Salp swarm algorithm based on golden section and adaptive and its application in target tracking

IET Image ProcessingVolume 16, Issue 9 p. 2321-2337 ORIGINAL RESEARCHOpen Access Salp swarm algorithm based on golden section and adaptive and its application in target tracking Zhimin Guo, Zhimin Guo State Grid Henan Electric Power Research Institute, Zhengzhou, P.R. ChinaSearch for more papers by this authorYangyang Tian, Yangyang Tian State Grid Henan Electric Power Research Institute, Zhengzhou, P.R. ChinaSearch for more papers by this authorYuxing Feng, Yuxing Feng College of Electric and Information Engineering, Zhengzhou University of Light Industry, Hengzhou, P.R. ChinaSearch for more papers by this authorHuanlong Zhang, Corresponding Author Huanlong Zhang [email protected] orcid.org/0000-0002-5130-5555 College of Electric and Information Engineering, Zhengzhou University of Light Industry, Hengzhou, P.R. China Correspondence Huanlong Zhang, College of Electric and Information Engineering, Zhengzhou University of Light Industry, Zhengzhou 450002, P.R. China. Email: [email protected]Search for more papers by this authorJunfeng Liu, Junfeng Liu College of Electric and Information Engineering, Zhengzhou University of Light Industry, Hengzhou, P.R. ChinaSearch for more papers by this authorZanfeng Wang, Zanfeng Wang Foxconn Technology Group, Henan Fuchi Technology Co., Ltd, Xinzheng, P.R. ChinaSearch for more papers by this author Zhimin Guo, Zhimin Guo State Grid Henan Electric Power Research Institute, Zhengzhou, P.R. ChinaSearch for more papers by this authorYangyang Tian, Yangyang Tian State Grid Henan Electric Power Research Institute, Zhengzhou, P.R. ChinaSearch for more papers by this authorYuxing Feng, Yuxing Feng College of Electric and Information Engineering, Zhengzhou University of Light Industry, Hengzhou, P.R. ChinaSearch for more papers by this authorHuanlong Zhang, Corresponding Author Huanlong Zhang [email protected] orcid.org/0000-0002-5130-5555 College of Electric and Information Engineering, Zhengzhou University of Light Industry, Hengzhou, P.R. China Correspondence Huanlong Zhang, College of Electric and Information Engineering, Zhengzhou University of Light Industry, Zhengzhou 450002, P.R. China. Email: [email protected]Search for more papers by this authorJunfeng Liu, Junfeng Liu College of Electric and Information Engineering, Zhengzhou University of Light Industry, Hengzhou, P.R. ChinaSearch for more papers by this authorZanfeng Wang, Zanfeng Wang Foxconn Technology Group, Henan Fuchi Technology Co., Ltd, Xinzheng, P.R. ChinaSearch for more papers by this author First published: 29 April 2022 https://doi.org/10.1049/ipr2.12490AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract In order to solve the problem that the conventional tracker is not adapted to the abrupt motion, a tracking algorithm based on the improved salp swarm algorithm (ISSA) was proposed. Visual tracking is considered to be a process of locating the optimal position through the interaction between leaders and followers in successive images. Firstly, the adaptive mechanism of leader and follower is introduced into the original salp swarm algorithm (SSA) to balance the exploitation and exploration of the algorithm. This method can improve the accuracy and effect of tracking. Secondly, the golden-sine algorithm was used to update the position of followers, considering that the SSA had a single spatial search mode for followers and was easy to fall into the local optimum. By comparing with 19 classical tracking algorithms, qualitative and quantitative analysis is carried out to verify the tracking effect of the proposed method. A large number of experimental results show that the algorithm proposed here has good performance in visual tracking, especially for mutation motion tracking. 1 INTRODUCTION Visual tracking refers to the detection, extraction, recognition and tracking of moving objects in image sequences, and further obtains the motion parameters of moving objects for further processing and analysis. In short, it is to locate the moving target in the video. In recent years, vision-based target tracking methods have developed rapidly. The correlation filtering (CF) algorithm is discriminant tracking, which mainly uses the given samples to train a discriminant classifier to judge whether the target is tracked or the surrounding background information. The rotation matrix is used to collect the samples and the fast Fourier transform is used to accelerate the algorithm. Deep learning mostly uses the general target feature expression extracted after off-line training with non-tracking data set for online target tracking. Tracking algorithms based on CF [1, 2] and deep learning [3, 4] have been successfully put forward one after another, which makes a big breakthrough in target tracking. Tracking algorithms based on CF and deep learning have achieved good results in visual tracking. However, most of these methods assume that the target has a smooth motion environment. In the real world, due to the complex tracking environment, the large scale of target movement and the influence of camera imaging function, the target displacement between two adjacent images is very large, which leads to the failure of target tracking. In order to solve the problem of target mutation, we transform the visual tracking problem into an optimization problem. In the visual tracking, each frame in the video sequence is optimized to find the target, so as to realize the target tracking in the video. Optimization algorithms have been widely used in visual tracking. For example, Júnior et al. [5] introduced particle swarm optimization (PSO) into visual tracking, which improved the operation efficiency a lot. Lin et al. [6] used ant colony algorithm +Mean Shift improved particle filtering method for target tracking. It not only improves the timeliness and accuracy of target tracking, but also enhances the artificial intelligence attribute of target tracking. Deng et al. [7] applied non-dominated solution sorting genetic algorithm (NSGA-II) to visual tracking, which well balanced the convergence and diversity of decision space and object space. In addition, Clustering Algorithm [8, 9], Monarch Butterfly Optimization (MBO) [10, 11], Slime Mould Algorithm (SMA) [12, 13], Moth Search Algorithm (MSA) [14, 15], Hunger Games Search (HGS) [16, 17], Runge Kutta Method (RUN) [18, 19], Colony Predation Algorithm (CPA) [20, 21] and Harris Hawks Optimization (HHO) [22, 23] etc., these algorithms have been applied to visual tracking and achieved great success. But up to now, there is no single algorithm that can solve all the problems perfectly. SSA [24] is a new swarm intelligence optimization algorithm proposed by Australian scholars in 2017. The algorithm simulates the behaviour of the sea creature salp. Different from other creatures, salp feeds in the form of ‘chain’, which is connected end to end. And the leader will not directly influence the moving direction of the whole population. This ensures the diversity of the population, which leads to the better results of the algorithm in most optimization problems. Compared with Ant Colony Optimization Algorithm, Genetic Algorithm, Moth-flame Optimization Algorithm and other traditional optimization algorithms, Salp Swarm Algorithm has the advantages of fewer parameters, simple structure, easy implementation and better optimization directivity. SSA has been widely concerned by researchers from all walks of life [25-28]. For example, Zhao et al. [29] proposed a new SSA that introduced adaptive control parameter and elite grey wolf domination strategy, which effectively solved the problems that original optimizer was ease to fall into local optimum and low convergence accuracy. In image segmentation, Wang et al. [30] applied SSA in multi-threshold image segmentation (MIS). Zhang et al. [31] designed an effective segmentation method based on a non-local mean two-dimensional (2D) histogram and 2D Kapur's entropy called SSA with Gaussian Barebone and Stochastic Fractal Search (GBSFSSSA) by combining GBSFSSSA mechanism. However, the SSA also has some shortcomings, such as being unable to search accurately in the later iteration period, poor local development ability and global exploration ability, slow convergence speed and so on. In order to solve the above problems, we improved salp swarm algorithm (SSA) in Section 2. In the ISSA, the idea of golden sine is introduced to update the follower's position. In addition, in order to enhance the development and exploration ability of the algorithm, the algorithm introduces the mechanism of adaptive change of the number of leaders and followers. In Section 3, we apply the ISSA to video tracking and introduce correlation filter to enhance the tracking effect. The fourth part discusses the improved algorithm and the visual tracking effect of the tracker. Finally, the contents of this section is summarized. The innovation of this study includes the following three aspects: (1) a new optimization algorithm ISSA is proposed based on SSA, which has better optimization effect and can effectively avoid local optimization. ISSA is well adapted to the problem of mutant motion. (2) The ISSA is applied to video tracking, and the CF operation is used for the optimal position generated by the iteration, which improves the robustness and efficiency of tracking. (3) The validity of the algorithm is verified by 19 benchmark functions. Extensive experimental results show that compared with SSA tracker, CSK tracker and seven state-of-the-art trackers, the proposed tracker has stronger advantages, especially for mutation motion tracking. 2 SALP SWARM ALGORITHM AND ITS IMPROVEMENT Salp swarm algorithm (SSA) SSA is a new intelligent optimization algorithm put forward by Mirjalili et al. [32] in 2017. SSA simulates the behaviour of salp community in the ocean to find food. In the deep sea, salp chain is formed continuously by the salp group. There are two kinds of salps: leader and follower. The leader's position update is influenced by the food source, and the leader will guide the follower to move and get closer to the food source. The salp leader updates the position P j i = F j + c 1 u b j − l b j c 2 + l b j , c 3 ≥ 0.5 F j − c 1 u b j − l b j c 2 + l b j , c 3 < 0.5 ( i = 1 , 2 , 3 , … , L 1 ) ( j = 1 , 2 , 3 , … , D ) \begin{eqnarray} P_j^i = \left\{ \def\eqcellsep{&}\begin{array}{l} {F_j} + {c_1}\left( {\left( {u{b_j} - l{b_j}} \right){c_2} + l{b_j}} \right),{c_3} \ge 0.5\\[9pt] {F_j} - {c_1}\left( {\left( {u{b_j} - l{b_j}} \right){c_2} + l{b_j}} \right),{c_3} < 0.5 \end{array} \right\}\nonumber\\ (i = 1,2,3, \ldots ,L1)(j = 1,2,3, \ldots ,D) \end{eqnarray} (1) P j i $P_j^i$ indicates the position of the leader in j $j$ dimension space. L 1 $L1$ represents the total number of leaders. u b j $u{b_j}$ and l b j $l{b_j}$ are expressed as the upper and lower bounds of the j dimension space. F j ${F_j}$ indicates the location of the food source. c 2 ${c_2}$ and c 3 ${c_3}$ are random numbers in the range 0 to 1. c 2 ${c_2}$ determines the leader's moving step size in the j-dimensional space and c 3 ${c_3}$ determines the leader's moving direction (forward or backward). c 1 = a e − ( b l L ) 2 ( a = 2 , b = 4 ) \begin{equation}{c_1} = a{e^{ - {{(\frac{{bl}}{L})}^2}}}(a = 2,b = 4)\end{equation} (2) l $l$ is the current iteration number and L $L$ represents the maximum iteration number. a $a$ and b $b$ is are the parameters in Equation 4. As the number of iterations increases, c 1 ${c_1}$ gradually decreases from 2 to 0, and c 1 ${c_1}$ balances the exploration and development of the algorithm. The salp follower updates the position. According to Newton's equation Δ P j i = 1 2 a t 2 + v 0 t ( i = L 1 + 1 , L 1 + 2 , … , n ) ( j = 1 , 2 , 3 , … , D ) \begin{equation}\Delta P_j^i = \frac{1}{2}a{t^2} + {v_0}t(i = L1 + 1,L1 + 2, \ldots ,n)(j = 1,2,3, \ldots ,D)\end{equation} (3) Δ P j i $\Delta P_j^i$ indicates the movement displacement of the ith follower in the jth dimension, t $t$ represents the optimization time, and the difference between iterations is 1, that is t = 1 $t = 1$ . v 0 ${v_0}$ is the initial velocity, v 0 = 0 ${v_0} = 0$ , acceleration a = v − v 0 t $a = \frac{{v - {v_0}}}{t}$ and velocity v = p j i − 1 − p j i t $v = \frac{{p_j^{i - 1} - p_j^i}}{t}$ , and n $n$ is the total number of salp. L 1 $L1$ represents the total number of leaders. Equation (3) can be simplified as:(see 4) Δ P j i = 1 2 ( P j i − 1 − P j i ) ( i = L 1 + 1 , L 1 + 2 , … , n ) ( j = 1 , 2 , 3 , … , D ) \begin{eqnarray} \Delta P_j^i &=& \frac{1}{2}(P_j^{i - 1} - P_j^i)(i = L1 + 1,L1 + 2,\ldots,n)\nonumber\\ &&\qquad\qquad\qquad\qquad(j = 1,2,3,\ldots,D) \end{eqnarray} (4) Then the follower's position update formula is P j n i = P j i + Δ P j i = 1 2 ( P j i − 1 + P j i ) ( i = L 1 + 1 , L 1 + 2 , … , n ) ( j = 1 , 2 , 3 , … , D ) \begin{eqnarray} \mathop P\limits _j^{ni} = P_j^i + \Delta P_j^i &=& \frac{1}{2}(P_j^{i - 1} + P_j^i)(i = L1 + 1,L1 + 2,\ldots,n)\nonumber\\ &&\qquad\qquad\qquad\quad(j = 1,2,3,\ldots,D) \end{eqnarray} (5) P j n i $P_j^{ni}$ is the position of the follower obtained from this iteration, P j i $P_j^i$ and P j i − 1 $P_j^{i - 1}$ are the positions of the i $i$ and i − 1 $i - 1$ followers in the last iteration, respectively. SSA has very few control parameters and is very exploratory. Parameters c 2 ${c_2}$ and c 3 ${c_3}$ are random numbers from 0 to 1, no external control is required, c 1 ${c_1}$ only needs to be given the value of a and b at the beginning. However, there are some problems, for example, the follower's search method is single, and it is easy to fall into local optimization in the search process. In order to solve these problems, this study introduces golden section and self-adaptive change mechanism to improve the original SSA. Improvement for salp swarm algorithm (ISSA) 2.2.1 Updating follower's position based on golden sine In the SSA, the position of followers in each iteration is the average of the current position and the previous position in the previous iteration. Within the local search range of the follower, the search mode is relatively single and the motion mode is linear motion. In this way, the follower's partial exploration is not sufficient, and the optimal value in the surrounding space may be missed, which leads to slow optimization convergence speed and may reduce the search accuracy of the algorithm for the optimal target position. Therefore, in order to improve the search diversity of the follower, avoid falling into the local minimum and improve the search ability of the algorithm, the golden sine algorithm is used to modify the position updating formula of the follower. Golden Sine Algorithm [33] is a new meta-heuristic search algorithm based on population inspired by mathematics. Its mathematical idea stems from the characteristic of sine function in trigonometric function, that is, the unit circle of sine function value can be searched by using sine function. It can be seen from Figure 1 that the golden sine algorithm explores the search space in a sinusoidal manner, and its effect is to traverse all the values on the corresponding unit circle. In addition, combined with the golden section search method, the search space is adaptively reduced in the iterative process, so that the particles in the algorithm gradually approach the optimal position. Through the combination of these two points, the golden sine algorithm is put forward, which has the advantages of fast convergence speed, less adjustment parameters and good robustness. FIGURE 1Open in figure viewerPowerPoint Golden sine algorithm Using the idea of golden sine algorithm, we can first use the golden section coefficient to carry out the golden section on the space around the follower, and fully explore the space by using the sine function search method. Considering the original mechanism of SSA, we compare the fitness value of the last iteration particle with that of the previous one, that is, evaluate the fitness of the follower, so as to get a better particle position. Guided by this, we explore in the search space in a sinusoidal way to get the local best position. The improved follower position formula is shown in (6) P j n i = P j i ∗ | sin ( r 1 ) | + r 2 ∗ sin ( r 1 ) ∗ | x 1 ∗ P j i − 1 − x 2 ∗ P j i | , f ( P j i − 1 ) < f ( P j i ) P j i ∗ | sin ( r 1 ) | + r 2 ∗ sin ( r 1 ) ∗ | x 1 ∗ P j i − x 2 ∗ P j i − 1 | , f ( P j i − 1 ) > f ( P j i ) \begin{equation} \mathop P\limits _j^{ni} = \left\{ \def\eqcellsep{&}\begin{array}{l} P_j^i*|\sin ({r_1})| + {r_2}*\sin ({r_1})*|{x_1}*P_j^{i - 1} - {x_2}*P_j^i|,\\[6pt] \qquad\qquad\qquad\qquad\qquad\qquad\qquad{\rm{ }}f(P_j^{i - 1}) < f(P_j^i)\\[9pt] P_j^i*|\sin ({r_1})| + {r_2}*\sin ({r_1})*|{x_1}*P_j^i - {x_2}*P_j^{i - 1}|,\\[6pt] \qquad\qquad\qquad\qquad\qquad\qquad\qquad{\rm{ }}f(P_j^{i - 1}) > f(P_j^i) \end{array} \right. \end{equation} (6) P j n i $P_j^{ni}$ is the position of the i $i$ follower in the j $j$ dimension space in this iteration. P j i $P_j^i$ is the position of the i $i$ follower in the last iteration. P j i − 1 $P_j^{i - 1}$ represents the position of the i − 1 $i - 1$ follower in the last iteration. r 1 ${r_1}$ is a random number from 0 to 2 π $2\pi $ , which determines the moving direction. r 2 ${r_2}$ is a random number from 0 to π $\pi $ , which determines the moving length of the particle. x 1 ${x_1}$ and x 2 ${x_2}$ are golden section coefficients. ( g o l d = ( 5 − 1 ) / 2 $gold = (\sqrt 5 - 1)/2$ , x 1 = − π + ( 1 − g o l d ) × 2 π ${x_1} = - \pi + (1 - gold) \times 2\pi $ , x 2 = − π + g o l d × 2 π ${x_2} = - \pi + gold \times 2\pi $ ) The introduction of golden sine algorithm enables the followers to fully develop the surrounding environment in local development. Figure 2 shows the convergence curve of salps population in the iterative process. According to the comparison between the ISSA and the original SSA in Figure 2, we can see that the SSA with the addition of golden sine not only improves the convergence accuracy a lot, but also achieves high accuracy in a few iterations. FIGURE 2Open in figure viewerPowerPoint Convergence curves 2.2.2 Introduction of adaptive change mechanism In the process of optimization, the leader plays the role of global exploration in the early stage and the follower plays the role of local development in the later stage. In the original SSA, the number of leaders and followers is always fixed, which is half of the total number of salps. When the algorithm conducts global exploration, it needs more leaders, while when it conducts local development in the later stage, it needs more followers. For this reason, we put forward the adaptive change mechanism of leaders and followers. The specific formula is N _ leader = c e i l ( N × ( 1 − ( l − 1 ) / max _ iter ) ) \begin{equation}N\_{\rm{leader}} = ceil(N \times (1 - (l - 1)/\max \_{\rm{iter}}))\end{equation} (7) N $N$ indicates the total number of bottles, l $l$ is the current iteration number, max _ iter $\max \_{\rm{iter}}$ represents the maximum iteration number. N _ leader $N{\rm{\_leader}}$ indicates the number of the leader. The c e i l ( x ) $ceil( x )$ function is used to find the smallest integer not less than x. With the increase in iteration times, the number of leaders decreases gradually, and the total number of salps remains unchanged, which makes the optimization algorithm more comprehensive exploration in the early stage and more detailed development in the later stage. 2.2.3 Steps of the improved salp swarm algorithm In the improved salp group algorithm, golden section algorithm and adaptive change mechanism are introduced to improve the algorithm. The adaptive mechanism is used to balance the development and exploration ability of the algorithm, and the golden section algorithm is used to improve the search diversity of followers, thus improving the search ability of the algorithm. Exploration and development are two important parts of optimization algorithm. Balancing the exploration and development ability of salp group algorithm is the key to improve the convergence speed and accuracy of the algorithm. The ISSA used golden-sine algorithm to update the position of followers, which solved the problem that the spatial search mode of followers was single and easy to fall into local optimization. To sum up, the improved algorithm has a good search effect. The ISSA pseudo-code is shown in Figure 3. FIGURE 3Open in figure viewerPowerPoint The pseudo-code of the improved salp swarm algorithm 3 VISUAL TRACKING BASED ON IMPROVED SALP SALP ALGORITHM In the process of visual tracking, there are often sudden changes or rapid movements of objects. In older to solve this problem, it has been suggested that the tracking problem should be transformed into an optimization problem. Visual tracking is considered to be a process of finding the best solution in continuous images. Taking ISSA as an example, this study puts forward a tracking framework, as shown in Figure 4. FIGURE 4Open in figure viewerPowerPoint The flowchart of the ISSA tracker. ISSA, improved salp swarm algorithm As shown in Figure 4, the user first selects the target in the first frame and initializes the state vector X = [x, y], where [x, y] represents the position of the target pixel coordinates. According to the dynamic model, the predicted position of the target in the next frame is given. Then, the characteristics of candidate samples and templates generated by ISSA are extracted. The target candidate with the maximum similarity value is found through the similarity function, that is the target. Calculation of fitness value Fitness function is a function used to calculate the similarity value between target and candidate target block. In addition, the hog [34] will represent the gradient amplitude and direction of each pixel in an image, so it can have better effects on illumination changes and shadows. The similarity calculation method is as follows: ρ ( X , Y ) = C o v ( X , Y ) D ( X ) D ( Y ) \begin{equation}\rho (X,Y) = \frac{{Cov(X,Y)}}{{\sqrt {D(X)} \sqrt {D(Y)} }}\end{equation} (8)Of them, X $X$ and Y $Y$ are respectively expressed as HOG features of target blocks and candidate target blocks. ρ ( X , Y ) $\rho (X,Y)$ represents the correlation coefficient X $X$ with Y $Y$ , C o v ( · ) $Cov( \cdot )$ represents covariance and D ( · ) $D( \cdot )$ refers to variance. The fitness function is E = 2 + 2 × ρ ( X , Y ) \begin{equation}E = 2 + 2 \times \rho (X,Y)\end{equation} (9) Video tracking with correlation filtering (CFSSA) In recent years, CF has become very popular in the field of target tracking, and various algorithms based on CF have appeared one after another. The basic idea of CF is to design a filtering template, and use this template to do correlation operation with the target candidate region, and the position of the maximum output response is the target position of the current frame. In SSA and ISSA mentioned in the second part, the position of food source plays a key role, and its correctness will directly affect the update of leaders and indirectly affect the update of followers' position. In order to get a more accurate position of food source, the convergence speed and tracking effect of the algorithm are improved. We introduce CF operation in target tracking. The introduction of convolution operator in the filter transforms the spatial correlation filter into the frequency domain filter, which improves the operation speed. In addition, the filter generates a cyclic matrix by cyclic sampling at the target location, so that a large number of training samples can be obtained by intensive sampling at the Achieve the goal location, and the resolution of the classifier is greatly improved. Based on this, we introduce CF operation in ISSA, so as to further optimize the position of food source locally and obtain a more accurate position of food source. The specific process is shown in Figure 5. FIGURE 5Open in figure viewerPowerPoint Flowchart of algorithm tracking 4 SIMULATION EXPERIMENT AND ANALYSIS In order to verify the feasibility of the ideas proposed in this chapter, the visual tracking effects of ISSA and CF are analyzed and discussed in experiments. Optimization algorithm effect analysis 4.1.1 Time complexity analysis In computer science, we usually use time complexity to describe the running time of an algorithm. Time complexity is an important index to measure the efficiency of an algorithm. The ISSA is analyzed by the big O asymptotic method. The time complexity of the ISSA is the same as that of the original SSA. The specific experimental results are shown in Table B.1. Under the condition of constant time complexity, we improve the efficiency and accuracy of the algorithm. Since the number of algorithm cycles in the iterative process is related to the spatial dimension of the test function, we added the spatial dimension dim in Table B.1. 4.1.2 Benchmark test experiment In this chapter, the improved Salp Swarm Algorithm is tested. Relevant information of these 19 functions, such as specific functions, variable dimensions and upper and lower limits, are shown in Appendix 1. Table A.1, Table A.2 and Table A.3 in Appendix 1 correspond to the details of single peak, multi-peak high-dimensional and low-dimensional multi-peak benchmark functions of 19 functions respectively. In order to compare, in 19 benchmark test functions, the ISSA, SSA, chaotic salp swarm algorithm (CSSA) [35], moth-flame optimization algorithm (MFO) [36], grasshopper optimization algorithm (GOA) [37] and ant lion optimization (ALO) [38] 30 times simulation experiment was carried out. The experimental results include the average and standard deviation of the optimal values obtained 30 times. We use the same maximum iteration times and population size as other algorithms to verify the efficiency of this algorithm (maximum iteration times t (t = 1000), population size n (n = 30)). In order to ensure the transparency of the experiment, the corresponding parameters of each algorithm are listed. The experimental parameters of each algorithm are shown in Table B.2 (in Appendix 2). After 30 independent runs, the experimental results are shown in Table B.3 (in Appendix 2), and the best results are expressed in bold black font. Among the benchmark functions, F1–F7 are uni-modal functions, and they have no local minimum. Uni-modal functions are often used to test the ability of algorithm development and exploration.F8–F13 are multimodal and high-dimensional test functions with multiple local minima, which are suitable for detecting the ability of the algorithm to jump out of the local optimal solution. The low dimensional multi-peak test function of F14–F19 is very close to the real search space. As can be seen from Table B.3 (in Appendix 2), when the maximum number of iterations is 1000 and the population size is 30, the optimization accuracy of the ISSA is better than that of the original SSA in 19 test functions. Among the test functions of F1–F13, the ISSA has 12 test functions that are superior to the comparison algorithm. Although the low-dimensional multi-peak reference function of the ISSA is similar to other algorithms, the algorithm still has good optimization accuracy. In conclusion, compared with the original SSA, the ISSA has better experimental results. 4.1.3 Statistical analysis The ISSA used golden sine algorithm to update the position of followers, which solved the problem that the spatial search mode of followers was single and easily fell into local optimization. In addition, the introduction of leader and follower adaptation mechanism balances the algorithm development and exploration, and improves the optimization accuracy of the algorithm. As can be seen from Figure 6, among the 19 test functions, the ISSA has the largest number of optimal experimental results of searching accuracy and standard deviation. Among them, the number of optimal results of searching accuracy reaches 17, accounting for 89% of the total number of functions. In addition, the ISSA also has the largest number of optimal results of standard deviation. Therefore, compared with the other five algorithms (SSA,CSSA, MFO, ALO and GOA), the ISSA has better stability. These results show that the ISSA is feasible and effective. FIGURE 6Open in figure viewerPowerPoint The optimal result number of 19 functions Tracking effect analysis In this work, we analyze the tracking effect of the tracker proposed from