(-(m(m-1){)}^2{\displaystyle {\sum }_{k=1}^2{h}_{ki}^{-4}}{\displaystyle {\prod }_{k_1=1,k_1\ne k}^3{h}_{k_{1}i}^{m-2}(d^2(x_i,v_k)+H_{ki})}{)}_{h_i=h_i^{\ast },{\lambda }_i={\lambda }_i^{\ast }}<0\end{array}$$

(61)

$$\left|{\overline{H}}_{c+1}(H_i^{\ast },{\lambda }_i^{\ast })\right|=-({\displaystyle \sum _{k=1}^c{\left(\frac{{\partial }^{2}L}{\partial {\lambda }_i\partial h_{ki}}\right)}^2{\displaystyle \prod _{k_1=1,k_1\ne k}^c\frac{{\partial }^{2}L}{\partial h_{k_{1}i}\partial h_{k_{1}i}}}}{)}_{h_i=h_i^{\ast },{\lambda }_i={\lambda }_i^{\ast }}<0$$

(62)

According to the following bordered Hessian matrix theorem (local sufficient conditions), $h_{ki}^{\ast }={\displaystyle \frac{{\displaystyle {\sum }_{l=1}^c{(d^2(x_i,v_l)+H_{li})}^{\frac{1}{m+1}}}}{{(d^2(x_i,v_k)+H_{ki})}^{\frac{1}{m+1}}}}$ is the local minimum point in the condition of $1=1/{\displaystyle {\sum }_{k=1}^{c}1/h_{ki}},\forall i$. □

Proposition 2.

Given $h={[h_{ki}]}_{c\times n}$, let $F(V)=J_{HLICM}(H,V,\lambda )$ if and only if Equation (40) is solved for $v_k^{\ast }$. $V^{\ast }$ is a minima of the function $F(V)$.

Proof. 

Given $h_{ij}$, if it is assumed that $v_j^{\ast }$ is a minima of the function $F(V)$, then

$${\displaystyle \frac{\partial F(V^{\ast })}{\partial v_k}}=(-2){\displaystyle {\sum }_{i=1}^n{h}_{ki}^m[(x_i-v_k)+{\displaystyle {\sum }_{j\in N_i,j\ne i}\frac{1}{d_{ji}+1}{(h_{kj}/(h_{kj}-1))}^m(x_j-v_k)]}=0}$$

(63)

$$v_k={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n{h}_{ki}^m(x_i+{\displaystyle {\sum }_{j\in N_i,j\ne i}\frac{1}{d_{ji}+1}{(h_{kj}/(h_{kj}-1))}^m{x}_j)}}}{{\displaystyle {\sum }_{i=1}^n{h}_{ki}^m(1+{\displaystyle {\sum }_{j\in N_i,j\ne i}\frac{1}{d_{ji}+1}{(h_{kj}/(h_{kj}-1))}^m)}}}}$$

(64)

Finally, we obtain the Hessian matrix of function $F(V)$ in $V=V^{\ast }$

$${\displaystyle \frac{\partial }{\partial v_l}}({\displaystyle \frac{\partial F(v^{\ast })}{\partial v_k}})=\left\{\begin{array}{cc}0,\hfill & \hfill l\ne k\\ 2{\sum }_{i=1}^n{h}_{ki}^m\left[1+{\displaystyle {\sum }_{j\in N_i,j\ne i}{(d_{ji}+1)}^{-1}{\left(h_{kj}/(h_{kj}-1)\right)}^m}\right],\hfill & \hfill l=k\end{array}\right.$$

(65)

According to Equation (65), all diagonal elements of this Hessian matrix are greater than 0 and all off-diagonal elements are 0. Therefore, this Hessian matrix is positive definite. □

Through the previous deduction, this paper can conclude that the objective function of the HLICM algorithm decreases accordingly. In addition, the proof of the proposed algorithms with continuity and compactness constraints is similar to the proof of the FCM algorithm [54,55]. Overall, according to Zangwill’s theorem, the proposed HLICM algorithm is locally convergent. Since the IHLICM and KWHLICM algorithms are improved algorithms of HLICM and their derivation principles are similar to those of the HLICM algorithm, after proving that the HLICM algorithm is locally convergent, we can also show that the two improved algorithms proposed in this paper are also locally convergent, which can be proved in the same way as above.

4. Experimental Results and Analysis

To verify the effectiveness of the proposed algorithm, we selected grayscale images and color natural images for segmentation testing and compared the test results of the proposed algorithm to the following algorithms: ARFCM [57], FLICMLNLI [58], FCM-VMF [59], DSFCMN [60], KWFLICM [23], PFLSCM [61], FCM-SICM [62], FSC-LNML [63], and FLICM [21]. In addition, Gaussian noise, Rician noise, speckle noise, and salt and pepper noise were added to these images and the segmentation results of these noisy images were manually observed for evaluation. Therefore, we used many evaluation metrics, such as accuracy (Acc), sensitivity (Sen), the Jaccard coefficient, segmentation accuracy (SA), the Kappa coefficient (Kappa), mean intersection over union (mIoU), peak signal noise ratio (PSNR), and the Dice similarity coefficient (DICE) to objectively evaluate the segmentation results. During the testing process, the local neighborhood window size of the HLICM algorithm was 3 × 3, with a fuzzy weighting exponent of 2.0. The local neighborhood window size of the IHLICM and KWHLICM algorithms was 3 × 3, with a range of fuzzy weighting exponents of [5, 10]. A hardware platform with CPU-i7, 2.70 GHz, and 16.0 GB RAM was tested on the Matlab r2021b software platform in the Windows 11 environment. For the comparison algorithms, the local neighborhood window sizes of the ARFCM, FLICMLNLI, FCM_VMF, DSFCM_N, KWFLICM, PFLSCM, FCM_SICM, FSC_LNML, and FLICM algorithms were 5 × 5, 3 × 3, 4 × 4, 3 × 3, 3 × 3, 3 × 3, 7 × 7, 7 × 7, and 3 × 3, respectively. The number of clusters in fuzzy clustering algorithm for image segmentation is determined according to the Matlab-based clustering validity index toolbox [64]. For the convenience of description, this section uses $\mu$ to denote the mean value of Gaussian noise, ${\sigma }_n^2$ represents the normalized variance of Gaussian noise, $p$ denotes the intensity level of salt and pepper noise, and $\sigma$ denotes the standard deviation of Rician noise. For example, Gaussian noise with a mean value of 0 and normalized variance of 0.1 is represented as GN(0, 0.1), salt and pepper noise with an intensity of 30% is represented as SPN(0.3), speckle noise with a normalized variance of 0.2 is represented as SN(0.2), and Rician noise with a standard deviation of 80 is represented as RN(80).

4.1. Evaluation Indicators

4.1.1. Peak Signal-to-Noise Ratio (PSNR) [65]

Peak signal-to-noise ratio (PSNR) is an indicator used to measure the quality of an image or video. It is usually used to compare the difference between the original image and a compressed or processed image. PSNR is calculated using the following formula:

$$PSNR=10\cdot {\log }_{10}\left({\displaystyle \frac{MA{X}^2}{MSE}}\right)$$

(66)

$$MSE={\displaystyle \frac{1}{M_1{N}_1}}{\displaystyle {\sum }_{x=1}^{M_1}{\displaystyle {\sum }_{y=1}^{N_1}{‖I(x,y)-O(x,y)‖}^2}}$$

(67)

where $MAX$ represents the maximum possible value of image pixels and $MSE$ is the mean squared error (MSE), which represents the square of the average difference in pixel values between the original image and the processed image. The higher the PSNR value, the better the image quality, as this means that the difference between the original image and the processed image is smaller.

4.1.2. Segmentation Accuracy (SA) [57]

Image segmentation is the process of dividing an image into different regions or objects, and segmentation accuracy is the evaluation of the degree of similarity between the segmentation results and the ground truth.

$$SA={\displaystyle {\sum }_{j=1}^c\frac{|A_j\cap C_j|}{{{\displaystyle \sum }}_{l=1}^c\left|C_l\right|}}$$

(68)

4.1.3. Mean Intersection over Union (mIoU) [66]

The average intersection and merger ratio is an indicator used to evaluate the quality of image segmentation, which is the average of IoU (intersection and merger ratio). The mIoU index integrates the degree of similarity between the segmentation results and the ground truth across all categories and is a commonly used indicator for segmentation accuracy.

$$mIoU={\displaystyle \frac{1}{c}}{\displaystyle {\sum }_{i=1}^c\frac{|A_i\cap C_i|}{|A_i\cup C_i|}}$$

(69)

In image segmentation tasks, the higher the mIoU value, the closer the segmentation result is to the ground truth, which means that the algorithm has better segmentation performance in different categories.

4.1.4. Accuracy (Acc), Sensitivity (Sen), the Jaccard Coefficient, and the DICE [67]

TP denotes true positive, TN denotes true negative, FP denotes false positive, and FN denotes false negative. The accuracy value ranges from 0 to 1. The closer the value is to 1, the higher the accuracy of classification or segmentation.

Accuracy

$$Acc={\displaystyle \frac{TP+TN}{TP+TN+FN+FP}}$$

(70)

Sensitivity

$$Sen={\displaystyle \frac{TP}{TN+TP}}$$

(71)

Jaccard coefficient

$$Jaccard={\displaystyle \frac{TP}{TP+FP+FN}}$$

(72)

Dice similarity coefficient (DICE)

$$DICE={\displaystyle \frac{2TP}{2TP+FP+FN}}$$

(73)

Accuracy is the ratio of the number of correctly classified or segmented samples to the total number of samples in a classification or segmentation task. Sensitivity is the proportion of true cases correctly identified as positive cases, also known as the true case rate. The Jaccard coefficient, also known as Intersection over Union (IoU), is used to evaluate the similarity between two sets. In segmentation tasks, the Jaccard coefficient index is often used to evaluate the degree of overlap between the segmentation results and the ground truth. The Dice similarity coefficient (DICE) is a commonly used indicator to evaluate the quality of image segmentation and is also widely used to assess the performance of medical image segmentation tasks.

4.1.5. The Kappa Coefficient [67]

The Kappa coefficient considers the consistency between the classification or segmentation results and the ground truth, avoiding the limitation of relying solely on superficial indicators such as accuracy to evaluate performance.

$$Kappa={\displaystyle \frac{p_o-p_e}{1-p_e}}$$

(74)

4.2. Test and Analysis of Algorithm Robustness

4.2.1. Synthetic Images

To verify the robustness of the proposed algorithm, four synthetic images shown in Figure 3 were selected and various types of noise were added to these synthetic images for testing the proposed algorithm and all the comparison algorithms.

RN(80) was added to Figure 3a, GN(0, 0.1) was added to Figure 3b, SPN(0.2) was added to Figure 3c, and SN(0.2) was added to Figure 3d. These images with noise were segmented using various fuzzy clustering algorithms. The segmentation results are shown in Figure 4.

Table 1. Evaluation indexes of different algorithms for the noisy synthetic images.

Image	Algorithm	Acc	Sen	Jaccard	PSNR	SA	Kappa	mIoU	DICE
Figure 3a + RN(80)	ARFCM	0.6673	0.501	0.3342	8.5511	0.501	0.2973	11.1393	0.501
	FLICMLNLI	0.9808	0.9712	0.944	21.437	0.9712	0.9539	31.4661	0.9712
	FCM_VMF	0.5678	0.3516	0.2133	7.3093	0.3516	0.0346	7.111	0.3516
	DSFCM_N	0.6787	0.5181	0.3496	9.2906	0.5181	0.3578	11.6542	0.5181
	KWFLICM	0.9937	0.9906	0.9813	26.2168	0.9906	0.9849	32.7095	0.9906
	PFLSCM	0.988	0.9819	0.9645	23.3581	0.9819	0.9713	32.1502	0.9819
	FCM_SICM	0.9793	0.9689	0.9396	21.2332	0.9689	0.9511	31.321	0.9689
	FSC_LNML	0.9803	0.9704	0.9426	21.2592	0.9704	0.953	31.4195	0.9704
	FLICM	0.9928	0.9892	0.9787	25.6239	0.9892	0.9828	32.6226	0.9892
	HLICM	0.5902	0.3854	0.2387	7.5407	0.3854	0.1477	7.9558	0.3854
	IHLICM	0.9914	0.9871	0.9746	25.0162	0.9871	0.9795	32.4868	0.9871
	KWHLICM	0.9946	0.992	0.9841	26.9084	0.992	0.9872	32.8021	0.992
Figure 3b + GN(0, 0.1)	ARFCM	0.7701	0.7221	0.4223	12.4469	0.5043	0.3984	11.239	0.5939
	FLICMLNLI	0.8876	0.9746	0.6688	18.9598	0.6806	0.5796	17.196	0.8015
	FCM_VMF	0.7177	0.6096	0.3345	4.9834	0.4257	0.2299	9.0141	0.5013
	DSFCM_N	0.6667	0.5	0.2588	10.2282	0.3492	0.2582	7.0511	0.4112
	KWFLICM	0.8966	0.9938	0.6911	22.6226	0.694	0.5962	17.7149	0.8173
	PFLSCM	0.8963	0.9933	0.6905	22.6854	0.6937	0.5958	17.7012	0.8169
	FCM_SICM	0.8954	0.9912	0.688	20.8113	0.6922	0.5895	17.6434	0.8152
	FSC_LNML	0.8926	0.9852	0.681	21.641	0.688	0.5882	17.481	0.8102
	FLICM	0.8938	0.9878	0.6841	21.0176	0.6899	0.5885	17.552	0.8124
	HLICM	0.4431	0.0199	0.0083	2.467	0.0139	0.3469	0.2333	0.0164
	IHLICM	0.9923	0.9885	0.9772	22.8343	0.9885	0.9827	32.5731	0.9885
	KWHLICM	0.999	0.9985	0.997	29.6905	0.9985	0.9977	33.2318	0.9985
Figure 3c + SPN(0.2)	ARFCM	0.7889	0.6834	0.519	10.4928	0.6834	0.5153	17.3007	0.6834
	FLICMLNLI	0.7883	0.6825	0.518	5.2762	0.6825	0.5176	17.2669	0.6825
	FCM_VMF	0.7418	0.6127	0.4417	9.0293	0.6127	0.4084	14.7226	0.6127
	DSFCM_N	0.9945	0.9918	0.9836	26.8906	0.9918	0.9876	32.7879	0.9918
	KWFLICM	0.9954	0.9931	0.9862	23.8601	0.9931	0.9896	32.8736	0.9931
	PFLSCM	0.9922	0.9884	0.9770	21.768	0.9884	0.9825	32.5664	0.9884
	FCM_SICM	0.9896	0.9843	0.9691	23.8148	0.9843	0.9765	32.3046	0.9843
	FSC_LNML	0.9915	0.9872	0.9747	24.9698	0.9872	0.9808	32.49	0.9872
	FLICM	0.9365	0.9047	0.8260	16.1069	0.9047	0.8575	27.5323	0.9047
	HLICM	0.3421	0.0132	0.0066	2.2015	0.0132	0.4843	0.2211	0.0132
	IHLICM	0.8928	0.9938	0.6799	22.0172	0.6828	0.5844	17.2772	0.8094
	KWHLICM	0.9974	0.9962	0.9923	28.3798	0.9962	0.9942	33.0781	0.9962
Figure 3d + SN(0.2)	ARFCM	0.8624	0.7248	0.5684	12.4959	0.7248	0.5005	14.2102	0.7248
	FLICMLNLI	0.8724	0.7448	0.5934	9.2425	0.7448	0.6128	14.834	0.7448
	FCM_VMF	0.8692	0.7384	0.5853	9.5715	0.7384	0.5829	14.6329	0.7384
	DSFCM_N	0.9288	0.8576	0.7507	16.5031	0.8576	0.7746	18.7673	0.8576
	KWFLICM	0.9474	0.8948	0.8097	19.1373	0.8948	0.8387	20.2421	0.8948
	PFLSCM	0.9098	0.8197	0.6945	16.2311	0.8197	0.7056	17.3613	0.8197
	FCM_SICM	0.8865	0.7729	0.6299	15.8017	0.7729	0.66	15.7477	0.7729
	FSC_LNML	0.9573	0.9145	0.8425	22.6646	0.9145	0.8664	21.0621	0.9145
	FLICM	0.9233	0.8466	0.7339	18.4919	0.8466	0.7704	18.3485	0.8466
	HLICM	0.5539	0.1077	0.0569	3.504	0.1077	0.138	1.4233	0.1077
	IHLICM	0.9368	0.8736	0.7756	18.0537	0.8736	0.7879	19.3901	0.8736
	KWHLICM	0.9658	0.9316	0.872	20.9993	0.9316	0.8926	21.8009	0.9316

N(80) is Rician noise with standard deviation of 80, SPN(0.3) is salt and pepper noise with the intensity of 30%, GN(0, 0.1) is Gaussian noise with mean value of 0 and the normalized variance of 0.1; and SN(0.2) is speckle noise with the normalized variance of 0.1.

gives the evaluation metrics corresponding to the various algorithms.

From Figure 4, it can be seen that when the image contains high noise, the segmentation results of the FCM_VMF and FLICM algorithms are poor, indicating that they do not have good noise resistance. The ARFCM algorithm has good noise resistance, but over-segmentation may occur. The PFLSCM and HLICM algorithms cannot effectively suppress the noise, while the FLICMLNLI algorithm can suppress the noise, but it can also cause over-segmentation of the image. The FCM_SICM and FSC_LNML algorithms can suppress noise, but their results have uneven edges. The DSFCM_N algorithm is robust to salt and pepper noise but sensitive to other types of noises. The KWFLICM and IHLICM algorithm can effectively suppress a large amount of noise, but the segmentation results are poor. The KWHLICM algorithm is more effective in synthetic images and is more suitable than the other comparison algorithms. From Table 1, it can be seen that the evaluation indexes of the KWHLICM algorithm are all higher than the other algorithms; therefore, the KWHLICM algorithm is universal best choice for segmenting images contaminated by noise.

4.2.2. Natural Images

To further test the performance of the algorithm, four natural images from BSDS500 [68] and VOC2010 [69] were selected for segmentation testing.

GN(0, 0.1) was added to Figure 5a,b and SPN(0.3) was added to Figure 5c,d. These images with noise were segmented using various fuzzy clustering algorithms. The segmentation results are shown in Figure 6.

Table 2. Evaluation indexes of different algorithms for natural images with noise.

Image	Algorithm	Acc	Sen	Jaccard	PSNR	SA	Kappa	mIoU	DICE
35010 + GN(0, 0.1)	ARFCM	0.8071	0.7107	0.5512	9.7139	0.7107	0.5357	18.3738	0.7107
	FLICMLNLI	0.879	0.8185	0.6928	13.1209	0.8185	0.7268	23.0924	0.8185
	FCM_VMF	0.6083	0.4125	0.2599	3.886	0.4125	0.0399	8.6619	0.4125
	DSFCM_N	0.9416	0.9123	0.8388	16.4171	0.9123	0.8656	27.9603	0.9123
	KWFLICM	0.9217	0.8826	0.7898	14.8158	0.8826	0.8213	26.3282	0.8826
	PFLSCM	0.9359	0.9038	0.8245	16.2609	0.9038	0.8525	27.4819	0.9038
	FCM_SICM	0.8672	0.8007	0.6677	12.8065	0.8007	0.6992	22.2559	0.8007
	FSC_LNML	0.9054	0.8582	0.7516	13.986	0.8582	0.7831	25.052	0.8582
	FLICM	0.8984	0.8475	0.7354	14.1298	0.8475	0.7709	24.5143	0.8475
	HLICM	0.6619	0.4928	0.327	8.8141	0.4928	0.228	10.8985	0.4928
	IHLICM	0.9333	0.8999	0.818	15.9346	0.8999	0.8484	27.2667	0.8999
	KWHLICM	0.9426	0.914	0.8416	16.6862	0.914	0.8694	28.0524	0.914
2007_000738 + GN(0, 0.1)	ARFCM	0.8456	0.7684	0.624	11.7094	0.7684	0.6417	20.7985	0.7684
	FLICMLNLI	0.9496	0.9244	0.8594	16.5232	0.9244	0.8858	28.6478	0.9244
	FCM_VMF	0.5614	0.3421	0.2064	3.7281	0.3421	0.0047	6.8784	0.3421
	DSFCM_N	0.8134	0.7201	0.5626	11.5132	0.7201	0.5601	18.7527	0.7201
	KWFLICM	0.9524	0.9286	0.8667	16.8869	0.9286	0.8917	28.8896	0.9286
	PFLSCM	0.9539	0.9308	0.8706	17.7253	0.9308	0.8954	29.0215	0.9308
	FCM_SICM	0.9412	0.9118	0.8379	16.0725	0.9118	0.866	27.929	0.9118
	FSC_LNML	0.9518	0.9277	0.8652	16.734	0.9277	0.8903	28.839	0.9277
	FLICM	0.9487	0.923	0.857	16.7076	0.923	0.8834	28.566	0.923
	HLICM	0.713	0.5695	0.3981	9.4704	0.5695	0.3391	13.2694	0.5695
	IHLICM	0.9408	0.9112	0.8368	16.4846	0.9112	0.8668	27.8941	0.9112
	KWHLICM	0.9592	0.9388	0.8847	18.0792	0.9388	0.9078	29.4916	0.9388
2009_003974 + SPN(0.3)	ARFCM	0.9331	0.9331	0.8746	11.7456	0.9331	0.8473	43.7294	0.9331
	FLICMLNLI	0.7359	0.7359	0.5822	5.7823	0.7359	0.2075	29.1076	0.7359
	FCM_VMF	0.8383	0.8383	0.7216	7.9127	0.8383	0.6381	36.0802	0.8383
	DSFCM_N	0.9716	0.9716	0.9448	15.4732	0.9716	0.9355	47.2424	0.9716
	KWFLICM	0.9809	0.9809	0.9626	17.1941	0.9809	0.9559	48.1277	0.9809
	PFLSCM	0.9523	0.9523	0.9089	13.2147	0.9523	0.8908	45.447	0.9523
	FCM_SICM	0.9654	0.9654	0.9331	14.6104	0.9654	0.9216	46.6565	0.9654
	FSC_LNML	0.9791	0.9791	0.959	16.7933	0.9791	0.9513	47.9504	0.9791
	FLICM	0.9627	0.9627	0.928	14.2793	0.9627	0.9156	46.4012	0.9627
	HLICM	0.9587	0.9587	0.9206	13.8372	0.9587	0.9041	46.0309	0.9587
	IHLICM	0.9779	0.9779	0.9567	16.5499	0.9779	0.9486	47.8348	0.9779
	KWHLICM	0.9824	0.9824	0.9654	17.539	0.9824	0.9591	48.2681	0.9824
2007_003330 + SPN(0.3)	ARFCM	0.7441	0.6161	0.4452	6.8606	0.6161	0.2414	14.8401	0.6161
	FLICMLNLI	0.4389	0.1583	0.086	2.6778	0.1583	0.0797	2.8659	0.1583
	FCM_VMF	0.5663	0.3495	0.2117	7.8681	0.3495	0.136	7.0578	0.3495
	DSFCM_N	0.8804	0.8206	0.6957	13.4025	0.8206	0.7106	23.1903	0.8206
	KWFLICM	0.8942	0.8313	0.7117	13.4062	0.8313	0.7205	24.0562	0.8313
	PFLSCM	0.7245	0.5868	0.4152	9.0832	0.5868	0.2914	13.8393	0.5868
	FCM_SICM	0.8343	0.7515	0.6019	11.92	0.7515	0.6019	20.0634	0.7515
	FSC_LNML	0.8334	0.75	0.6001	11.6917	0.75	0.6072	20.0017	0.75
	FLICM	0.6228	0.4342	0.2773	8.2936	0.4342	0.2558	9.2436	0.4342
	HLICM	0.5408	0.3112	0.1843	7.0116	0.3112	0.1161	6.1429	0.3112
	IHLICM	0.8913	0.8369	0.7196	13.4113	0.8369	0.7212	23.9863	0.8369
	KWHLICM	0.91	0.865	0.7622	14.371	0.865	0.7754	25.4052	0.865

gives the evaluation metrics corresponding to various algorithms.

From Figure 6, it can be seen that the ARFCM and KWFLICM algorithms can effectively suppress noise, but their segmentation results are not satisfactory. The FCM_VMF, DSFCM_N, and PFLSCM algorithms cannot effectively suppress noise. The FLICMLNLI algorithm can suppress most of the noise, but it is prone to over-segmentation. The FCM_SICM and FSC_LNML algorithms can still suppress noise, but they are prone to losing detailed information in the image during segmentation. The HLICM and FLICM algorithms can suppress some noise, but their segmentation results are also unsatisfactory. The KWHLICM algorithm can suppress a large amount of noise, indicating its robustness, and its segmentation results are the best among all the algorithms. Therefore, in summary, the KWHLICM algorithm has good application prospects for natural image segmentation.

4.2.3. Remote Sensing Images

To illustrate the effectiveness of the algorithm, four remote sensing images from the UC Merced Land Use dataset [70] were selected for segmentation testing.

SN(0.2) was added to Figure 7a,b, SPN(0.4) was added to Figure 7c,d, and these noisy images were segmented using various fuzzy clustering algorithms. The segmentation results are shown in Figure 8.

Table 3. Evaluation indexes of different algorithms for noisy remote sensing images.

Image	Algorithm	Acc	Sen	Jaccard	PSNR	SA	Kappa	mIoU	DICE
buildings69 + SN(0.2)	ARFCM	0.6864	0.5295	0.3601	7.7277	0.5295	0.2995	12.0035	0.5295
	FLICMLNLI	0.7429	0.6143	0.4433	9.5792	0.6143	0.4146	14.7778	0.6143
	FCM_VMF	0.5672	0.3508	0.2127	7.8796	0.3508	0.023	7.091	0.3508
	DSFCM_N	0.8233	0.7349	0.5809	11.603	0.7349	0.6034	19.3644	0.7349
	KWFLICM	0.8097	0.7146	0.5559	11.4341	0.7146	0.5696	18.5312	0.7146
	PFLSCM	0.7758	0.6637	0.4966	9.9682	0.6637	0.4971	16.5544	0.6637
	FCM_SICM	0.782	0.673	0.5072	9.949	0.673	0.5126	16.9066	0.673
	FSC_LNML	0.7873	0.6809	0.5162	10.9229	0.6809	0.5184	17.2059	0.6809
	FLICM	0.7943	0.6914	0.5284	11.0962	0.6914	0.5334	17.6131	0.6914
	HLICM	0.527	0.2905	0.1699	4.6904	0.2905	0.0755	5.6633	0.2905
	IHLICM	0.6965	0.5448	0.3743	9.3695	0.5448	0.3168	12.4779	0.5448
	KWHLICM	0.8267	0.7401	0.5874	11.6808	0.7401	0.6094	19.5808	0.7401
tenniscourt95 + SN(0.2)	ARFCM	0.6902	0.5353	0.3655	9.1703	0.5353	0.3213	12.182	0.5353
	FLICMLNLI	0.9519	0.9278	0.8653	17.0663	0.9278	0.8577	28.8447	0.9278
	FCM_VMF	0.548	0.322	0.1919	7.2992	0.322	0.1678	6.3963	0.322
	DSFCM_N	0.5571	0.3357	0.2017	7.9079	0.3357	0.2106	6.723	0.3357
	KWFLICM	0.9542	0.9314	0.8715	17.5641	0.9314	0.8575	29.0516	0.9314
	PFLSCM	0.7513	0.6269	0.4565	10.3653	0.6269	0.4455	15.2183	0.6269
	FCM_SICM	0.9607	0.941	0.8886	18.2487	0.941	0.8771	29.6188	0.941
	FSC_LNML	0.9693	0.954	0.9121	19.3537	0.954	0.9056	30.4021	0.954
	FLICM	0.5647	0.347	0.2099	8.0035	0.347	0.1758	6.9974	0.347
	HLICM	0.5433	0.3149	0.1869	7.5337	0.3149	0.1772	6.229	0.3149
	IHLICM	0.8934	0.8401	0.7242	13.8614	0.8401	0.6497	24.1415	0.8401
	KWHLICM	0.9707	0.956	0.9157	19.5066	0.956	0.9097	30.5232	0.956
storagetanks42 + SPN(0.4)	ARFCM	0.7013	0.5519	0.3811	7.1462	0.5519	0.178	12.7038	0.5519
	FLICMLNLI	0.5905	0.3858	0.239	3.0518	0.3858	0.2223	7.9657	0.3858
	FCM_VMF	0.7182	0.5774	0.4058	9.6883	0.5774	0.267	13.528	0.5774
	DSFCM_N	0.8015	0.7023	0.5412	10.6572	0.7023	0.5443	18.0396	0.7023
	KWFLICM	0.9145	0.8718	0.7727	12.0773	0.8718	0.7832	25.7568	0.8718
	PFLSCM	0.7302	0.5953	0.4238	8.0759	0.5953	0.4086	14.1271	0.5953
	FCM_SICM	0.6502	0.4753	0.3117	8.0735	0.4753	0.2375	10.3897	0.4753
	FSC_LNML	0.7775	0.6662	0.4995	10.1503	0.6662	0.4959	16.6491	0.6662
	FLICM	0.4582	0.1873	0.1033	6.2131	0.1873	0.1096	3.4439	0.1873
	HLICM	0.4845	0.2268	0.1279	6.8809	0.2268	0.1515	4.2636	0.2268
	IHLICM	0.8344	0.7516	0.602	11.7069	0.7516	0.5862	20.0678	0.7516
	KWHLICM	0.9187	0.878	0.7826	12.2518	0.878	0.7958	26.0855	0.878
runway22 + SPN(0.4)	ARFCM	0.6441	0.4661	0.3039	7.6938	0.4661	0.1317	10.1287	0.4661
	FLICMLNLI	0.4776	0.2165	0.1214	3.4518	0.2165	0.0048	4.0455	0.2165
	FCM_VMF	0.4977	0.2466	0.1406	3.6175	0.2466	0.0388	4.688	0.2466
	DSFCM_N	0.9033	0.8549	0.7466	14.0114	0.8549	0.7759	24.8875	0.8549
	KWFLICM	0.934	0.901	0.8198	13.6366	0.901	0.8476	27.3262	0.901
	PFLSCM	0.7454	0.6181	0.4473	8.3616	0.6181	0.4375	14.9095	0.6181
	FCM_SICM	0.8067	0.71	0.5504	8.3693	0.71	0.5678	18.3477	0.71
	FSC_LNML	0.8454	0.768	0.6234	11.2611	0.768	0.6391	20.7808	0.768
	FLICM	0.4353	0.1529	0.0828	3.8178	0.1529	0.2488	2.7594	0.1529
	HLICM	0.6178	0.4267	0.2712	6.4005	0.4267	0.1023	9.0416	0.4267
	IHLICM	0.8736	0.8104	0.6813	13.2393	0.8104	0.6918	22.7091	0.8104
	KWHLICM	0.9366	0.905	0.8264	14.5244	0.905	0.8539	27.5469	0.905

gives the evaluation metrics corresponding to the various algorithms.

From Figure 8, it can be seen that the ARFCM, KWFLICM, and FLICM algorithms can effectively suppress the noise, but their segmentation results are poor. The FCM_VMF, DSFCM_N, PFLSCM, and HLICM algorithms cannot effectively suppress the noise. The FLICMLNLI algorithm cannot effectively segment the image while suppressing the noise. The FCM_SICM and FSC_LNML algorithms can still suppress noise, but they are prone to losing detailed information in the image during segmentation. The KWHLICM algorithm can suppress a large amount of noise, indicating its robustness, and the segmentation results are the best among all the algorithms. In addition, from Table 3, for all the evaluation indexes, the proposed KWHLICM algorithm’s values are the highest among all the algorithms. In summary, the KWHLICM algorithm has good application prospects for remote sensing image segmentation.

4.2.4. Medical Images

To illustrate the effectiveness of the algorithm, four MR images were selected from the Brain Tumor MRI Dataset [71] for segmentation testing.

RN(80) was added to Figure 9a, RN(90) was added to Figure 9b, and GN(0, 0.1) was added to Figure 9c,d. These images with noise were segmented using various fuzzy clustering algorithms. The segmentation results are shown in Figure 10.

Table 4. Evaluation indexes of different algorithms for MR images with noise.

Image	Algorithm	Acc	Sen	Jaccard	PSNR	SA	Kappa	mIoU	DICE
Tr-me_ 0180 + RN(80)	ARFCM	0.8089	0.7134	0.5544	11.0986	0.7134	0.5777	18.481	0.7134
	FLICMLNLI	0.869	0.8036	0.6716	12.9467	0.8036	0.6929	22.3873	0.8036
	FCM_VMF	0.5256	0.2885	0.1685	4.8038	0.2885	0.0528	5.618	0.2885
	DSFCM_N	0.9296	0.8944	0.809	15.7027	0.8944	0.8343	26.9663	0.8944
	KWFLICM	0.9368	0.9052	0.8269	16.0956	0.9052	0.8487	27.5619	0.9052
	PFLSCM	0.8279	0.7419	0.5897	11.8537	0.7419	0.6183	19.6554	0.7419
	FCM_SICM	0.9275	0.8912	0.8037	15.5499	0.8912	0.8261	26.7907	0.8912
	FSC_LNML	0.9353	0.9029	0.823	16.0768	0.9029	0.8438	27.4346	0.9029
	FLICM	0.519	0.2785	0.1618	7.1822	0.2785	0.0242	5.3927	0.2785
	HLICM	0.6549	0.4823	0.3178	8.6043	0.4823	0.3064	10.5927	0.4823
	IHLICM	0.9388	0.9083	0.8319	16.3982	0.9083	0.8481	27.731	0.9083
	KWHLICM	0.9494	0.9241	0.8589	17.1511	0.9241	0.8782	28.6309	0.9241
Te-no_ 0013 + RN(90)	ARFCM	0.7977	0.6965	0.5343	10.3466	0.6965	0.4949	17.811	0.6965
	FLICMLNLI	0.7946	0.6919	0.5289	10.9236	0.6919	0.4889	17.6306	0.6919
	FCM_VMF	0.4321	0.1482	0.08	3.3136	0.1482	0.0272	2.6673	0.1482
	DSFCM_N	0.6493	0.4739	0.3106	8.5867	0.4739	0.2995	10.3524	0.4739
	KWFLICM	0.859	0.7884	0.6508	12.6522	0.7884	0.6064	21.6926	0.7884
	PFLSCM	0.7539	0.6308	0.4607	10.1671	0.6308	0.4521	15.3577	0.6308
	FCM_SICM	0.8704	0.8055	0.6744	13.0409	0.8055	0.6793	22.4799	0.8055
	FSC_LNML	0.838	0.757	0.609	12.0918	0.757	0.6167	20.2996	0.757
	FLICM	0.4315	0.1472	0.0795	5.9744	0.1472	0.4162	2.6484	0.1472
	HLICM	0.621	0.4316	0.2752	7.9845	0.4316	0.2397	9.1717	0.4316
	IHLICM	0.8536	0.7804	0.6399	12.6573	0.7804	0.6222	21.3286	0.7804
	KWHLICM	0.9009	0.8514	0.7412	14.2151	0.8514	0.757	24.7082	0.8514
37 no + GN(0, 0.1)	ARFCM	0.8391	0.7586	0.6111	10.7429	0.7586	0.5924	20.3684	0.7586
	FLICMLNLI	0.9226	0.8838	0.7918	14.3431	0.8838	0.7997	26.395	0.8838
	FCM_VMF	0.5679	0.3519	0.2135	5.4027	0.3519	0.1204	7.1161	0.3519
	DSFCM_N	0.809	0.7135	0.5546	10.6189	0.7135	0.5205	18.4869	0.7135
	KWFLICM	0.9356	0.9035	0.8239	15.5901	0.9035	0.8283	27.4647	0.9035
	PFLSCM	0.9456	0.9184	0.8491	16.813	0.9184	0.8565	28.3044	0.9184
	FCM_SICM	0.9253	0.888	0.7985	15.0557	0.888	0.8038	26.6163	0.888
	FSC_LNML	0.9464	0.9196	0.8512	16.7574	0.9196	0.8587	28.3749	0.9196
	FLICM	0.9447	0.917	0.8467	16.7861	0.917	0.8529	28.2228	0.917
	HLICM	0.7885	0.6827	0.5183	10.6503	0.6827	0.3551	17.2767	0.6827
	IHLICM	0.9454	0.918	0.8485	16.8143	0.918	0.8562	28.2827	0.918
	KWHLICM	0.9529	0.9294	0.8681	17.4791	0.9294	0.8759	28.9369	0.9294
Tr-me_ 0235 + GN(0, 0.1)	ARFCM	0.7905	0.6858	0.5218	9.558	0.6858	0.5328	17.3946	0.6858
	FLICMLNLI	0.8931	0.8397	0.7237	13.1501	0.8397	0.7484	24.1233	0.8397
	FCM_VMF	0.5382	0.3073	0.1815	6.542	0.3073	0.0512	6.0514	0.3073
	DSFCM_N	0.929	0.8935	0.8075	15.7062	0.8935	0.8329	26.9177	0.8935
	KWFLICM	0.9375	0.9063	0.8286	16.3113	0.9063	0.8498	27.6196	0.9063
	PFLSCM	0.9483	0.9224	0.856	17.1174	0.9224	0.879	28.5339	0.9224
	FCM_SICM	0.8748	0.8123	0.6839	13.168	0.8123	0.6949	22.7958	0.8123
	FSC_LNML	0.9216	0.8824	0.7895	15.2904	0.8824	0.8134	26.3183	0.8824
	FLICM	0.9108	0.8662	0.764	14.7483	0.8662	0.7831	25.4668	0.8662
	HLICM	0.6067	0.4101	0.2579	7.9267	0.4101	0.1141	8.5969	0.4101
	IHLICM	0.9348	0.9022	0.8219	16.1389	0.9022	0.8415	27.3966	0.9022
	KWHLICM	0.9549	0.9323	0.8731	17.7196	0.9323	0.8919	29.1048	0.9323

gives the evaluation metrics corresponding to the various algorithms.

From Figure 10, it can see that the ARFCM, KWFLICM, FLICMLNLI algorithms can effectively suppress noise, but their segmentation results are not satisfactory. The FCM_VMF, PFLSCM, HLICM, and DSFCM_N algorithms cannot effectively suppress noise. The FCM_SICM and FSC_LNML algorithms can suppress noise, but often lose the detailed information of the image during the segmentation process. The KWFLICM algorithm can suppress noise while losing a lot of detail information, which is not what we want. The KWHLICM algorithm can suppress a large amount of noise, which indicates that it is robust and the segmentation result is the best among all algorithms. From Table 4, all the evaluation indexes of the proposed KWHLICM algorithm are the highest among all the algorithms. Therefore, in summary, the KWHLICM algorithm has good application prospects for MR image segmentation.

4.3. Testing and Analyzing the Effect of Noise Intensity on Algorithm Performance

Rician noise with a standard deviation of 50 to 80 was added to the medical image of Figure 9a. Noisy images were processed using 12 fuzzy clustering algorithms and the variation curves of the performance metrics of different algorithms with a standard deviation of Rician noise were obtained as shown in Figure 11. The variation of the evaluation metrics of most of the algorithms was not significant, and the FCM_VMF, HLICM and ARFCM algorithms have low evaluation metrics on medical images with Rician noise. With the increase of the standard deviation of the Rician noise, the evaluation metrics of the proposed KWHLICM algorithm have higher values than all the comparison algorithms, and therefore the KWHLICM algorithm is more stable and robust to changes in Rician noise.

Gaussian noise with different normalized variances was added to the synthetic images in Figure 3b. Segmentation tests were performed on these images using 12 algorithms and the variation curves of the evaluation metrics with the normalized variance of Gaussian noise were obtained, as shown in Figure 12. Most of the algorithms have insignificant variation of the evaluation metrics with noise, among which the HLICM, FCM_VMF, and ARFCM algorithms have lower evaluation metrics. Most of the performance metrics of the proposed KWHLICM method are better than all the comparison algorithms. In addition, the performance curve of the KWHLICM algorithm changes slowly with the normalized variance of Gaussian noise. Therefore, the proposed KWHLICM algorithm outperforms all the comparative algorithms in the presence of Gaussian noise.

Salt and pepper noise with different intensity levels was added to the natural image in Figure 5c, 12 fuzzy clustering algorithms were used to segment these noisy images, and the curves of the evaluation metrics of the different algorithms varying with the strength of salt and pepper noise were obtained, as shown in Figure 13. Overall, the metrics of all the algorithms decrease with the increase of the intensity level of salt and pepper noise. When the noise intensity is less than 0.21, the metrics of the PFLSCM algorithm are higher than other algorithms, but with the increase of noise (when the noise intensity is stronger than 0.21), all the evaluation metrics of the KWHLICM algorithm are higher than other algorithms. In addition, the metrics of the ARFCM and FCM_VCM algorithms are very low. Therefore, the proposed KWHLICM algorithm outperforms all the comparison algorithms in the presence of salt and pepper noise.

Speckle noise with different normalized variances was added to the remote sensing image in Figure 9a, 12 fuzzy clustering algorithms were used to segment these noisy images, and the curves of the evaluation metrics of the different algorithms with the intensity of the speckle noise were obtained, as shown in Figure 14. As the normalized variance of the speckle noise increases, most of the evaluation metrics of all the algorithms decrease. The evaluation index value of the DSFCM_N algorithm is higher when the noise is lower than 0.05, but as the noise increases, the KWHLICM algorithm obtains higher evaluation values than all the comparison algorithms. The evaluation index values of the HLICM and FCM_VMF algorithms are generally low. Overall, the segmentation performance of the proposed KWHLICM algorithm is minimally affected by the variation of the noise, and it is more robust to speckle noise than all the comparison algorithms.

4.4. Analysis and Testing of Algorithm Complexity

Algorithm complexity is an important indicator for measuring algorithm performance.

Table 5. Computational complexity of different fuzzy clustering algorithms.

Algorithm	Computational Complexity
ARFCM	$O(n\times c\times w\times t)$
FLICMLNLI	$O(n\times w_2\times w_3+c\times n\times w^2\times t)$
FCM_VMF	$O(n\times w+n\times c\times t)$
DSFCM_N	$O(c\times n\times w\times t)$
KWFLICM	$O(n\times w+w\times n\times c\times t)$
PFLSCM	$O(n\times w^2+n\times c\times w^2\times t)$
FCM_SICM	$O(n\times w\times {\log }_2(n\times w)+n+n\times c\times t)$
FSC_LNML	$O(n\times w^2+n\times w+n\times c\times t)$
FLICM	$O(n\times c\times t\times w)$
HLICM	$O(n\times c\times t\times w)$
KWHLICM	$O(n\times w+w\times n\times c\times t)$
IHLICM	$O(n\times w+w\times n\times c\times t)$

gives the computational complexity of all the algorithms in this paper, where $\gamma$ is the number of pixels in image, $w$ is the size of the local neighborhood window under robust fuzzy clustering, $c$ is the number of classes, $t$ is the number of iterations for algorithm convergence, $w_1$ is the size of the local neighborhood window under Gaussian and bootstrap bilateral filtering, $w_2$ is the size of the local neighborhood window under non-local mean filtering, and $w_3$ is the size of the local search window under non-mean filtering.

As shown in Table 5, the computational complexities of the ARFCM, FLICMLNLI, and PFLSCM algorithms are significantly higher than other compared algorithms. To verify the computational complexity of these algorithms, we tested and analyzed them using different images. By analyzing the time cost of these algorithms for processing noisy images, we confirmed the computational complexity of the algorithms covered in this paper. RN(80), GN(0, 0.1), SPN(0.3), and SN(0.2) were added to the two synthetic images in Figure 3b,d; SPN(0.3) and GN(0, 0.3) were added to the two natural images in Figure 5b,d; SPN(0.2) and SN(0.2) were added to the two remote sensing images in Figure 7a,c; and RN(80) and GN(0, 0.1) were added to two medical images in Figure 9a,d. Figure 15 shows the time cost bar chart of the different algorithms for these noisy images, where RN is Rician noise, SPN is salt and pepper noise, GN is Gaussian noise, and SN is speckle noise.

As shown in Figure 15, the ARFCM, FLICMLNLI and PFLSCM algorithms consume significantly more time in processing noisy images than the other algorithms, and all three algorithms are relatively ineffective in segmenting images. However, the proposed KWHLICM, HLICM, and IHLICM algorithms consume less time but still struggle to meet the requirements of large-scale real-time image processing. In the near future, we will study fast algorithms related to the proposed algorithms [72,73] to meet the requirements of real-time image processing.

4.5. Impact of Neighborhood Size on Algorithm Performance

To investigate the influence of local neighborhood window size on the algorithm, we selected local domain windows of 3 × 3, 5 × 5, 7 × 7, 9 × 9, and 11 × 11 to test images with various types of noise and analyzed the segmentation results to determine the impact of local neighborhood window size on the algorithm.

GN(0, 0.1) was added to the synthetic image in Figure 3b, SPN(0.3) was added to the natural image in Figure 5d, SN(0.2) was added to the remote sensing image in Figure 7b, and RN(80) was added to the medical image in Figure 9a. The segmentation results of three proposed algorithms HLICM, IHLICM, and KWHLICM are shown in Figure 16, Figure 17 and Figure 18, respectively.

4.5.1. Impact Testing of Neighborhood Window Size on HLICM

From Figure 16, it can be seen that although the tested image contains a small amount of noise, the segmentation results deteriorated as the neighborhood window size increased. From

Table 6. Evaluation indexes of HLICM varying with neighborhood size for noisy images.

Image	Window Size	Acc	Sen	Jaccard	PSNR	SA	Kappa	mIoU	DICE
Figure 3b + GN(0, 0.1)	3 × 3	0.7846	0.677	0.5117	10.5861	0.677	0.5039	17.0561	0.677
	5 × 5	0.7561	0.6342	0.4643	9.2284	0.6342	0.4394	15.4771	0.6342
	7 × 7	0.7139	0.5708	0.3994	7.6556	0.5708	0.3446	13.3123	0.5708
	9 × 9	0.5783	0.3675	0.2251	4.3919	0.3675	0.0473	7.5036	0.3675
	11 × 11	0.585	0.3774	0.2326	4.6463	0.3774	0.0606	7.754	0.3774
Figure 5d + SPN(0.3)	3 × 3	0.7794	0.6691	0.5028	10.4415	0.6691	0.4586	16.7591	0.6691
	5 × 5	0.7449	0.6173	0.4465	9.4334	0.6173	0.3764	14.882	0.6173
	7 × 7	0.7388	0.6082	0.437	8.8233	0.6082	0.3526	14.5654	0.6082
	9 × 9	0.7262	0.5893	0.4177	8.237	0.5893	0.3325	13.9249	0.5893
	11 × 11	0.6984	0.5476	0.3771	7.3094	0.5476	0.2761	12.5687	0.5476
Figure 7b + SN(0.2)	3 × 3	0.8707	0.8061	0.6751	11.9614	0.8061	0.5579	22.5042	0.8061
	5 × 5	0.9221	0.8832	0.7908	13.3359	0.8832	0.7728	26.3599	0.8832
	7 × 7	0.8849	0.8274	0.7056	11.9199	0.8274	0.6739	23.5193	0.8274
	9 × 9	0.8475	0.7712	0.6276	10.8084	0.7712	0.5805	20.9212	0.7712
	11 × 11	0.8107	0.716	0.5577	9.8215	0.716	0.4913	18.5891	0.716
Figure 9a + RN(80)	3 × 3	0.7709	0.6564	0.4885	9.3521	0.6564	0.4212	16.2841	0.6564
	5 × 5	0.7216	0.5824	0.4109	8.8048	0.5824	0.3115	13.6955	0.5824
	7 × 7	0.7135	0.5703	0.3989	9.2374	0.5703	0.2749	13.2959	0.5703
	9 × 9	0.6884	0.5325	0.3629	8.5715	0.5325	0.2317	12.0962	0.5325
	11 × 11	0.6671	0.5007	0.334	8.5147	0.5007	0.1775	11.1324	0.5007

, it can be seen that when the neighborhood window size is 3 × 3, the segmentation evaluation index of the tested image was the highest. Therefore, the proposed HLICM algorithm can obtain satisfactory segmentation results when the neighborhood window size is 3 × 3.

4.5.2. Impact Testing of Window Size on IHLICM

From Figure 17, although the tested image contains a small amount of noise, the segmentation results deteriorated as the window size increased. From

Table 7. Evaluation indexes of IHLICM varying with neighborhood size for noisy images.

Image	Window Size	Acc	Sen	Jaccard	PSNR	SA	Kappa	mIoU	DICE
Figure 3b + GN(0.1)	3 × 3	0.9924	0.9886	0.9774	21.9386	0.9886	0.9828	32.58	0.9886
	5 × 5	0.9864	0.9796	0.9601	19.5093	0.9796	0.9694	32.0034	0.9796
	7 × 7	0.9809	0.9713	0.9442	17.94	0.9713	0.9569	31.4723	0.9713
	9 × 9	0.9755	0.9632	0.9291	16.8368	0.9632	0.9448	30.9687	0.9632
	11 × 11	0.9696	0.9544	0.9128	15.9371	0.9544	0.9316	30.4254	0.9544
Figure 5d + SPN(0.3)	3 × 3	0.8884	0.8326	0.7132	13.4479	0.8326	0.7156	23.7723	0.8326
	5 × 5	0.8578	0.7867	0.6484	12.1451	0.7867	0.6384	21.6143	0.7867
	7 × 7	0.8399	0.7599	0.6128	11.286	0.7599	0.5936	20.4254	0.7599
	9 × 9	0.8189	0.7283	0.5727	10.2177	0.7283	0.5372	19.0915	0.7283
	11 × 11	0.7987	0.6981	0.5362	9.5646	0.6981	0.4826	17.8723	0.6981
Figure 7b + SN(0.2)	3 × 3	0.8933	0.84	0.7241	13.8606	0.84	0.6496	24.1377	0.84
	5 × 5	0.8851	0.8277	0.706	13.4943	0.8277	0.604	23.5348	0.8277
	7 × 7	0.8724	0.8086	0.6787	13.0383	0.8086	0.55	22.6222	0.8086
	9 × 9	0.8684	0.8026	0.6703	12.9038	0.8026	0.5318	22.3441	0.8026
	11 × 11	0.865	0.7975	0.6632	12.7963	0.7975	0.5166	22.1082	0.7975
Figure 9a + RN(80)	3 × 3	0.9388	0.9082	0.8318	16.3974	0.9082	0.848	27.7261	0.9082
	5 × 5	0.9309	0.8964	0.8123	15.8696	0.8964	0.8258	27.0755	0.8964
	7 × 7	0.9172	0.8758	0.7791	15.0826	0.8758	0.7892	25.97	0.8758
	9 × 9	0.9014	0.8521	0.7424	14.3348	0.8521	0.7462	24.746	0.8521
	11 × 11	0.8844	0.8266	0.7045	13.652	0.8266	0.699	23.483	0.8266

, it can be seen that when the neighborhood window size is 3 × 3, the segmentation evaluation index of the tested image was the highest. Therefore, the proposed IHLICM algorithm can obtain satisfactory segmentation results when the neighborhood window size is 3 × 3.

4.5.3. Impact Testing of Window Size on KWHLICM

From Figure 18, it can be seen that when the neighborhood window size was 3 × 3, the segmentation results of the remote sensing images and medical images still contained a small amount of noise, which is not what we want, and when the neighborhood window size was 5 × 5, the noise was suppressed and the segmentation results were satisfactory. As the neighborhood window size continues to increase, the algorithm’s ability to suppress the noise is enhanced; however, some details are lost in image segmentation. From

Table 8. Evaluation indexes of KWHLICM varying with window size for noisy images.

Image	Window Size	Acc	Sen	Jaccard	PSNR	SA	Kappa	mIoU	DICE
Figure 3b + GN(0.1)	3 × 3	0.999	0.9985	0.997	29.6905	0.9985	0.9977	33.2318	0.9985
	5 × 5	0.9984	0.9977	0.9953	27.7219	0.9977	0.9965	33.1781	0.9977
	7 × 7	0.9977	0.9965	0.993	26.227	0.9965	0.9947	33.1012	0.9965
	9 × 9	0.9972	0.9957	0.9915	25.2658	0.9957	0.9936	33.0507	0.9957
	11 × 11	0.9962	0.9943	0.9886	24.0446	0.9943	0.9914	32.953	0.9943
Figure 5d + SPN(0.3)	3 × 3	0.8845	0.8267	0.7046	13.423	0.8267	0.7185	23.4873	0.8267
	5 × 5	0.8854	0.828	0.7065	13.5333	0.828	0.7184	23.5514	0.828
	7 × 7	0.8768	0.8152	0.688	13.2182	0.8152	0.6981	22.9335	0.8152
	9 × 9	0.8662	0.7994	0.6658	12.8518	0.7994	0.6737	22.1923	0.7994
	11 × 11	0.8557	0.7835	0.6441	12.5009	0.7835	0.6495	21.4689	0.7835
Figure 7b + SN(0.2)	3 × 3	0.9574	0.9361	0.8799	18.0612	0.9361	0.8772	29.3299	0.9361
	5 × 5	0.9768	0.9652	0.9327	20.5707	0.9652	0.9298	31.0901	0.9652
	7 × 7	0.9671	0.9507	0.906	18.9988	0.9507	0.8996	30.201	0.9507
	9 × 9	0.96	0.9399	0.8867	18.1137	0.9399	0.8768	29.5563	0.9399
	11 × 11	0.9542	0.9314	0.8715	17.5149	0.9314	0.8584	29.0516	0.9314
Figure 9a + RN(80)	3 × 3	0.932	0.898	0.8149	15.8557	0.898	0.838	27.1649	0.898
	5 × 5	0.9494	0.9241	0.8589	17.1482	0.9241	0.8783	28.6316	0.9241
	7 × 7	0.9479	0.9219	0.8551	17.0072	0.9219	0.8745	28.5048	0.9219
	9 × 9	0.9414	0.9121	0.8384	16.4742	0.9121	0.8588	27.9469	0.9121
	11 × 11	0.9314	0.8971	0.8133	15.757	0.8971	0.8352	27.1113	0.8971

, it can be seen that when the neighborhood window size is 3 × 3, the segmentation evaluation index of the synthetic image is the highest, while when the neighborhood window size is 5 × 5, the segmentation evaluation index of the natural image, remote sensing image, and medical image is the highest. Therefore, it can be concluded that the KWHLICM algorithm can achieve satisfactory segmentation results when the window size is between 3 × 3 and 5 × 5.

4.6. Impact of the Fuzzy Weighting Exponent on Algorithm Performance

The fuzzy weighting exponent is an important parameter in fuzzy clustering algorithms that has a certain impact on clustering performance. The range of the fuzzy weighting exponent varies depending on the algorithm. In general, the range of the fuzzy weighting exponent in FCM-related algorithms is between [1.5, 2.5] [74]. For the proposed harmonic fuzzy C-means clustering algorithm, we have conducted extensive experimental tests to confirm that its fuzzy weighting exponent is reasonably selected as [5,10]. Therefore, the HLICM, IHLICM, and KWHLICM algorithms proposed in this paper select values within this interval for testing and objectively analyze the impact of the fuzzy weighting exponent on its algorithm segmentation performance.

4.6.1. Fuzzy Weighting Exponent in HCM

This paper selected four representative numerical datasets (https://github.com/milaan9/Clustering-Datasets) (accessed on 15 October 2023) for clustering testing. These numerical data included Iris with 150 samples and four features, Triangle1 with 1000 samples and two features, Seeds with 210 samples and seven features, and Wine with 178 samples and thirteen features. The common accuracy (ACC) and normalized mutual information (NMI) indexes are used to evaluate the clustering performance of the proposed HCM algorithm, and the test results are displayed in Figure 19.

As shown in Figure 19, it can be seen that when $m\ge 5$, the ACC and NMI values basically do not change with the change of $m$. Therefore, we conclude that the fuzzy weighting exponent in the HCM algorithm can be taken from 5.0.

In addition, we also selected four different types of images for segmentation testing. These images included an MRI brain image (https://www.kaggle.com/preetviradiya/brian-tumor-dataset) (accessed on 15 October 2023), two MSRC natural images (https://www.microsoft.com/en-us/research/project/image-understanding) (accessed on 15 October 2023), and an uneven illumination image. The common accuracy (ACC) and Jaccard similarity coefficient (Jaccard) were used to evaluate the segmentation performance of the proposed HCM algorithm, and the test results are displayed in Figure 20.

As shown in Figure 20, it can be seen that when $m\ge 5$ the two indicators do not change much, and it can be concluded that for image data the reasonable range of the weighting exponent $m$ in the HCM algorithm should be greater than or equal to 5.

Based on the experimental results of Figure 19 and Figure 20, it can be seen that with the change of $m$, the values of the ACC, NMI, and Jaccard change or unchanged within a small range, with basic fluctuations within 0.1. This indicates that the fuzzy weighting exponent in the HCM algorithm has almost no impact on the algorithm, so the reasonable range of the weighting exponent $m$ in the HCM algorithm for numerical data and image data is [5, 10].

Overall, it is precisely because the range of the fuzzy weighting exponent in the HCM algorithm is [5, 10] that we will also select the value range of the HLICM, IHLICM, and KWHLICM algorithms to be [5, 10].

4.6.2. The Fuzzy Weighting Exponent in the HLICM, IHLICM, and KWHLICM Algorithms

GN(0, 0.1) was used to corrupt the synthetic image in Figure 3b, SPN(0.3) was used to corrupt the natural image in Figure 5d, SN(0.2) was used to corrupt the remote sensing image in Figure 7b, and RN(80) was used to corrupt the MR image in Figure 9a. The proposed algorithm was tested with different fuzzy weighting exponents using corresponding noisy images. Figure 21 shows the curve of the algorithm performance as a function of the fuzzy weighting exponent.

As shown in Figure 21, the comparison of four types of noise shows that the three proposed algorithms are more sensitive to speckle noise, while for other three types of noise, these three algorithms are less sensitive to changes in the fuzzy weighting exponent. Overall, as the fuzzy weighting exponent changes, the performance of three proposed algorithms remains stable, and the range of fuzzy weighting exponents provided in the proposed algorithms is reasonable.

4.7. Impact of Initial Clustering Centers on Algorithm Performance

To verify the sensitivity of the algorithm to the initial clustering centers, the gray levels within the maximum and minimum values of the noisy image were equally divided into $c$ segments, and the gray levels with frequency 0 were removed. At each execution, a set of values from $c$ segment was selected as the initial clustering centers [75], and five sets of initial clustering centers were selected for segmentation testing. We selected the synthetic image in Figure 3b, which was polluted by Gaussian noise; the natural image in Figure 5d, which was corrupted by salt and pepper noise; the remote sensing image in Figure 7b, which was corrupted by speckle noise; and the MR image in Figure 9a, which was corrupted by Rician noise. Figure 22, Figure 23, Figure 24 and Figure 25 show a series of box plots of algorithm performance as a function of the initial clustering centers.

As shown in Figure 22, Figure 23, Figure 24 and Figure 25, compared with images with different types of noise, the IHLICM and KWHLICM algorithms are more sensitive to the initial clustering centers on images containing speckle noise and salt and pepper noise, while the HLICM algorithm is more sensitive to the initial clustering centers on images containing Rician noise and speckle noise. These three proposed algorithms are insensitive to Gaussian noise, and overall, three proposed algorithms are not very sensitive to initial clustering centers.

4.8. Testing of the Algorithms’ Generalization Performance

To verify the adaptability of the algorithms, we select a large number of images from the BSDS500 dataset to verify the effectiveness and generalizability of the HLICM, IHLICM and KWHLICM algorithms. Considering the spatial limitation of this paper, only eight images from the BSDS500 dataset are provided and segmented using 12 fuzzy clustering algorithms. GN(0, 0.1) was used to corrupt #120003, #296028, #385028, and #353013 and SPN(0.3) was used to corrupt #56028, #140075, #223061, and #97033. The segmentation results of these algorithms for noisy natural images are shown in Figure 26.

From Figure 26, it can be seen that the segmentation results of the FCM_VMF and FLICMLNLI algorithms are poor, indicating that these algorithms lack a certain robustness to noise. The FCM_VMF and FLICMLNLI algorithms have the worst segmentation effect on noisy images; the ARFCM algorithm loses a large amount of information in the image during segmentation; the segmentation results of the DSFCM_N, PFLSCM, FLICM, and HLICM algorithms contain a large amount of noise, which is unsatisfactory; the FCM_SCIM and FSC_LNML algorithms suppress a large amount of noise, but the edges of the segmented image are not smooth; and although the KWFLCM and IHLICM algorithms have a good performance in image segmentation, there is still noise in the segmentation results. Therefore, compared with other algorithms, the segmentation results of KWHLICM algorithm are closer to the ground truth and preserve many details. From Figure 27, it can be seen that KWHLICM outperforms other algorithms in terms of segmentation performance. Therefore, the KWHLICM algorithm has a significant advantage in segmenting natural images with noise.

To further verify the generalizability of the algorithm, we continued by selecting a large number of images from the BSDS500 dataset for noise-free segmentation testing. The experimental results still show that KWHLICM still has good segmentation performance when segmenting noiseless images. Figure 28 show the segmentation results of eight noiseless natural images.

From Figure 28, it can be seen that when segmenting natural images without noise, the ARFCM and FCM_VMF algorithms severely lose the detailed information of the image, while the FCM_SICM and FSC_LNML algorithms still blur the boundaries of the image when processing it. For #306005, the FCM_SICM algorithm failed to segment and for #106020 and #21077, the FSC_LNML algorithm cannot perform reasonable segmentation. Using the HLICM algorithm resulted in misclassification. For the FLICMLNLI, DSFCM_N, KWFLICM, PFLSCM, FLICM, KWHLICM and IHLICM algorithms, the difference in image segmentation results is small, and the naked eye cannot distinguish their differences well. From Figure 29, it can be seen that the ACC and PSNR values of the KWHLIMC algorithm are better than those of the other algorithms. Overall, the proposed KWHLICM algorithm for noise-free images has better generalization than existing fuzzy clustering-related algorithms.

4.9. Testing and Analysis of the Algorithms for Color Image

To verify the adaptability of the algorithm to color images, a large number of images from BSDS500 and UC Merced Land Use remote sensing datasets were selected to verify the generalizability and effectiveness of the proposed algorithm. SPN(0.2) was added to #296059# 304034, GN(0, 0.1) was added to #189011# 232038, and SN(0.2) was added to buildings18. Using 12 algorithms to segment these noise-free and noisy images, the segmentation results are shown in Figure 30.

From Figure 30, it can be seen that for noise-free color images, all algorithms except the ARFCM, FCM_VMF and FCM_SICM algorithms can effectively extract the target in the image. For color images contaminated by noise, the DSFCM-N, PFLSCM, FLICMLLI, FLICM and HLICM algorithms cannot completely suppress the noise, and the segmentation results still contain some noise. The ARFCM, FCMSCIM and FSC_LNML algorithms can suppress most of the noise in the image, but the edges of the segmented image are not smooth and satisfactory. The KWHLICM algorithm achieves better results, which are closer to the ground truths and also retain some details. From Figure 31, it can be seen that for both noiseless and noisy color images, the KWHLICM algorithm has higher metrics than other algorithms. Therefore, the proposed KWHLICM algorithm also has good universality for color image segmentation.

4.10. Statistical Comparisons by the Friedman Test

To systematically compare different algorithms, this paper this paper uses the Friedman test [76] to evaluate the running efficiency (time) and segmentation quality (ACC, PSNR, and mIoU) of the 12 algorithms on the 16 images in Figure 3, Figure 5, Figure 7 and Figure 9. More specifically, the Friedman test rejects the null hypothesis that exhibits the same level of significance, which requires the use of the Nemenyi test for verification. If the average rankings of two algorithms differ by at least one key difference, their performance will be significantly different. The key difference is defined as $\mathrm{CD}=q_{\alpha }\sqrt{K(K+1)/6N}$. The CD plots are shown in Figure 32.

According to the CD plots in Figure 32, the KWHLICM algorithm for 16 images has a statistical advantage over the other compared algorithms in terms of running efficiency and segmentation quality. In Figure 32a, the FLICMLNLI, PFLSCM, ARFCM, and DSFCM_N algorithms outperform the KWHLICM algorithm in terms of running efficiency. Therefore, the KWHLICM algorithm does not perform much better than these comparison algorithms in terms of runtime, but achieves a good compromise between segmentation quality and statistical running efficiency.

4.11. Algorithm Convergence Test

In this article, we monitor the convergence of the algorithm by calculating the number of iterations to determine if the change in the clustering centers that the algorithm converges to is below a preset threshold. Updating the cluster center is crucial during the clustering iteration process. When the change in the cluster center is less than the preset threshold, it can be considered that the iterative algorithm has converged and the cluster center has reached a stable state. To test the convergence of different algorithms, four images from the BSDS500, UC Merced land use, and brain tumor MRI datasets were selected for segmentation testing.

Table 9. Iteration number of different algorithms for noisy images.

Algorithm	#36046 +SPN(0.3)	#36046 +GN(0.1)	#37 no +RN(80)	Buildings18 +SN(0.2)
ARFCM	100	39	100	100
FLICMLNLI	20	20	20	20
FCM_VMF	33	40	41	45
DSFCM_N	695	387	190	608
KWFLICM	42	43	42	32
PFLSCM	5	5	5	5
FCM_SICM	22	31	17	24
FSC_LNMI	61	65	31	75
FLICM	38	39	37	29
HLICM	4	4	4	4
IHLICM	69	72	79	71
KWHLICM	69	72	79	71

gives the algorithm iterations for different noisy images.

From Table 9, it can be seen that the HLICM algorithm has fewer iterations than the other algorithms, while the IHLICM and KHLICM algorithms have more iterations. However, by comparing the iteration times of different algorithms, it can be seen that the HLICM algorithm is significantly better than other compared algorithms in terms of convergence speed, while the convergence speed of the KWHLICM and IHLICM algorithms is not significantly different from other algorithms. Overall, the proposed KWHLICM and IHLICM algorithms not only have good segmentation performance, but also have high computational efficiency.

5. Conclusions and Outlook

This paper first defined the new concepts of harmonic fuzzy sets and harmonic fuzzy partition. Then, based on these basic concepts, a new harmonic fuzzy local information C-mean clustering algorithm was proposed, and the local convergence of this algorithm was rigorously proved using Zangwill’s theorem. Meanwhile, in order to improve the generalizability of the algorithms, as inspired by the IFLICM and KWFLCM algorithms, two enhanced robust image segmentation algorithms were proposed using local information, and their good performances were verified through experiments. In conclusion, the originality of our work lies in the integration of harmonic fuzzy membership and local spatial information in the clustering process:

(1)

The use of harmonic fuzzy partition to constrain fuzzy membership of all the algorithms;

(2)

In the proposed algorithm, sample clustering is performed by using harmonic fuzzy membership and neighborhood information of pixels;

(3)

The fuzzy membership degree in the proposed algorithm has a value range of $(1,+\infty )$, which is only supported by the harmonic fuzzy set theory proposed in this paper.

Overall, the proposed algorithms have satisfactory clustering results. These algorithms and related findings not only help to reveal the intrinsic structure of data, but also promote the development of fuzzy clustering related theories. Meanwhile, they can be widely used in data mining, image processing, and medical diagnosis. Clustering algorithms have important applications in various fields and can help people better understand data and make effective decisions.

There are many directions in which future work can be expanded. The following are some examples:

(1)

Explore the theory of harmonic fuzzy graphs [77] used to solve complex engineering problems;

(2)

Constructing harmonic fuzzy logic to solve problems related to fuzzy inference [78];

(3)

Construction of harmonic fuzzy neural networks [79,80] for solving complex engineering problems such as intelligent transportation and industrial automation.