OTSU Multi-Threshold Image Segmentation Based on Improved Particle Swarm Algorithm

Abstract

In view of the slow convergence speed of traditional particle swarm optimization algorithms, which makes it easy to fall into local optimum, this paper proposes an OTSU multi-threshold image segmentation based on an improved particle swarm optimization algorithm. After the particle swarm completes the iterative update speed and position, the method of calculating particle contribution degree is used to obtain the approximate position and direction, which reduces the scope of particle search. At the same time, the asynchronous monotone increasing social learning factor and the asynchronous monotone decreasing individual learning factor are used to balance global and local search. Finally, chaos optimization is introduced to increase the diversity of the population to achieve OTSU multi-threshold image segmentation based on improved particle swarm optimization (IPSO). Twelve benchmark functions are selected to test the performance of the algorithm and are compared with the traditional meta-heuristic algorithm. The results show the robustness and superiority of the algorithm. The standard dataset images are used for multi-threshold image segmentation experiments, and some traditional meta-heuristic algorithms are selected to compare the calculation efficiency, peak signal to noise ratio (PSNR), structural similarity (SSIM), feature similarity (FSIM), and fitness value (FITNESS). The results show that the running time of this paper is 30% faster than other algorithms in general, and the accuracy is also better than other algorithms. Experiments show that the proposed algorithm can achieve higher segmentation accuracy and efficiency.

1. Introduction

Image segmentation is widely used as the basis of computer vision. Image segmentation refers to describing an image as a collection of some connected areas so that the image features are different in different areas. At present, image segmentation methods mainly include the threshold method, edge detection method, region method, morphological watershed method, and so on [1]. The threshold method is at the core of image segmentation applications because of its simple implementation and fast calculation speed [2]. Thresholds can be divided into two forms, namely, two-stage thresholds (BT) and multilevel thresholds (MT). The two-level threshold uses a single threshold to divide the image into two categories, and the multi-level threshold uses multiple thresholds to divide the image into more than two uniform segments [3]. The threshold can be determined by kapur entropy [4], Tsallis entropy [5], fuzzy entropy [6], and OTSU variance [7]. This method uses the information in the histogram and does not need any ground live to classify pixels. However, threshold policy is a time-consuming process, especially when increasing the number of thresholds. The running time will greatly increase, so this has become a problem that must be solved in future research.

Recently, Houssein et al. proposed an improved balance optimizer to solve the problem of imbalance between the exploration stage and the development stage of the balance optimizer, which solves the problem that the algorithm optimization is easy to fall into local optimization and shows the excellent performance of this algorithm [8]. Sharma, Sushmita, and others improved the original butterfly optimization algorithm by combining the mutualism and parasitic stages of the traditional butterfly algorithm and the symbiotic biological search algorithm. Using kapur entropy as the fitness function, they selected a group of benchmark images to find the optimal threshold. The results show that the improved butterfly algorithm is superior to other algorithms in all evaluation indicators [9]. In addition, Elaziz et al., in view of the shortcomings of the Harris Hawks Optimizer algorithm, proposed an improved version of the HHO algorithm, which solved the problem of poor search ability of traditional algorithms and makes it easy to fall into local optimization [10]. Zhang et al. proposed an improved PSO algorithm to solve the problem of premature convergence of traditional PSO and effectively realize adaptive image segmentation [11]. Zhao et al. proposed a cross strategy-based ant colony algorithm (CCACO) in order to solve the continuity problem of the ant colony algorithm, which uses Kapur entropy as the objective function for image segmentation. Experiments show that the proposed CCACO achieved excellent segmentation results in both low-threshold and high-threshold [12].

Of course, in the process of image segmentation, it is far from enough to rely only on threshold segmentation. Because of the low processing speed and segmentation accuracy of threshold segmentation, people will use optimization algorithms to optimize the segmentation process. Among many optimization algorithms, the meta-heuristic optimization algorithm is widely used because of its low cost, high accuracy, and fast speed. So far, various optimization algorithms have been introduced to deal with nonlinear and practical applications, such as genetic algorithm (GA) [13], particle swarm optimization (PSO) [14], whale optimization algorithm (WOA) [15], butterfly optimization algorithm (BOA) [9], sine cosine optimization algorithm (SCA) [16], crow optimization algorithm (CSA) [17], gray wolf optimization algorithm (GWO) [18], and bee colony optimization algorithm (ABC) [19]. In recent years, Raj et al. proposed the Whale Optimization Algorithm (WOA) to optimize TCSC and SVC reactive power planning for the problems of transmission loss and high operating costs. The results show that this method has fewer iterations, will not fall into local minimum, and has good convergence characteristics [15]. Shiva and Gudadappanavar proposed an opposite crow search algorithm for transmission loss, which has better performance in reducing active power and system operation cost [16]. In order to solve the problems of insufficient reactive power and unstable voltage of transmission lines, Babu and Kumar et al. proposed an improved sine cosine optimization algorithm, which uses the techniques of the sine cosine algorithm (SCA) and quasi inverse sine cosine algorithm (QOSCA) to minimize transmission losses and operating costs [17]. Xu et al. proposed an improved hunger game search algorithm (IHGS) for solar photovoltaic system parameter identification, which solved the problem of stability of the algorithm when solving the global optimal solution. It shows the feasibility and effectiveness of the improved HGS algorithm [20]. In addition, Trojovsky et al. proposed a new population-based optimization algorithm, the Pelican Algorithm (POA). This algorithm has high development and strong search ability [21]. Shabani and Asgarian put forward the search and rescue optimization algorithm (SAR) for the problem of single objective continuous optimization and demonstrated the feasibility of the algorithm through experiments [22]. Oliva and Elizabeth put forward a new solution to the problem that the traditional algorithm population is small and easy to fall into the local optimum. They used chaos mapping and opposite learning to initialize the solution of the given problem and improve the diversity of the population by constantly following the new initial population position through the interference operator. The experimental results show that the proposed method has high efficiency in dealing with the optimal solution problem [23].

In order to improve the performance of PSO on image threshold segmentation, a new OTSU multi-threshold segmentation method based on Improved PSO was proposed. The selection process of thresholds were optimized and compared with many classical PSO methods. The results show that our algorithm reduces the running time and improves the accuracy in threshold segmentation. The main contributions in this paper can be summarized as follows:

(1)

An Improved PSO algorithm is proposed, in which (a) chaos optimization was added to reduce premature convergence; (b) elite particle search strategy to particle swarm optimization algorithm was used to reduce optimization time and improve efficiency; (c) learning factors were improved to balance local search and global search.

(2)

Combining PSO with OTSU algorithm, a gray image segmentation algorithm based on improved particle swarm optimization is proposed. The proposed improved particle swarm optimization segmentation algorithm can search for a more accurate threshold, thus promoting better component division of gray-scale images.

(3)

Some classical test functions are selected to verify the robustness and development of the algorithm in solving single peak, multi-peak, and multi-peak fixed dimension functions.

(4)

Compared with the multi threshold segmentation of some standard algorithms, the performance of the improved PSO segmentation algorithm was verified, and the effectiveness of the algorithm on images was verified through multi-threshold image segmentation experiments on PASCAL 2012 dataset images. Experiments show that the method in this paper is faster than other meta heuristic algorithms in OTSU threshold segmentation, and PSNR FSIM and SSIM performance indicators verify that the algorithm in this paper has higher accuracy in image segmentation.

The rest of this paper is organized as follows: Section 2 summarizes the multi-threshold OTSU segmentation model. Section 3 describes the PSO algorithm. Section 4 proposes a multi-threshold segmentation algorithm based on the Improved PSO algorithm. Section 5 describes the experimental results of the segmentation method based on the improved particle swarm optimization. Section 6 presents conclusions and future work.

2. Multi-Threshold OTSU Segmentation Model

The OTSU image segmentation method is the maximum inter-class variance method, which aims to determine the optimal threshold for image segmentation. In OTSU, the image is divided into foreground and background by threshold, and the threshold with the largest class variance in the foreground and background regions is taken as the best segmentation threshold. Assume that the gray scale range is $\{0,1,2,\cdots ,L-1\}$, representing L different gray levels in a digital image, whose size is $M\times N$ pixels. Let $B\left(i\right)$ be the proportion of grayscale i in the entire image, calculated as

$$B\left(i\right)=a\left(i\right)/(M\times N),$$

(1)

where $a\left(i\right)$ is the number of pixels with grayscale i. Let the threshold t divide the image into two parts, foreground and background.

$$\left\{\begin{array}{c}\hfill h_0=\sum _{i=0}^{t}B\left(i\right)\hfill \\ \hfill h_1=\sum _{i=t+1}^{L-1}B\left(i\right)\hfill \end{array}\right.$$

(2)

$$\left\{\begin{array}{c}\hfill {\rho }_0=\sum _{i=0}^{t}i\times B\left(i\right)/h_0\hfill \\ \hfill {\rho }_1=\sum _{i=t+1}^{L-1}i\times B\left(i\right)/h_1\hfill \end{array}\right.$$

(3)

where ${\rho }_T$ is the average grayscale of the image, which can be expressed as:

$${\rho }_T=\sum _{i=0}^{L}i\times B\left(i\right)$$

(4)

where $h_0$ and $h_1$ are the proportion of the foreground and background in the image, respectively, and ${\rho }_0$ and ${\rho }_1$ are the average grayscale of the image foreground and background, respectively. ${\sigma }_B^2$ is the inter class variance, which can be expressed as:

$${\sigma }_B^2=h_0{({\rho }_0-{\rho }_T)}^2+h_1{({\rho }_1-{\rho }_T)}^2$$

(5)

when the interclass variance ${\sigma }_B^2$ reaches the maximum, t is the optimal threshold. In the case of multiple thresholds, a threshold set of $\{t_1,t_2,\cdots ,t_{K-1}\}$ is the optimal combination of thresholds. The optimization problem can be expressed as:

$$T^{\ast }(t_1,t_2,\cdots ,t_{K-1})=argmax{\sigma }_B^2(t_1,t_2,\cdots ,t_{K-1})$$

(6)

The OTSU multi-threshold method uses the exhaustive method to solve the optimal thresholds combination with a total computational cost of O($L^K$). It can be seen that the computational cost will increase exponentially with the increase in the number of thresholds. In order to improve the efficiency of image segmentation, this paper proposes an improved particle swarm optimization algorithm to optimize the optimal threshold combination and then perform multi-threshold image segmentation.

3. Particle Swarm Algorithm

Particle swarm optimization (PSO) is a popular intelligent optimization algorithm [24] whose origins originate from the research on the predatory behaviors of birds. The PSO algorithm is similar to the genetic algorithm in that the solution of the objective function needs to be initialized randomly and optimized by iteration, without the crossover and mutation of chromosomes. Each particle is abstracted as a bird in the search space, which represents the solution of the optimization problem. The particle has two attributes: speed and position. The former represents the speed of movement and the latter represents the direction of movement. Substituting particle position information and velocity information into the fitness function to be optimized, the fitness value can be obtained. In the process of optimization, the particle determines its next motion through its own flight experience (individual extreme value) and group experience (group extreme value).

In a D-dimensional search space, the particle population size m, particle position $x_i=(x_{i1},x_{i2},\cdots ,x_{iD})$, and velocity $v_i=(v_{i1},v_{i2},\cdots ,v_{iD})$ are initialized, where D = 1, m = 50, range of position X is [0, 255], and the range of velocity V is $[-10,10]$. Particle i individual extremes $p_i=(p_{i1},p_{i2},\cdots ,p_{id},\cdots ,p_{iD})$ and Global Extremums $p_g=(p_{g1},p_{g2},\cdots ,p_{gd},\cdots ,p_{gD})$$p_i$ is the optimal position currently searched by the particle, and $p_g$ is the optimal position currently searched by all of the particles. Over the course of each iteration, particles update their velocity and position according to the following equation:

$$v_{id}^{k+1}=\omega v_{id}^k+c_1{r}_1(p_{id}-z_{id}^k)+c_2{r}_2(p_{gd}-z_{id}^k)$$

(7)

$$x_{id}^{k+1}=x_{id}^k+v_{id}^{k+1}$$

(8)

where $i=1,2,\cdots ,m$, $d=1,2,\cdots ,D$, k is the number of iterations, and $r_1$ and $r_2$ are random numbers between $[0,1]$; these two parameters are used to maintain the diversity of the population. Furthermore, $c_1$ and $c_2$ are learning factors. Take $c_1=c_2=2$, inertia weight $\omega$ reflects the ability of particles to inherit the previous velocity, with a linear decrement method, as follows:

$$\omega ={\omega }_{max}-\frac{{\omega }_{max}-{\omega }_{min}}{k_{max}}\times k$$

(9)

where ${\omega }_{max}=0.9$, ${\omega }_{min}=0.4$, k is the number of iterations, $k_{max}$ is the total number of iterations, and $k_{max}=100$.

4. Multi-Threshold Segmentation Algorithm Based on Improved PSO

The classical PSO algorithm is an effective method to deal with the optimization problem of threshold segmentation. However, due to the mechanism of particle swarm optimization algorithm, when used in image processing, a large number of calculations will make the image processing time longer, and premature convergence will fall into local optimization, thus reducing the overall work efficiency and affecting the accuracy of threshold segmentation. This paper proposes an improved particle swarm optimization algorithm to optimize OTSU multi-threshold image segmentation. The speed and accuracy of multi-threshold image segmentation are improved by optimizing the basic parameters of the particle swarm optimization algorithm and proposing an elite particle search strategy based on the classical particle swarm optimization algorithm.

4.1. Linear Optimization Learning Factor

For the problem of low segmentation accuracy of the classical particle swarm algorithm, a linear function is introduced in the process of parameter optimization of the particle swarm algorithm to update the learning factor of particles, and the changes in learning factors $c_1$ and $c_2$ affect individual and swarm particles to approach the optimal solution, respectively. Most of the search results from global search to local search are particles, and the centers of gravity of particle searches are also different at different stages. To mitigate this issue, this paper uses asynchronous linear decrement and asynchronous linear incrementing learning factors to balance global search and local search, so that the learning factor can change with the procedure of iterations. According to the characteristics of the PSO algorithm, in the early iterations, it is more likely to rely on the mobile experiences of particle individuals for global search, while in the later iteration, it mainly relies on the mobile experience of the group for local search. The PSO algorithm can enhance the global search ability by taking the larger value of the learning factor $c_1$ the smaller value of $c_2$ in the early iterations, and the smaller value of $c_1$ the larger value of $c_2$ in the later stages can enhance the local search ability. In order to better play the global search ability and local search ability at different stages and improve the accuracy of the segmentation, we use the asynchronous change learning factor. The formula is:

$$c_1\left(k\right)=c_{min}+(c_{max}-c_{min})(1-k/k_{max})$$

(10)

$$c_1\left(k\right)=c_{min}+(c_{max}-c_{min})(1-k/k_{max})$$

(11)

where $c_{max}$ and $c_{min}$ are the maximum and minimum values of the learning factor, $c_{max}=2.5$, $c_{min}=0.5$; k and $k_{max}$ are the current number of iterations and the maximum number of iterations, respectively. The improved speed update formula is:

$$v_{id}^{k+1}=\omega v_{id}^k+c_1\left(k\right)(p_{id}-z_{id}^k)+c_2\left(k\right)(p_{gd}-z_{id}^k)$$

(12)

4.2. Elite Particle Search Strategy

In view of the low efficiency and slow speed of classical particle swarm optimization in multi-threshold segmentation, this paper improves classical particle swarm optimization by updating the speed and position of particles and proposes a search strategy for elite particles, which is divided into three steps. Firstly, the threshold segmentation contribution of each particle is calculated; secondly, 20% elite particles are selected according to the threshold segmentation contribution of each particle; and finally, the optimal solution of this iteration is found through chaos optimization.

In this paper, 20% of the particles are selected as elite particles according to the contribution. The contribution of each particle is obtained by comparing the fitness value of each particle with the fitness value of the global extremum in the current iteration. We calculated the distance difference between the location of a single particle and the location of the global limit particle. The closer the distance was to the optimal threshold, the greater the contribution to the optimization. Assuming that the global extremum particle is $(e_{gbest},s_{gbest})$ in each iteration, the distance difference between the location of a single particle and the location of the global extreme particle and the threshold segmentation contribution value of each single particle are obtained according to the following formula:

$$z=\sqrt{{(e_{gbest}-e_i)}^2+{(s_{gbest}-s_i)}^2}$$

(13)

$$ACD=\frac{1}{z}$$

(14)

where z represents the distance between the individual particle of the ith particle and the particle individual, corresponding to the global extremum, $e_{gbest}$ is the x-axis of the particle individual, corresponding to the global extremum in the target search space, $s_{gbest}$ is the y-axis coordinate of the particle individual, corresponding to the global extremum in the target search space, $e_i$ is the x-axis of the i-th particle individual in the target search space, $s_i$ is the y-axis coordinate of the ith particle individual in the target search space, $ACD$ (actual contribution degree) is the threshold segmentation contribution of each particle, threshold segmentation contribution, and distance difference of z, which is inversely proportional. The closer the i-th particle individual is to the particle individual, corresponding to the global extreme value, the greater the contribution of the i-th particle individual to the threshold segmentation of this optimization. The top 20% of elite particles are selected in order of contribution from largest to smallest by threshold segmentation, reducing optimization time. For details, see Figure 1. Where the arrows represent the change in the number of thresholds, and the three circles indicate the optimization when the threshold is a different value.

Figure 1. Particle swarm narrows the search range according to the contribution of particles.

Taking the gray value range of the segmented image as the coordinate range of x and y values, the black dot as all particles, and the central red dot as the global optimal solution $gbest$ of the current iteration, the search range can be reduced to a black circular region, that is, the region where the elite particles are located, and k is the number of threshold values. Each threshold requires a corresponding particle swarm to determine, while n threshold values require n particle swarms. In the same iteration, determine the global optimal particle of the n-particle swarm in turn.

Put the selected elite particles into chaos optimization. During each iteration, the $gbest$ is scrambled and used as the update position of the particles, which makes them search locally for the global optimal solution. The purpose is to increase the diversity of the population, enhance the local optimization ability of the algorithm, and realize the local depth search of the particle swarm algorithm. In order to solve the problem of premature convergence of particle swarm optimization algorithm and easy to fall into local optimization, so as to improve the accuracy of image segmentation chaos, the specific steps of chaos optimization elite particles are as follows:

Step 1. Set the number of chaotic iterations to M and map $gbest$ through Equation (15) to the defining domain of the Logistic equation [0, 1];

$$y_1^k=\frac{g_{best}-R_{min}^k}{R_{max}^k-R_{mi{n}^k}}$$

(15)

where, $y_1^k$ is the sequence element obtained by the first iteration, $g_{best}$ is the global extremum in the whole population, $R_{min}^k$ is the minimum value of the particle position, and $R_{max}^k$ is the maximum value of the particle position.

Step 2. $y_1^k$ is obtained after M iterations through Equation (16), and the chaotic sequence $y_k=(y_1^k,y_2^k,\cdots ,y_M^k)$ is obtained.

$$y_{n+1}^k=\mu y_n^k(1-y_n^k)$$

(16)

where $y_{n+1}^k$ is the sequence element obtained by the (n+1)-thi iteration, $\mu$ is the control parameter, and $y_M^k$ is the sequence element obtained by the M-th iteration.

Step 3. The chaotic sequence is inversely mapped back to the original solution space through Equation (17) to obtain the feasible solution sequence of chaotic variables $g_{best}^{\ast k}=(g_{best,1}^{\ast k},g_{best,2}^{\ast k},\cdots ,g_{best,M}^{\ast k})$

$$g_{best,m}^{\ast k}=R_{min}^k+(R_{max}^k-R_{min}^k)y_m^k$$

(17)

where $y_m^k$ is the sequence element obtained by the m-th iteration, $m=(1,2,\cdots ,M)$.

Step 4. Calculate the fitness value of each feasible solution vector in the feasible solution sequence and retain the optimal vector, denoted as $g_{best}^{\ast k}$.

Step 5. Randomly select a particle from the current particle swarm and replace the position vector of the selected particle with a $g_{best}^{\ast k}$.

Step 6. Iterate until the maximum number of iterations is reached or gets a sufficiently satisfactory solution.

4.3. Our Method and Process

Compared with the classical particle swarm optimization algorithm, our improved particle swarm optimization algorithm is superior to the classical algorithm in terms of both speed and accuracy. At the same time, it also avoids the problem that the particle swarm optimization algorithm is prone to fall into local optimization, greatly improving the efficiency of image segmentation. The flow chart of our improved particle swarm optimization algorithm is shown in Figure 2.

Figure 2. Improved particle swarm algorithm flowchart.

To a certain extent, linear learning factors and elite particle search strategies are added to the classical particle swarm algorithm, which improves the problems of slow optimization speed, low precision, low efficiency, and easy optimization in the classical particle swarm. The detailed process based on the improved particle swarm algorithm is as follows:

Step 1. Set the parameters, which include the population size (m), dimension $\left(D\right)$, velocity range $\left(v\right)$, position range $\left(x\right)$, and maximum number of iterations $\left(k_{m}ax\right)$.

Step 2. Initialize the population. Initialize the dimensions and vectors of particle individuals according to the threshold number $\left(k\right)$. The range of position vectors is [0255].

Step 3. Update $c_1$ and $c_2$ with asynchronous monotonically decreasing individual learning factor and asynchronous monotonically increasing social learning factor, $c_1\left(k\right)$ and $c_2\left(k\right)$, instead of Equations (10) and (11) to individual, thereby improving the accuracy of particle optimization. Calculate the fitness values for individual particles, updating the individual extremums of particles and the global extremums of particles.

Step 4. Iteratively update the position of the particle individual and the speed of the particle individual.

Step 5. 20% elite particles are selected according to the threshold segmentation contribution of individual particles.

Step 6. Use chaotic iterative optimization to find the best fitness value for elite particles by comparing the chaotic optimization results.

Step 7. Determine whether the current number of chaotic iterations meets the set maximum number of chaos iterations or whether the chaos optimization reaches preset accuracy. If the current number of chaos iterations meets the maximum number of chaotic iterations or the chaos optimization reaches the set accuracy, go to step 8; otherwise, go to step 2.

Step 8. Map the optimal fitness value inversely back into the particle cluster group, calculate the fitness value of all particles in the particle cluster, and output the optimal solution if the set conditions are met; otherwise, go to step 1.

5. Experimental Analysis

5.1. Experimental Environment

Experiments demonstrate the effectiveness of our improved multi-threshold image segmentation algorithm. The test image is taken from the PASCAL 2012 data set [25]. The experimental environment consists of MATLAB 2018a, Windows 10, and an Intel(R) Core(TM) i5-5600H @ 2.4GHz CPU with 8GB of RAM. We set population size $m=50$, maximum number of iterations $k_{max}=100$, particle position range $x_i\in [0,255]$, velocity range $v_i\in [-10,10]$, individual learning factor $c_1\in [0.5,2.5]$, and social learning factor $c_2\in [0.5,2.5]$.

5.2. Benchmark Function

This paper selects 12 standard test functions for comparative experiments to test the robustness, exploration, and development performance of the functions. These functions were derived from the CEC benchmark function [26], covering unimodal functions (F1–F4), multimodal functions (F5–F8), and fixed dimensional multimodal functions (F9–F12), all of which had minimum values. The algorithm compared with this paper uses the traditional meta-heuristic algorithms, which are whale optimization algorithm (WOA) [27], butterfly optimization algorithm [28], and particle swarm optimization algorithm [14]. In order to make a fair evaluation, this paper conducted 30 experiments on four optimization algorithms at the same time, and the experimental results less than ${10}^{-8}$ are represented by 0.

Table 1 shows the selected test functions: Dim is the number of variables and N\A is unsolvable. Table 2 shows the average value and standard deviation of the benchmark function calculated by this method and the traditional meta heuristic algorithm. According to the data in Table 2, our method is robust in dealing with the mechanism of benchmark function, whether it is unimodal, multimodal, or a fixed-dimensional multimodal function. From Table 2, we can see that our method has better performance than WOA, BOA, and PSO. In particular, there is a smaller error in multimodal function, which shows that the method in this paper has better exploration capability. We found that our method did not find the global minimum in several functions, but the error was smaller than other algorithms. To sum up, our method was more exploitative and exploratory than other algorithms. The specific experiments are as follows:

Table 1. Benchmark Function.

Test Function	Type	Dim	Range	${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$
$F_1\left(x\right)$ = ${\sum }_{i=1}^n{x}_i^2$	US	30	$[-100,100]$	0
$F_2\left(x\right)$ = ${\sum }_{i=1}^n|x_i|+{\prod }_{i=1}^n\left|x_i\right|$	US	30	$[-10,10]$	0
$F_3\left(x\right)$ = ${\sum }_{i=1}^n\left(\right|x_i+0.5{\left|\right)}^2$	US	30	$[-100,100]$	0
$F_4\left(x\right)$ = ${\sum }_{i=1}^{n}i{x}_i^4+random[0,1)$	US	30	$[-128,128]$	0
$F_5\left(x\right)$ = $-20exp(-0.2\sqrt{\frac{1}{n}{\sum }_{i=1}^n{x}_i^2})-exp\left(\frac{1}{n}{\sum }_{i=1}^{n}cos\left(2\pi x_i\right)\right)+20+e$	UN	30	$[-32,32]$	0
$F_6\left(x\right)$ = $\frac{1}{4000}{\sum }_{i=1}^n{x}_i^2-{\prod }_{i=1}^{n}cos\left(\frac{x_i}{\sqrt{i}}\right)+1$	UN	30	$[-600,600]$	0
$F_7\left(x\right)$ = $\frac{\pi }{n}(10sin\left(\pi y_1\right)+{\sum }_{i=1}^{n-1}{(y_i-1)}^2(1+10si{n}^2\left(\pi y_{i+1}\right))+{(y_n-1)}^2)$	UN	30	$[-50,50]$	0
$+{\sum }_{i=1}^{n}u(x_i,10,100,4)$
$F_8\left(x\right)$ = $0.1(si{n}^2\left(3\pi x_1\right)+{\sum }_{i=1}^n{(x_i-1)}^2(1+si{n}^2(3\pi x_i+1))+{(x_n-1)}^2$	UN	30	$[-50,50]$	0
$(1+si{n}^2\left(2\pi x_n\right)))+{\sum }_{i+1}^{n}u(x_i,5,100,4)$
$F_9\left(x\right)$ = $4x_1^2-2.1x_1^4+\frac{1}{3}{x}_1^6+x_1{x}_2-4x_2^2+4x_2^4$	FDM	2	$[-5,5]$	$-1.0316$
$F_{10}\left(x\right)$ = ${(x_2-\frac{5.1}{4{\pi }^2}{x}_1^2+\frac{5}{\pi }{x}_1-6)}^2+10(1-\frac{1}{8\pi })cos{x}_1+10$	FDM	2	$[-5,5]$	0.398
$F_{11}\left(x\right)$ = $(1+{(x_1+x_2+1)}^2(19-14x_1+3x_1^2-14x_2+6x_1{x}_2+3x_2^2))$	FDM	2	$[-2,2]$	3
$\times (30+{(2x_1-3x_2)}^2\times (18-32x_1+12x_1^2+48x_2-36x_1{x}_2+27x_2^2))$
$F_{12}\left(x\right)$ = $-{\sum }_{i=1}^{1}0{((X-a_i){(X-a_i)}^T+c_i)}^{-1}$	FDM	4	[0,10]	$-10.5363$

Table 2. Compare the mean and variance of different algorithms through benchmark functions.

Function	Mean (WOA)	Std (WOA)	Mean (BOA)	Std (BOA)	Mean (PSO)	Std (PSO)	Mean (OURS)	Std (OURS)
$f_1$	0	0	0	0	$7.23\times {10}^{-5}$	$6.30\times {10}^{-5}$	0	0
$f_2$	0	0	0	0	0.23241	$8.23\times {10}^{-2}$	0	0
$f_3$	0	0	0.43910	0.18334	5.61222	0.73845	0	0
$f_4$	0.00511	0.00429	0.00935	0.00903	0.69190	0.21219	0.00149	0.00065
$f_5$	0	0	0	0	2.10830	0.27572	0	0
$f_6$	0	0	0	0	0.00105	0.00168	0	0
$f_7$	0.02203	0.01070	0.69756	0.08152	4.41429	1.81612	0.03898	0.07815
$f_8$	0	0	0.01205	0.01832	0.50179	0.24452	0	0
$f_9$	$-1.0316$	0	N\A	N\A	$-1.0316$	0	$-1.0316$	0
$f_{10}$	0.39790	$9.22\times {10}^{-6}$	0.54290	0.30599	0.39789	0	0.39789	$2.48\times {10}^{-5}$
$f_{11}$	3.00001	$3\times {10}^{-5}$	3.04893	0.05603	3	0	3.00017	0.00025
$f_{12}$	$-7.25931$	3.34554	$-4.01282$	0.38772	$-8.91402$	3.24476	$-10.49534$	0.12228

5.3. Segmentation Experiment and Results

We took the optimization time and image segmentation quality to evaluate the performance of algorithms for comparison. The segmentation quality was evaluated with peak signal-to-noise ratio (PSNR) [29,30], structural similarity (SSIM) [31,32], and feature similarity (FSIM) [33,34], and fitness values of all algorithms were calculated. PSNR was used to evaluate the degree of image distortion, and the higher its value, the smaller the image distortion. SSIM evaluated image segmentation performance from image contrast and structural information. The larger the value, the better the performance. FSIM was used to evaluate the feature similarity between the original image and the segmented image. The larger the $FSIM\in [0,1]$, the better the image segmentation quality.

In this paper, the images in PASCAL 2012 [25] were selected as the experimental data set, including Figure 3a–f (numbered 000019, 000436, 001478, 003579, 004423, 006404). They were typical images with relatively centralized gray distribution and small inter class variance, while Figure 3g–l (numbered 001236, 001876, 002036, 004231, 004610, 006946) selected images with obvious differences in gray distribution, that is, the inter class variance was relatively large, as shown in Figure 3. Theoretically, the greater the number of segmentation thresholds, the greater the accuracy of the image segmentation. In order to quantitatively analyze the performance of segmentation, in the experiment, four groups of different segmentation thresholds were used. k is, 2, 4, 6, and 8 for index analysis. Our improved particle swarm optimization algorithm was compared with the classical particle swarm optimization algorithm [13], whale optimization algorithm [25], and butterfly optimization algorithm [26]. With several evaluation indicators, such as the classical particle swarm optimization algorithm, the whale optimization algorithm and the butterfly optimization algorithm are both bionic algorithms. The differences are that the whale optimization algorithm achieves the goal of searching by searching, surrounding, chasing, and attacking prey, while the butterfly algorithm locates the source of the target through its own perception. In this way, we can better understand the segmentation effect of the improved particle swarm optimization OTSU image segmentation method compared with other algorithms under different thresholds. Table 3, Table 4, Table 5, Table 6, Table 7, Table 8, Table 9 and Table 10 show the comparison results of PSNR, SSIM, FSIM, and fitness values, respectively. It can be seen that with the increase of the number of thresholds k, the distortion after image segmentation gradually decreases, which is closer to the original image. SSIM and FSIM gradually increase, and the segmentation performance gets better and better, which shows the advantages of multi-threshold image segmentation methods. To verify the universality of the algorithm in this paper, we randomly selected 100 images from the PASCAL 2012 dataset for the same experiment and listed the average values of the experimental parameters in Table 11. In addition, the algorithm that performs best on the evaluation criteria is marked in bold.

Figure 3. Histogram distribution.

Table 3. Comparison of PSNR in images a–f.

Metric Name	Image	k	WOA	BOA	PSO	OURS
PSNR	Image1(000019)	2	11.4924	12.6732	13.0284	13.7823
		4	12.5075	13.7829	14.1673	14.7642
		6	16.5389	17.2659	17.8454	19.6037
		8	19.8595	20.4863	20.5269	21.3771
	Image2(000436)	2	13.1415	13.7986	14.5376	14.1383
		4	14.1361	14.5284	15.3228	15.6705
		6	17.9243	17.9988	18.3650	19.3411
		8	18.5029	19.2229	20.9130	21.9703
	Image3(001478)	2	13.3784	13.7194	13.7913	14.2379
		4	13.6647	14.1434	14.9181	15.4509
		6	15.8758	16.4661	17.4344	18.6340
		8	17.3109	17.6060	19.4388	20.3651
	Image4(003579)	2	13.6386	13.5726	14.4871	15.9936
		4	14.3408	14.7052	15.6358	15.7378
		6	15.9297	16.6595	17.2260	18.2881
		8	16.6926	16.8525	17.4432	19.6017
	Image5(004423)	2	15.1932	15.8189	15.5466	15.7275
		4	15.5115	16.2379	16.9289	16.8156
		6	17.5728	18.8819	20.8917	21.7370
		8	18.6900	19.9407	21.9854	23.3966
	Image6(006404)	2	13.8768	13.8550	14.9768	14.9706
		4	14.6173	14.7452	14.8361	15.7539
		6	16.5043	17.5922	18.7453	19.1893
		8	17.8543	18.5252	20.7008	21.1228

Table 4. Comparison of SSIM in images a–f.

Metric Name	Image	k	WOA	BOA	PSO	OURS
SSIM	Image1(000019)	2	0.4186	0.4628	0.4521	0.4955
		4	0.4447	0.4559	0.5443	0.5968
		6	0.4921	0.5161	0.6220	0.6973
		8	0.6899	0.6930	0.7011	0.7333
	Image2(000436)	2	0.4336	0.4922	0.5550	0.5806
		4	0.4781	0.5615	0.5975	0.6381
		6	0.5482	0.6102	0.6238	0.6676
		8	0.5720	0.6267	0.6541	0.7895
	Image3(001478)	2	0.4299	0.4474	0.5120	0.5506
		4	0.5219	0.5774	0.6484	0.6386
		6	0.5634	0.5971	0.6960	0.7235
		8	0.6563	0.6857	0.7436	0.8154
	Image4(003579)	2	0.5610	0.5961	0.5969	0.5904
		4	0.5796	0.6057	0.6172	0.6714
		6	0.6286	0.6411	0.6484	0.7172
		8	0.6764	0.6938	0.7036	0.7297
	Image5(004423)	2	0.3469	0.3755	0.3758	0.4954
		4	0.4443	0.4570	0.5714	0.5978
		6	0.5396	0.5680	0.6687	0.7178
		8	0.6817	0.6638	0.7430	0.7746
	Image6(006404)	2	0.4216	0.4419	0.4675	0.4990
		4	0.4429	0.4723	0.5482	0.5745
		6	0.4859	0.5186	0.5744	0.6482
		8	0.5632	0.6114	0.6367	0.6861

Table 5. Comparison of FSIM in images a–f.

Metric Name	Image	k	WOA	BOA	PSO	OURS
FSIM	Image1(000019)	2	0.7053	0.7214	0.7423	0.7711
		4	0.7278	0.7535	0.7861	0.8026
		6	0.7596	0.7974	0.8013	0.8250
		8	0.8058	0.8144	0.8224	0.8476
	Image2(000436)	2	0.6525	0.6934	0.7571	0.7918
		4	0.7120	0.7235	0.7610	0.8132
		6	0.7350	0.7441	0.7832	0.8317
		8	0.7582	0.7655	0.8086	0.8594
	Image3(001478)	2	0.6592	0.6763	0.7084	0.7135
		4	0.6795	0.6940	0.7288	0.7402
		6	0.7110	0.7417	0.7643	0.7732
		8	0.7390	0.7652	0.7864	0.8277
	Image4(003579)	2	0.7495	0.7634	0.7770	0.7953
		4	0.7679	0.7855	0.7921	0.8113
		6	0.7869	0.8024	0.8275	0.8322
		8	0.8139	0.8289	0.8411	0.8672
	Image5(004423)	2	0.7552	0.7695	0.7892	0.7859
		4	0.7723	0.7820	0.8118	0.8382
		6	0.7965	0.7998	0.8367	0.8547
		8	0.8185	0.8247	0.8559	0.8712
	Image6(006404)	2	0.6292	0.6467	0.6638	0.6779
		4	0.6482	0.6614	0.6735	0.7066
		6	0.6684	0.6876	0.6910	0.7134
		8	0.6835	0.7155	0.7334	0.7780

Table 6. Comparison of Fitness in images a–f.

Metric Name	Image	k	WOA	BOA	PSO	OURS
Fitness	Image1(000019)	2	1704.8574	1704.5298	1704.3760	1704.8259
		4	2539.6813	2539.4239	2539.2458	2539.7578
		6	2765.9773	2765.2783	2764.2456	2764.2568
		8	3858.8308	3858.1415	3857.7672	3858.3303
	Image2(000436)	2	2213.5065	2213.1738	2213.3147	2213.1353
		4	3450.9044	3450.6474	3450.6818	3450.8937
		6	4033.8449	4033.1927	4033.1099	4033.8446
		8	4905.7727	4905.6214	4904.3492	4905.7649
	Image3(001478)	2	2451.9890	2451.7517	2451.1717	2451.9397
		4	3375.8902	3375.4263	3375.1152	3375.8437
		6	3766.7624	3766.2709	3766.2389	3766.4151
		8	3986.8089	3986.3177	3984.3164	3986.7295
	Image4(003579)	2	1765.9298	1765.3507	1765.1458	1765.4344
		4	2276.9954	2276.8173	2276.3680	2276.9576
		6	3877.9674	3877.1473	3877.1218	3877.2627
		8	5090.7653	5090.5032	5090.2623	5090.6338
	Image5(004423)	2	1721.8701	1721.1817	1721.1124	1721.2721
		4	2551.9400	2551.7442	2551.4594	2551.8431
		6	2968.9682	2968.7638	2968.4520	2968.7339
		8	3859.5790	3859.1403	3857.1300	3859.3798
	Image6(006404)	2	2972.7388	2972.2542	2972.1044	2972.4020
		4	3387.7543	3387.3168	3387.2231	3387.6870
		6	4142.6126	4142.4051	4142.3022	4142.4612
		8	5984.9055	5984.7659	5984.3470	5984.8045

Table 7. Comparison of PSNR in Images g–l.

Metric Name	Image	k	WOA	BOA	PSO	OURS
PSNR	Image7(001236)	2	13.8293	14.5503	15.7167	15.7977
		4	14.1970	14.6582	15.6621	15.6767
		6	17.4146	17.5859	18.3812	20.5188
		8	18.2980	18.1377	19.1016	21.6592
	Image8(001876)	2	13.4009	14.5790	14.8465	14.7751
		4	14.5690	15.6121	15.6524	16.4362
		6	16.4996	16.7998	17.5997	19.7598
		8	17.9384	18.6226	19.5498	21.5051
	Image9(002036)	2	13.5557	14.4696	14.6034	14.7418
		4	15.5611	15.1427	16.4177	16.3583
		6	16.5306	17.3650	18.3565	19.1878
		8	18.4281	18.5254	20.5951	22.6777
	Image10(004231)	2	13.6632	14.3337	15.2033	15.9469
		4	14.2989	14.6908	15.5551	15.2502
		6	15.1205	16.5671	16.7517	17.4736
		8	15.7529	16.7325	17.2486	18.2805
	Image11(004610)	2	14.7697	14.9273	15.6025	16.4122
		4	14.8004	15.4323	15.8120	16.9533
		6	16.1080	17.5817	19.8235	20.4539
		8	18.6965	19.8851	20.8712	21.7442
	Image12(006946)	2	11.6113	12.9387	13.8889	13.3067
		4	12.3074	12.4025	14.6690	14.1064
		6	15.1121	15.2394	16.3323	17.7174
		8	16.7195	16.8900	17.9451	18.3580

Table 8. Comparison of SSIM in Images g–l.

Metric Name	Image	k	WOA	BOA	PSO	OURS
SSIM	Image7(001236)	2	0.5571	0.5861	0.5875	0.6273
		4	0.6012	0.6302	0.6561	0.6567
		6	0.6231	0.6519	0.7253	0.7461
		8	0.6685	0.6918	0.7335	0.7824
	Image8(001876)	2	0.4474	0.4610	0.4619	0.4849
		4	0.4971	0.5300	0.5719	0.5826
		6	0.5496	0.5826	0.6138	0.6611
		8	0.6215	0.6291	0.6736	0.6884
	Image9(002036)	2	0.3660	0.3920	0.4136	0.4650
		4	0.3878	0.4012	0.4896	0.5109
		6	0.4127	0.4322	0.5483	0.5705
		8	0.4818	0.4845	0.5912	0.6318
	Image10(004231)	2	0.3309	0.3387	0.3567	0.3671
		4	0.3726	0.3862	0.4269	0.4262
		6	0.4245	0.4527	0.4565	0.5049
		8	0.4708	0.5428	0.5565	0.5778
	Image11(004610)	2	0.6188	0.6238	0.6454	0.6840
		4	0.6297	0.6632	0.6925	0.6918
		6	0.7249	0.7650	0.7693	0.7891
		8	0.7368	0.7728	0.7881	0.8017
	Image12(006946)	2	0.4310	0.4429	0.4523	0.5013
		4	0.4984	0.5204	0.5363	0.5699
		6	0.5377	0.6239	0.6737	0.6875
		8	0.6507	0.6918	0.7303	0.7471

Table 9. Comparison of FSIM in Images g–l.

Metric Name	Image	k	WOA	BOA	PSO	OURS
FSIM	Image7(001236)	2	0.7066	0.7277	0.7510	0.7889
		4	0.7397	0.7645	0.7916	0.8246
		6	0.7560	0.8089	0.8433	0.8760
		8	0.8030	0.8279	0.8660	0.8982
	Image8(001876)	2	0.6461	0.6516	0.6747	0.6850
		4	0.6642	0.6737	0.6927	0.7071
		6	0.6823	0.6957	0.7182	0.7419
		8	0.7128	0.7246	0.7584	0.8142
	Image9(002036)	2	0.6726	0.6949	0.7191	0.7347
		4	0.6938	0.7144	0.7341	0.7694
		6	0.7220	0.7379	0.7550	0.7977
		8	0.7421	0.7518	0.7796	0.8084
	Image10(004231)	2	0.6381	0.6679	0.6980	0.7130
		4	0.6526	0.6729	0.7111	0.7368
		6	0.6932	0.7274	0.7510	0.7787
		8	0.7349	0.7564	0.7860	0.8072
	Image11(004610)	2	0.7283	0.7484	0.7585	0.7467
		4	0.7379	0.7513	0.7685	0.7760
		6	0.7451	0.7710	0.7819	0.8029
		8	0.7670	0.7850	0.8129	0.8558
	Image12(006946)	2	0.7375	0.7582	0.7955	0.8250
		4	0.7541	0.7763	0.8152	0.8430
		6	0.7796	0.8097	0.8421	0.8654
		8	0.8016	0.8217	0.8427	0.8693

Table 10. Comparison of Fitness in Images g–l.

Metric Name	Image	k	WOA	BOA	PSO	OURS
Fitness	Image7(001236)	2	2750.8992	2750.7616	2750.1240	2750.7641
		4	3517.7437	3517.7256	3517.3212	3517.7290
		6	4509.9853	4509.7814	4509.5764	4509.7829
		8	6495.5661	6495.5018	6493.2953	6495.5538
	Image8(001876)	2	2338.4961	2338.3209	2338.2888	2338.4402
		4	3251.9846	3251.4460	3251.2548	3251.7075
		6	3637.9811	3637.4199	3637.1219	3637.6122
		8	4735.8050	4735.3475	4734.1491	4735.5701
	Image9(002036)	2	2777.9633	2777.8345	2777.1929	2777.9091
		4	3709.8962	3709.2028	3709.1977	3709.6104
		6	4837.6270	4837.2012	4837.1285	4837.5268
		8	5453.6888	5453.5749	5453.3442	5453.6073
	Image10(004231)	2	1699.8490	1699.3038	1699.1601	1699.7288
		4	2878.5953	2878.2350	2878.1299	2878.2976
		6	3708.6870	3708.4114	3708.4016	3708.4181
		8	4828.7905	4828.5773	4825.4886	4828.9096
	Image11(004610)	2	1343.9485	1343.6048	1343.4410	1343.7570
		4	2561.8768	2561.4219	2561.3565	2561.2494
		6	2905.9755	2905.5645	2905.5418	2905.7108
		8	4101.8497	4101.5807	4100.4235	4101.6487
	Image12(006946)	2	2200.6262	2200.1469	2200.1288	2200.5102
		4	3164.7962	3164.2352	3164.2936	3164.7918
		6	3932.8209	3932.2591	3932.2245	3932.3805
		8	4459.9945	4459.8215	4458.1837	4459.9166

Table 11. Average of 100 image segmentation results.

Metric Name	k	WOA	BOA	PSO	OURS
PSNR	2	13.2343	14.7236	14.9630	15.8237
	4	14.5273	14.9263	15.4328	15.9417
	6	16.3862	16.9732	17.7237	19.6872
	8	18.4687	19.2433	20.2934	21.6278
SSIM	2	0.4327	0.4938	0.5019	0.5823
	4	0.4825	0.5349	0.5835	0.6132
	6	0.5926	0.6419	0.6723	0.6835
	8	0.6324	0.6638	0.6924	0.7247
FSIM	2	0.6842	0.7154	0.7369	0.7437
	4	0.7147	0.7519	0.7830	0.8039
	6	0.7422	0.7751	0.7984	0.8241
	8	0.7645	0.7957	0.8226	0.8469
Running Tinme	2	1.35	1.35	1.35	1.28
	4	1.64	1.65	1.64	1.52
	6	2.85	3.12	2.97	2.25
	8	3.82	4.83	3.94	2.46

When the number of thresholds is small (k = 2 or k = 4), the PSNR, SSIM, and FSIM values of the improved particle swarm optimization algorithm are slightly better than other algorithms. However, after the threshold number increases (k = 6 or k = 8), the advantages of improved particle swarm optimization are reflected. The calculated PSNR, SSIM, and FSIM values were the highest among the four algorithms, which indicates that the segmentation, when the improved particle swarm algorithm solves the optimal threshold combination, has the highest similarity with the original image, and the optimization accuracy performance of the algorithm can be reflected. The algorithm runtime is shown in Figure 4 and Figure 5.

Figure 4. Algorithm running time (Images a–f in Figure 3).

Figure 5. Algorithm running time (Images g–l in Figure 3).

From Table 3, Table 4, Table 5, Table 6, Table 7, Table 8, Table 9 and Table 10, it can be seen that the segmentation effect of the proposed algorithm is better. Image 1 (000019) was selected as an example, and the PSNR value of the image, k = 8, was 21.3771, which was significantly better than other comparison algorithms. The SSIM value of the image was 0.7333, which was also slightly better than other algorithms, indicating that the image distortion is low and ensures the image quality after segmentation. Under the premise of maintaining the segmentation effect, it can be seen from Figure 4 that the running time of the proposed scheme was shorter than that of other algorithms, especially in the case of a large number of thresholds. When k = 8, the algorithm segmentation running time was shortened from 3.7 s to 2.8 s, which decreased by more than 30%, which greatly improved the running speed. To verify the universality of the algorithm in this paper, we randomly selected 100 images from the PASCAL 2012 dataset for parameter comparison tests. The experimental results show that our method ia faster and more accurate than other algorithms in terms of PSNR, SSIM, FSIM, and runtime parameters. Experimental results show that the proposed method can improve the segmentation accuracy and segmentation efficiency. Table 6 and Table 10 show the fitness values obtained by different algorithms under different thresholds. The segmented image is shown in Figure 6 and Figure 7.

Figure 6. When threshold $k=8$, the original image and the segmented image of different algorithms are shown (images a–f in Figure 3).

Figure 7. When threshold $k=8$, the original image and the segmented image of different algorithms are shown (images g–l in Figure 3).

6. Conclusions

This paper proposes an improved particle swarm optimization algorithm and applies it to the field of multi-threshold imaging. A new elite particle search strategy is introduced to narrow the scope of particle search, asynchronous monotone increasing social learning factor and asynchronous monotone decreasing individual learning factor are introduced to balance global and local search, and chaos optimization is introduced to increase population diversity. An optimal multi-threshold combination of image segmentation based on an improved particle swarm optimization algorithm is used for optimization, and multi-threshold image segmentation is realized. The test function proves that the algorithm has good development and exploration. The experimental results show that the proposed algorithm can achieve higher segmentation accuracy and efficiency.

Although our method has a faster speed and higher accuracy in image segmentation, compared with threshold segmentation methods, such as fuzzy entropy, our algorithm has a greater speed improvement, but the segmentation accuracy is slightly insufficient, which is one of the directions we will study in the future. Next, it is important to consider applying the improved particle swarm optimization algorithm to image segmentation and image processing in other fields (such as new material image processing, target detection, lightweight embedded robots, and other fields), study the ability of the algorithm to solve practical problems, and further enhance the practicability of the algorithm.