IMATSA – an improved and adaptive intelligent optimization algorithm based on tunicate swarm algorithm

Abstract

Swarm intelligence optimization algorithm has been proved to perform well in the field of parameter optimization. In order to further improve the performance of intelligent optimization algorithm, this paper proposes an improved and adaptive tunicate swarm algorithm (IMATSA) based on tunicate swarm algorithm (TSA). IMATSA improves TSA in the following four aspects: population diversity, local search convergence speed, jumping out of local optimal position, and balancing global and local search. Firstly, IMATSA adopts Tent map and quadratic interpolation to initialize population and enhance the diversity. Secondly, IMATSA uses Golden-Sine algorithm to accelerate the convergence of local search. Thirdly, in the process of global development, IMATSA adopts Levy flight and the improved Gauss disturbance method to adaptively improves and coordinates the ability of global development and local search. Then, this paper verifies the performance of IMATSA based on 14 benchmark functions experiment, ablation experiment, parameter optimization experiments of Support Vector Machine (SVM) and Gradient Boosting Decision Tree (GBDT), Wilcoxon signed rank test and image multi-threshold segmentation experiment with the performance metrics are convergence speed, convergence value, significance level P-value, Peak Signal-to-Noise Ratio (PSNR) and Standard Deviation (STD). Experimental results show that IMATSA performs better in three kinds of benchmark functions; each component of IMATSA has a positive effect on the performance; IMATSA performs better in parameter optimization experiments of SVM experiment and GBDT; there is significant difference between IMATSA and other algorithms by Wilcoxon signed rank test; in image segmentation, the performance is directly proportional to the number of thresholds, and compared with other algorithms, IMATSA has better comprehensive performance.

Keywords

Intelligent optimization algorithm optimization tent map quadratic interpolation Golden-Sine levy flight improved Gauss disturbance

1. Introduction

Intelligent optimization algorithm [8,9,13] has been proved to be effective in solving parameter optimization problems [33,34,36], image segmentation [17,31], and other engineering application fields. Intelligent optimization algorithm has a good research and development prospect, including sooty tern optimization algorithm (STOA) [1,7], sailed fish optimizer (SFO) [20,28], grey wolf optimization (GWO) [25,30], butterfly optimization algorithm (BOA) [3,37], supply-demand-based optimization (SDO) [42], pathfinder algorithm (PFA) [40], whale optimization algorithm (WOA) [23] and other optimization algorithms, etc.

Kaur et al. [15] proposed tunicate swarm algorithm (TSA) in 2020. Kaur compared TSA with other swarm intelligence optimization algorithms, and the results show that TSA can find the optimal solution quickly. Based on gradient free optimization technology, TSA has strong development ability and performs well in the field of function optimization. However, the search method of TSA is lack of flexibility and mobility, so it is difficult to jump out of the local optimal position. By directly aggregating to the global optimum, the global search of TSA is insufficient. In the follow-up research, some scholars applied TSA to different research fields and achieved some results. In solving the problem of unknown search points, Sharma [29] used TSA to estimate the optimal value of unknown parameters of photovoltaic cell modules under standard temperature conditions, which effectively proves the superiority of TSA in extracting optimal parameters from photovoltaic cells. Daniel [6] applied TSA algorithm to feature selection to solve the scalability problem caused by large feature set. Fetouh [10] applied TSA to the distribution system to find the optimal solution to provide quantitative and qualitative power services and reduce energy consumption. In order to solve the problems of insufficient global exploration ability, low optimization accuracy and slow convergence speed of TSA, Li [19] improved the TSA algorithm by using chaos initialization and global search vector, then applied it to solve the dynamic economic emission scheduling problem. Houssein [12] integrated TSA with a local escaping operator (LEO) for overcoming the weaknesses of the original TSA, and proposed TSA-LEO. The LEO strategy in TSA-LEO prevents searching deflation in TSA and improves the convergence rate and local search efficiency of swarm agents.

Based on gradient free optimization technology, TSA has strong development capability and performs well in the field of function optimization. However, the search method of TSA lacks flexibility and mobility, and it is difficult to jump out of the local optimal position. Moreover, as a result of using the method of direct aggregation to the global optimal location, the global search of TSA is insufficient. In order to further improve the performance of TSA, this paper proposes IMATSA and improves TSA in four aspects: population diversity, local search convergence speed, ability to jump out of local optimal position, and balancing global and local search. Firstly, IMATSA proposes a method of combining Tent map and quadratic interpolation for population initialization to improve population diversity; secondly, IMATSA uses Golden-Sine algorithm to accelerate the convergence of local search and improve the local search capability; thirdly, IMATSA adopts Levy flight and the improved Gauss disturbance method to disturd the optimal position, which improves the ability of the algorithm to jump out of the local optimal position, and effectively balances the global and local search. Then, this paper applys IMATSA in the fields of image segmentation and other parameter optimization problems.

In order to verifies the performance of IMATSA, this paper designs a variety of experiments, including benchmark functions experiment, ablation experiment, parameter optimization experiment of Support Vector Machine (SVM), parameter optimization experiment of Gradient Boosting Decision Tree (GBDT), Wilcoxon signed rank test and image multi-threshold segmentation experiment. The performance metrics of benchmark functions experiment, ablation experiment and parameter optimization experiments are convergence speed and convergence value; the performance metrics of Wilcoxon signed rank test are convergence value and significance level P-value; and the performance metrics of image multi-threshold segmentation experiment are Peak Signal-to-Noise Ratio (PSNR) and Standard Deviation (STD).

Experiments show that IMATSA performs better in benchmark function experiment, and the experimental results are ahead of most orders of magnitude; each improvement has a certain positive effect; in the parameter optimization experiments of SVM and GBDT, the convergence values of IMATSA are better; the value of significance level P-value in Wilcoxon signed rank test are all less than 0.05, and there is significant difference between IMATSA and other algorithms; in image multi threshold segmentation experiments, IMATSA has better PSNR and MEAN in most scenes, and the performance is directly proportional to the number of thresholds.

2. Methods

2.1. Tunicate swarm algorithm

TSA simulates the jet propulsion behavior and group behavior of tunicates, and the theory is novel. Jet propulsion behavior is mainly composed of three parts: avoiding individual position conflict, approaching the best individual and moving to the best individual. In group behavior, the tunicates perceive the changes of water flow through the peripheral nerves, determine their positions based on the light emitted by their companions, and gather together to the target food source to achieve the effect of group foraging. The algorithm can truly simulate the foraging behavior of tunicates in the ocean, and has certain optimization ability and accuracy.

2.1.1. Jet propulsion behavior

Before jet propulsion, the tunicates need to avoid the conflict of individuals. In order to avoid the conflict, a new position vector will be calculated, and the functional equations are given as follows: $\begin{array}{c} (1) & \vec{A} = \frac{\vec{G}}{\vec{M}} \\ (2) & \vec{G} = c_{2} + c_{3} - \vec{F} \\ (3) & \vec{F} = 2 \cdot c_{1} \end{array}$

Where $\vec{G}$ represents gravity, $c_{1}$ , $c_{2}$ , $c_{3}$ are random numbers belonging to $[0, 1]$ , and $\vec{F}$ represents water advection in the deep sea. $\vec{M}$ is the force between individuals of the population, and the functional equation is given as follow: $\begin{matrix} (4) & \vec{M} = ⌊ P_{min} + c_{1} \cdot (P_{max} - P_{min}) ⌋ \end{matrix}$

Where $P_{min}$ represents the minimum value of individual action speed, $P_{max}$ represents the maximum value, $c_{1}$ is the random number belonging to $[0, 1]$ , $P_{min}$ and $P_{max}$ are set to 1 and 4.

After avoiding the conflict of domain search location, the individual moves towards the food source. In this process, the distance between the individual and the food source position needs to be calculated. The functional equation is given as follow: $\begin{matrix} (5) & \vec{{PD}_{i}^{t}} = | \vec{{FS}_{best}^{t}} - rand \cdot \vec{P {(x)}_{i}^{t}} | \end{matrix}$

Where t represents the number of iterations, $\vec{{PD}_{i}^{t}}$ represents the distance between the i individual and the food source at the t iteration, $\vec{{FS}_{best}^{t}}$ is the optimal position after the t iteration, $rand$ is the random number belonging to $[0, 1]$ , and $\vec{P {(x)}_{i}^{t}}$ represents the position of the i individual at the t iteration.

The search individual keeps moving to the position of food source. The functional equation is given as follow: $\begin{matrix} (6) & \vec{P {(x^{'})}_{i}^{t}} = \{\begin{array}{ll} \vec{{FS}_{best}^{t}} + \vec{A} \cdot \vec{{PD}_{i}^{t}}, & rand ⩾ 0.5 \\ \vec{{FS}_{best}^{t}} - \vec{A} \cdot \vec{{PD}_{i}^{t}}, & rand < 0.5 \end{array} \end{matrix}$

Where $\vec{{FS}_{best}^{t}}$ is the food source and optimal position of the population in the t iteration, $\vec{{PD}_{i}^{t}}$ represents the distance between the i individual and the food source, $\vec{A}$ is the position vector calculated in the step of avoiding individual conflict, and $\vec{P {(x^{'})}_{i}^{t}}$ represents the position after calculation and update.

2.1.2. Group behavior

After avoiding individual conflict and calculating the distance between the search individual and the food source, the individual converges to the food source position based on the group behavior, and adjusts the individual position according to the food source. The functional equation is given as follow: $\begin{matrix} (7) & \vec{P {(x)}_{i}^{t + 1}} = \frac{\vec{P {(x)}_{i}^{t}} + \vec{P {(x)}_{i}^{t - 1}}}{2 + c_{1}} \end{matrix}$

Where $\vec{P {(x)}_{i}^{t - 1}}$ represents the distance between the individual and the food source after $t - 1$ iteration, $c_{1}$ is a random number belonging to $[0, 1]$ .

2.2. Improvements

2.2.1. Tent map and quadratic interpolation

In the later stage of optimization, the population diversity of TSA decreases and the probability of falling into the local optimal position increases, resulting in insufficient convergence accuracy. In this paper, a Tent map and quadratic interpolation method is proposed to initialize the population and enhance the population diversity.

Chaotic map is based on nonlinear theory, which means that in a deterministic system, there is a seemingly random irregular motion. It can traverse all states according to the characteristics in a certain range, with the characteristics of ergodicity, nonlinearity, randomness and universality.

At present, logistic map and Tent map are the most common chaotic map methods. Research [21] shows that Tent map has better ergodicity than logistic map. Tent map is a piecewise linear map in the field of mathematics, with good correlation and uniform distribution function. The chaotic sequence of Tent map has uniform interval distribution and fast iteration speed. The functional equation of Tent map is given as follow: $\begin{matrix} (8) & x_{n + 1} = \{\begin{array}{ll} x_{n} / α, & x_{n} \in [0, α) \\ (1 - x_{n}) / (1 - α), & x_{n} \in [α, 1] \end{array} t = 0, 1, 2, 3, \dots, T \end{matrix}$ where $x_{n}$ is the chaotic value generated during iterations, α is the constant belonging to $[0, 1]$ , T is the maximum number of iterations. In this paper, $α = 0.75$ .

Figure 1 depicted the sequence distribution histogram of Tent map, in which the maximum number of iterations is 500 and the number of intervals is 20. As can be seen from Fig. 1, the Tent map is evenly distributed.

Fig. 1.

Tent map sequence distribution histogram.

After initializing the population with Tent map, this paper uses quadratic interpolation (QI) method to create new individuals to enhance the diversity and make the distribution more uniform. Experimental results [32] show that the application of quadratic interpolation as a local search operator to meta heuristic algorithm can effectively enhance the local search ability of the applied algorithm.

QI uses the curve to fit the shape of the quadratic function. It is a local search operator to fit the optimal value of the function with the calculated curve extreme points.

Supposing X and Y are the two random positions of the population in the D dimension problem, the current global optimal position is Z, the functional equation of parabola passing through $(x_{q}, F (X))$ , $(y_{q}, F (Y))$ and $(z_{q}, F (Z))$ is as follow: $\begin{matrix} (9) & p (t) = a_{0} + a_{1} t + a_{2} t^{2}, a_{2} \neq 0 \end{matrix}$

$X = (x_{1}, x_{2}, \dots, x_{D})$ , $Y = (y_{1}, y_{2}, \dots, y_{D})$ , $Z = (z_{1}, z_{2}, \dots, z_{D})$ , the fitness values of X, Y and Z are $F (X)$ , $F (Y)$ and $F (Z)$ respectively, q is the subscript of the dimension.

The functional equations of second-order polynomial are as follows: $\begin{array}{c} (10) & p (x_{q}) = a_{0} + a_{1} x_{q} + a_{2} x_{q}^{2} = F (X) \\ (11) & p (y_{q}) = a_{0} + a_{1} y_{q} + a_{2} y_{q}^{2} = F (Y) \\ (12) & p (z_{q}) = a_{0} + a_{1} z_{q} + a_{2} z_{q}^{2} = F (Z) \end{array}$

Based on the functional equations (10), (11) and (12), $a_{0}$ , $a_{1}$ and $a_{2}$ are solved, and the functional equation is given as follow: $\begin{matrix} (13) & \begin{aligned} p (t) & = F (X) \frac{(t - y_{q}) (t - z_{q})}{(x_{q} - y_{q}) (x_{q} - z_{q})} \\ + F (Y) \frac{(t - z_{q}) (t - x_{q})}{(y_{q} - z_{q}) (y_{q} - x_{q})} \\ + F (Z) \frac{(t - x_{q}) (t - y_{q})}{(z_{q} - x_{q}) (z_{q} - y_{q})} \end{aligned} \end{matrix}$

If $p^{'} (t) = 0$ , the quadratic interpolation method creates a new position according to the following functional equation: $\begin{matrix} (14) & x_{q} = \frac{(z_{q}^{2} - y_{q}^{2}) \times F (X) + (x_{q}^{2} - z_{q}^{2}) \times F (Y) + (y_{q}^{2} - x_{q}^{2}) \times F (Z)}{2 ((z_{q} - y_{q}) \times F (X) + (x_{q} - z_{q}) \times F (Y) + (y_{q} - x_{q}) \times F (Z))} \end{matrix}$

Based on the functional equation (14), a new individual $\overline{X} = (\overline{x_{1}}, \overline{x_{2}}, \dots, \overline{x_{D}})$ is created.

This paper proposes a Tent map and QI strategy. The new strategy firstly initializes the population according to the Tent map, and then generates new individuals according to QI. Then, all individuals of the two populations are ordered according to the fitness values, and the best individuals are selected as the final initialization population, which improves the quality of the whole population. The processes are summarized as follows:

Within the upper and lower limits, N positions are initialized according to the Tent map as the initial population positions.

New individuals are generated according to QI.

The initial and new individuals are merged, the fitness values after merging are reordered, and the first N individuals are selected as the final initial populations.

2.2.2. Golden-Sine

The Golden-Sine algorithm [35] is based on the design of sine function in mathematics, and improves the optimization performance by using the golden section coefficient. It is a new intelligent algorithm, which can be divided into three steps, including population initialization, golden section coefficient and updating position. In this paper, IMATSA uses Golden-Sine to reduce the search space in the local search stage to approximate the optimal solution of the algorithm and speed up the convergence speed.

The initial step of Golden-Sine is to update the initial space by randomly generating a uniform distribution of each dimension. The functional equation is given as follow: $\begin{matrix} (15) & V_{i} = rand * (ub - lb) + lb \end{matrix}$

Where $V_{i}$ is the initial value of the ith individual, $ub$ and $lb$ are the upper and lower limits.

In the process of updating position, the golden section coefficient is introduced to reduce the space, so as to guide the individual to approach the optimal value. $\begin{matrix} (16) & \begin{array}{c} x_{1} = a * (1 - τ) + b * τ \\ x_{2} = a * τ + b * (1 - τ) \end{array} \end{matrix}$

Where $x_{1}$ and $x_{2}$ are the golden section coefficients, which aim to balance the processes of search and development. The initial default values of a and B are $- π$ and π respectively, τ is the golden ratio, $τ = (1 - \sqrt{5}) / 2$ .

The Golden-Sine updates the individual position according to the following equation: $\begin{matrix} (17) & V_{i}^{t + 1} = V_{i}^{t} | sin (r_{1}) | - r_{2} sin (r_{1}) | x_{1} D_{i}^{t} - x_{2} V_{i}^{t} | \end{matrix}$

Where $V_{i}^{t}$ is the position of the i individual at t iteration; $D_{i}^{t}$ is the optimal position of the i individual at t times; $r_{1}$ is a random number in [0, 2 π]; $r_{2}$ is a random number in [0, π].

The principle of Golden-Sine is to use the unit circle scanning of sinusoidal function, which is similar to the spatial search of the solution of the problem to be optimized, and reduce the search space through the golden section ratio to approximate the optimal solution of the algorithm.

In this paper, after updating the individual position by using equation (7) of group behavior, IMATSA uses Golden-Sine to accelerate the convergence speed and update the individual position again. The equation is as follow: $\begin{matrix} (18) & \begin{aligned} \vec{P {(x)}_{i}^{t + 1}} & = \vec{P {(x)}_{i}^{t}} * | sin (r_{1}) | \\ + r_{2} sin (r_{1}) * | x_{1} * \vec{{FS}_{best}^{t}} - x_{2} * \vec{P {(x)}_{i}^{t}} | \end{aligned} \end{matrix}$

2.2.3. Levy flight

Levy flight is a random walking strategy [39], and its moving step size conforms to the stable distribution of a heavy tail, moving alternately for a short distance and a relatively long distance. Therefore, using Levy flight strategy in swarm intelligence optimization algorithm can effectively improve the search range, the diversity of search and the ability of jumping out of the local optical position.

In this paper, the equation for updating the optimal position according to Levy flight is as follows: $\begin{matrix} (19) & \vec{P {(x)}_{i j}^{t + 1}} = \vec{P {(x)}_{i j}^{t}} + S_{p} \end{matrix}$

Where $\vec{P {(x)}_{i j}^{t}}$ is the optimal position at the t iteration, j is the dimension, $S_{p}$ is the step size generated by Levy flight.

This paper constructs Levy flight step disturbance method, and the equations of are shown are as follows: $\begin{array}{c} (20) & S_{i j} = \frac{u}{{| v |}^{1 / β}} \\ (21) & u \sim N (0, δ_{u}^{2}) \\ (22) & v \sim N (0, δ_{v}^{2}) \\ (23) & δ_{u} = {\frac{Γ (1 + β) sin (π \times β / 2)}{Γ [(1 + β) / 2] \times β \times 2^{(β - 1) / 2}}}^{1 / β} \\ (24) & δ_{v} = 1 \\ (25) & S_{p} = \vec{{FS}_{best}^{t}} S_{i j} \end{array}$

Where i is the individual, j is the current dimension of the individual, u and v are generated according to the current individual dimension J, u and v belong to normal distribution, $β = 1.5$ , S is the random number generated according to the current individual dimension, $\vec{{FS}_{best}^{t}}$ is the current global optimal position.

2.2.4. Improved Gauss disturbance

Gauss disturbance strategy [11] refers to adding Gauss local disturbance term in the optimization process, so as to increase the probability of the algorithm jumping out of the local optimal position, maintain the population diversity and avoid premature convergence. In this paper, an improved Gauss disturbance strategy is proposed to disturb the optimal position, which enhances the search ability and convergence accuracy of the algorithm. In the iterations, Gauss disturbance is applied to the current optimal position. The functional equation of Gauss disturbance before improvement is as follow: $\begin{matrix} (26) & \vec{{FS}_{best}^{t}} = \vec{F S_{best}} (1 + randn (1)) \end{matrix}$ where $\vec{F S_{best}}$ is the current global optimal position, $\vec{{FS}_{best}^{t}}$ is the position after Gauss disturbance, then the fitness value after disturbance is calculated, and the optimal individual is selected according to the fitness values before and after disturbance.

In order to coordinate the global and local search ability of tunicate swarm algorithm, $randn (1)$ in the original functional equation is improved to dynamic Gauss radius σ. the improved functional equations are as follow: $\begin{array}{c} (27) & \vec{{FS}_{best}^{t}} = \vec{F S_{best}} (1 + σ) \\ (28) & σ = 0.05 / e^{(1 - t / T)} \end{array}$ where t is the current number of iterations, T is the maximum number of iterations, and σ decreases nonlinearly during the iteration process. In the early stage of the algorithm, σ is large and the algorithm has strong global exploration ability; In the middle and later stage, σ gradually decreases, and the algorithm will do more sufficient local search.

2.2.5. The processes of IMATSA

Firstly, after the parameters are setted, the initial population is created by Tent map method, and the new population is created by QI; then, the initial population and new population are reordered by fitness values, and the first D individuals are selected as the final initial population.

Secondly, IMATSA is executed based on parameters and population. After the j individual of the i iteration is executed, the Golden-Sine algorithm is executed for the j individual to accelerate the convergence of local search.

Thirdly, IMATSA adopts Levy flight and the improved Gauss disturbance method to disturd the optimal position, which can adaptively improve and coordinate the ability of global development and local search. In order to give full play to the effects of Levy flight and the improved Gauss disturbance, the probability parameter p is set in this paper. p belongs to $[0, 1]$ . When p < 0.5, Levy flight is adopted, and when p ⩾ 0.5, the improved Gauss disturbance is adopted.

Fourthly, when the number of iterations is equal to $Ite r_{max}$ , the algorithm jumps to the next step, otherwise it starts the next iteration. Finally, when the maximum number of iterations is reached, the algorithm ends and stores the fitness value and the optimal position. The processes of IMATSA are depicted in Fig. 2.

Fig. 2.

The processes of IMATSA.

3. Experimental results

In order to verify the performance of IMATSA, several type experiments are designed in this paper. The benchmark functions, parameter optimization experiments of SVM and GBDT can test the general performance of the algorithm. The Wilcoxon signed rank test can verify whether there are significant differences in performance between IMATSA and other algorithms. The experiment of image multi threshold segmentation can test the image segmentation performance of the algorithm. All experiments in this paper are operated on the PC with 1.19 GHz CPU and 16 GB memory, python development language and pycharm development software. The operating system is windows 10 and 64 bit.

3.1. The experiment of benchmark functions

14 benchmark functions [38] are used in the experiments. The parameters are depicted in Table 1, and the benchmark functions are depicted in Table 2.

Table 1
Parameters of the algorithms

Algorithm Parameter name Parameter value

STOA $Cf$ 2

u 1

v 1

GWO Synergy coefficient vector (A) [ $- α$ , α]

Synergy coefficient vector (C) $[0, 2]$

Optimal value ( $a_{max}$ ) 2

Optimal value ( $a_{min}$ ) 0

SFO A 4

e 0.01

SFpercent 0.3

BOA Search conversion probability (p) 0.6

Power exponent 0.1

Sensory modality 0.01

Search conversion probability (p) 0.6

SDO r $[0, 1]$

$r_{1}$ $[0, 1]$

PFA Step factor of pathfinder (r) $[0, 1]$

Interaction coefficient of followers (α) $[1, 2]$

Attraction coefficient of pathfinder to follower (β) $[1, 2]$

WOA $a_{max}$ 2

$a_{min}$ 0

b 1

TSA Minimum individual speed 1

Maximum individual speed 4

TSA-LEO Minimum individual speed 1

Maximum individual speed 4

Probability parameter of executing LEO ( $pr$ ) 0.3

IMATSA Minimum individual speed 1

Maximum individual speed 4

Algorithm	Parameter name	Parameter value
STOA	$Cf$	2
u	1
v	1
GWO	Synergy coefficient vector (A)	[ $- α$ , α]
Synergy coefficient vector (C)	$[0, 2]$
Optimal value ( $a_{max}$ )	2
Optimal value ( $a_{min}$ )	0
SFO	A	4
e	0.01
SFpercent	0.3
BOA	Search conversion probability (p)	0.6
Power exponent	0.1
Sensory modality	0.01
Search conversion probability (p)	0.6
SDO	r	$[0, 1]$
$r_{1}$	$[0, 1]$
PFA	Step factor of pathfinder (r)	$[0, 1]$
Interaction coefficient of followers (α)	$[1, 2]$
Attraction coefficient of pathfinder to follower (β)	$[1, 2]$
WOA	$a_{max}$	2
$a_{min}$	0
b	1
TSA	Minimum individual speed	1
Maximum individual speed	4
TSA-LEO	Minimum individual speed	1
Maximum individual speed	4
Probability parameter of executing LEO ( $pr$ )	0.3
IMATSA	Minimum individual speed	1
Maximum individual speed	4

In Table 2, $F_{1}$ – $F_{5}$ are unimodal functions, $F_{6}$ – $F_{8}$ are high-dimensional multimodal functions, and $F_{9}$ – $F_{13}$ are low-dimensional multimodal functions. The unimodal function can verify the convergence speed and accuracy; the high-dimensional multimodal function can verify the global search ability; and the low dimensional multimodal function can verify the development ability of the algorithm. In the experiment, the population of each algorithm is 100, the number of iterations is 500, and each algorithm runs 20 times independently. The Standard Deviation (STD) and MEAN are selected as the experimental indicators. The STD can reflect the robustness, and the MEAN can reflect the optimization accuracy and exploration ability.

Table 2

Benchmark functions

The experimental results of benchmark functions are depicted in Table 3 and Table 4. In unimodal functions $F_{1}$ – $F_{5}$ , MEAN and STD of IMATSA are several orders of magnitude ahead of other algorithms. In high-dimensional multimodal functions $F_{6}$ – $F_{8}$ and low dimensional multimodal functions $F_{9}$ – $F_{13}$ , MEAN of IMATSA are all the best. The overall performance of IMATSA is significantly better. The STD and MEAN of IMATSA are better than TSA-LEO, and the execution time of IMATSA is much less than TSA-LEO. Compared with TSA-LEO, the comprehensive performance of IMATSA is better.

Table 3

Experimental results of STOA, SFO, GWO, BOA, IMATSA

Fun	Metric	STOA	GWO	SFO	BOA	IMATSA
$F_{1}$	MEAN	5.1814E−117	7.9748E−41	2.6227E−9	8.8775E−12	0.0000
	STD	0.0000	8.4722E−41	6.5148E−9	8.1105E−13	0.0000
$F_{2}$	MEAN	2.4674E−88	4.3075E−24	0.0002	1.1158E9	0.0000
	STD	1.0165E−87	2.7928E−24	0.0002	4.8639E9	0.0000
$F_{3}$	MEAN	3.3366E−173	3.6514E−11	2.8761E−6	7.6887E−12	0.0000
	STD	0.0000	7.5610E−11	7.3530E−6	5.4763E−13	0.0000
$F_{4}$	MEAN	4.4891E−90	1.1339E−10	1.5905E−5	5.1159E−9	0.0000
	STD	1.2793E−89	8.5157E−11	3.2407E−5	3.9235E−10	0.0000
$F_{5}$	MEAN	0.0001	0.0004	0.0002	0.0007	4.9783E−7
	STD	0.0001	0.0001	0.0002	0.0003	3.4536E−7
$F_{6}$	MEAN	0.0000	1.1368	3.4191E−8	1.7087E−5	0.0000
	STD	0.0000	2.2997	1.0698E−7	7.3987E−5	0.0000
$F_{7}$	MEAN	−39.9999	−39.9999	−39.9999	−39.9999	−39.9999
	STD	0.0000	1.3759E−14	2.3730E−5	0.0000	0.0000
$F_{8}$	MEAN	0.0010	0.0033	1.1396E−9	3.2007E−13	0.0000
	STD	0.0030	0.0061	4.1928E−9	2.3100E−13	0.0000
$F_{9}$	MEAN	−1.0316	−1.0316	−1.0294	−1.0316	−1.0316
	STD	3.06759E−5	7.3612E−7	0.0051	1.7233E−5	3.5108E−7
$F_{10}$	MEAN	0.3981	0.3978	0.3978	0.3983	0.3978
	STD	0.0003	2.7916E−7	4.9156E−7	0.0005	2.0356E−7
$F_{11}$	MEAN	−2.8700	−3.2433	−2.5964	−3.1492	−2.9508
	STD	0.2337	0.0766	0.3503	0.0627	0.0535
$F_{12}$	MEAN	3.0009	3.0000	5.3106	3.0079	3.0000
	STD	0.0013	3.1023E−6	6.7269	0.0080	1.3413E−7
$F_{13}$	MEAN	0.5123	0.5897	1.8792	0.0010	0.5976
	STD	1.0803	2.1578	2.9800	0.0028	0.9128

Table 4

Experimental results of SDO, PFA, TSA, TSA-LEO, IMATSA

Fun	Metric	SDO	PFA	WOA	TSA	TSA-LEO	IMATSA
$F_{1}$	MEAN	529.7952	44.4553	1.1496E−94	2.6430E−117	1.6232E−122	0.0000
	STD	759.1679	24.3652	4.8656E−94	2.5203E−117	1.2466E−122	0.0000
$F_{2}$	MEAN	3.0301	6.9069	5.3634E−57	7.7851E−61	2.2623E−63	0.0000
	STD	2.6087	1.8993	2.0932E−56	2.2866E−61	1.5656E−63	0.0000
$F_{3}$	MEAN	2920.5902	5236.3226	1.5908E+4	1.3800E−102	2.0297E−105	0.0000
	STD	4329.0740	1287.3175	8.6655E+3	1.7480E−102	3.6891E−105	0.0000
$F_{4}$	MEAN	36.4657	19.4918	19.5327	2.7429E−53	1.0876E−55	0.0000
	STD	12.0592	4.6734	21.6561	1.3500E−53	1.1469E−55	0.0000
$F_{5}$	MEAN	0.0458	0.1817	0.0010	2.4837E−5	0.0001	4.9783E−7
	STD	0.1196	0.0636	0.0009	2.5351E−5	7.6152E−5	3.4536E−7
$F_{6}$	MEAN	25.3756	102.5270	0.0000	0.0000	0.0000	0.0000
	STD	32.4492	26.3855	0.0000	0.0000	0.0000	0.0000
$F_{7}$	MEAN	−34.9184	−38.3698	−39.9999	−39.9999	−39.9999	−39.9999
	STD	3.0922	1.2014	7.1054E−15	0.0000	0.0000	0.0000
$F_{8}$	MEAN	5.3651	1.3539	0.0000	0.0010	0.0000	0.0000
	STD	6.4613	0.2165	0.0000	0.0030	0.0000	0.0000
$F_{9}$	MEAN	−1.0208	−1.0316	−1.0316	−1.0316	−1.0316	−1.0316
	STD	0.0092	2.5316E−16	6.8541E−12	6.9169E−6	8.8117E−6	3.5108E−7
$F_{10}$	MEAN	0.4155	0.3978	0.3978	0.3981	0.3979	0.3978
	STD	0.0169	0.0000	1.7426E−7	0.0002	1.2736E−5	2.0356E−7
$F_{11}$	MEAN	−2.8373	−3.2625	−3.2350	−3.1302	−3.2552	−2.9508
	STD	0.2461	0.0594	0.0762	0.1293	0.0666	0.0535
$F_{12}$	MEAN	3.3206	3.0000	3.0000	3.0000	3.0000	3.0000
	STD	0.3221	6.5433E−15	1.8263E−6	4.7637E−7	4.9826E−6	1.3413E−7
$F_{13}$	MEAN	0.0069	1.1798E−25	0.4901	5.5034	4.4028E−9	0.5976
	STD	0.0290	5.1426E−25	2.1367	4.1290	8.6174E−9	0.9128

Figure 3 depicts the convergence graph of the benchmark functions. With the limited space, the convergence diagrams of $F_{1}$ , $F_{5}$ , $F_{7}$ , $F_{8}$ , $F_{11}$ and $F_{13}$ benchmark functions are selected in this paper. From Fig. 3, although the convergence speed of IMATSA is sometimes not the fastest, the convergence value of IMATSA is always the best. IMATSA has the best comprehensive performance.

The following is the description of the functions not shown in Fig. 3. For unimodal functions $F_{2}$ – $F_{4}$ , the convergence speed and value of IMATSA are better, and only the convergence value of IMATSA is 0. For high-dimensional multimodal function $F_{6}$ , although the convergence values of STOA, TSA and TSA-LEO are also 0, the convergence speed of IMATSA is better. For low-dimensional multimodal functions $F_{9}$ , $F_{10}$ and $F_{12}$ , Although STD of IMATSA is slightly worse than PFA and SDO occasionally, MEAN of IMATSA is always the best.

3.2. Ablation experiments

In this paper, the ablation experiments are designed to verify the effect of each improvement of IMATSA. The bench functions used in ablation experiments are the same as those in Table 2. The experimental results are depicted in Table 5. As shown in Table 4, with the addition of every improvement, the MEAN and STD are better and better.

Fig. 3.

The convergence graph of benchmark functions.

Table 5

The results of ablation experiments

TSA		√	√	√	√
Tent map and QI			√	√	√
Golden-Sine				√	√
Other improvements					√
$F_{1}$	MEAN	2.6430E−117	2.9512E−120	0.0000	0.0000
	STD	2.5203E−117	8.8536E−120	0.0000	0.0000
$F_{2}$	MEAN	7.7851E−61	1.1541E−63	0.0000	0.0000
	STD	2.2866E−61	3.4624E−63	0.0000	0.0000
$F_{3}$	MEAN	1.3800E−102	1.0307E−104	1.8988E−108	0.0000
	STD	1.7480E−102	3.0923E−104	5.6188E−108	0.0000
$F_{4}$	MEAN	2.7429E−53	0.0000	0.0000	0.0000
	STD	1.3500E−53	0.0000	0.0000	0.0000
$F_{5}$	MEAN	2.4837E−5	9.8913E−6	6.0368E−6	4.9783E−7
	STD	2.5351E−5	9.2237E−6	6.3815E−6	3.4536E−7
$F_{6}$	MEAN	0.0000	0.0000	0.0000	0.0000
	STD	0.0000	0.0000	0.0000	0.0000
$F_{7}$	MEAN	−39.9999	−39.9999	−39.9999	−39.9999
	STD	0.0000	0.0000	0.0000	0.0000
$F_{8}$	MEAN	0.0010	0.0000	0.0000	0.0000
	STD	0.0030	0.0000	0.0000	0.0000
$F_{9}$	MEAN	−1.0316	−1.0316	−1.0316	−1.0316
	STD	6.9169E−6	4.3708E−6	2.4012E−6	3.5108E−7
$F_{10}$	MEAN	0.3981	0.3981	0.3981	0.3978
	STD	0.0002	0.0001	0.0001	2.0356E−7
$F_{11}$	MEAN	−3.1302	−2.8810	−2.8922	−2.9508
	STD	0.1293	0.1028	0.0913	0.0535
$F_{12}$	MEAN	3.0000	3.0000	3.0000	3.0000
	STD	4.7637E−7	3.0010E−7	2.7256E−7	1.3413E−7
$F_{13}$	MEAN	4.6784	4.2412	0.5981	0.9960
	STD	3.6399	3.2976	0.9125	0.5060

3.3. Parameter optimization experiment of SVM

Support vector machines (SVM) can transformed the linear classification boundary into nonlinear classification boundary by means of kernel function, so as to obtain better generalization performance. SVM is usually used for classification or regression problems [4,22,43].

Based on different algorithms, this paper optimizes the C and gamma of SVM parameters to search the partition hyperplane that meets the maximum interval. The population size is 100, and the number of iterations is 100.

C [27] controls the punishment degree of sample misclassification, which reflects the balance between model classification performance and complexity. Gamma is the coefficient of RBF [16] kernel function, which implicitly changes the mapping function and controls the performance of feature space.

Figure 4 depicts the convergence graph of the classification error rate based on different algorithms. From Fig. 4, IMATSA has the fastest convergence rate and the lowest classification error rate, and the comprehensive performance of IMATSA is better.

3.4. Parameter optimization experiment of GBDT

Gradient Boosting Decision Tree (GBDT) is an iterative decision tree algorithm, which is composed of multiple decision trees. The conclusions of all trees are accumulated to make the final answer. The core idea is to fit the residual of the previous round of learners through the negative gradient of the loss function. GBDT is usually used for classification or regression problems [5,18,41].

Based on different algorithms, this paper optimizes four parameters of GBDT to reduce classification error rate, including max_depth, subsample, min_samples_leaf and min_samples_split. The population size is 5, and the number of iterations is 10.

max_depth is the maximum depth of the decision tree. subsample is the proportion of subsampling, which belongs to $(0, 1]$ . min_samples_leaf is the minimum number of samples of leaf nodes, which limits the minimum number of samples of leaf nodes. min_samples_split is the minimum number of samples required for internal node division.

Figure 5 depicts the convergence graph of the classification error rate based on different algorithms. From Fig. 5, IMATSA has the fastest convergence rate and the lowest classification error rate, and the comprehensive performance of IMATSA is better.

3.5. Wilcoxon signed rank test

Based on parameter optimization experiments of GBDT, the Wilcoxon signed rank test [2,14] is designed to verify whether there are significant differences in performance between IMATSA and other algorithms.

In the parameter optimization experiments of GBDT, the algorithms are executed 5 times and 10 times respectively. Then the Wilcoxon signed rank test with significance level being 0.05 is performed. The assumptions are given as follows:

H0: there is no significant difference about classification error rate between IMATSA and other algorithms.

H1: there is significant difference about classification error rate between IMATSA and other algorithms.

Table 6 depicts the results of Wilcoxon signed rank test. The values are less than 0.05. Therefore, H1 is accepted. There is significant difference about classification error rate between IMATSA and other algorithms.

3.6. The experiment of image multi threshold segmentation

3.6.1. Threshold segmentation methods

In the field of image segmentation, threshold segmentation method is a typical method. This paper extends the threshold segmentation methods based on Otsu and Renyi entropy to multi-threshold segmentation method.

The principle of multi-threshold segmentation is to determine the threshold group $[t_{1}, t_{2}, \dots, t_{n}]$ $(n > 0)$ according to certain rules, and to divide the image into $n + 1$ part according to the threshold group.

For the image with gray level $[0, 1, \dots, L - 1]$ (L is 256), i is the current gray level, the probability of gray level i is $p_{i}$ , and the total number of pixels is N. If the number of thresholds is n, the image is divided into $n + 1$ parts, whose pixel probabilities of each part is $w_{0}$ to $w_{n}$ respectively.

In 1979, a threshold segmentation method named Otsu [24] was proposed. Otsu dynamically determines the threshold based on the maximum inter class variance. In this paper, a multi-threshold segmentation method is realized based on Otsu, and the gray mean value of each part is $u_{0}$ to $u_{n}$ respectively.

The Renyi entropy [26] quantifys the value of entropy, and maximizes the sum of entropy between segmented target and background. In this paper, a multi-threshold segmentation method is realized based on Renyi entropy, and the gray mean value of each part is $H_{0}$ to $H_{n}$ respectively.

3.6.2. The processes of multi-threshold segmentation method based on IMATSA

If the threshold vector is $T = [t_{1}, t_{2}, \dots, t_{n}]$ , which satisfies $0 < t_{1} < t_{2} < \dots < t_{n} < L$ , and its subvalues are all positive integers, then the image will be segmented according to this vector. In IMATSA, Otsu and Renyi entropy are used as objective functions respectively, if the vector conforms to the functional equation (40) or (45), then it is the target threshold vector. The processes of multi-threshold segmentation method based on IMATSA are depicted in Fig. 6.

Fig. 4.

The SVM convergence graph of the classification error rate.

Fig. 5.

The GBDT convergence graph of the classification error rate.

Table 6

Wilcoxon signed rank test

Comparison	P-value (5 times)	P-value (10 times)
IMATSA versus GWO	3.6284E−7	3.1702E−13
IMATSA versus BOA	0.0001	6.3488E−8
IMATSA versus SFO	8.4333E−5	2.9148E−11
IMATSA versus STOA	0.0422	0.0004
IMATSA versus SDO	1.2568E−6	2.3556E−13
IMATSA versus PFA	0.0372	0.0348
IMATSA versus WOA	0.0005	6.2837E−10
IMATSA versus TSA	0.0007	4.3402E−6
IMATSA versus TSA-LEO	0.0047	6.8353E−8

Fig. 6.

The processes of multi-threshold segmentation method based on IMATSA.

3.6.3. The experiment of image multi-threshold segmentation

Baboon, Barbara, Goldhill, and Peppers in the classical threshold segmentation data set are selected as the detection images.

The comparitive algorithms are IMATSA, TSA-LEO, GWO, SFO, BOA and WOA, The population size is 100, the number of iterations is 100, and each algorithm run independently for 20 times. Peak Signal-to-Noise Ratio (PSNR) and Standard Deviation (STD) are used as experimental indicators, and the functional equations are as follows: $\begin{array}{c} (29) & PSNR = 10 lg (\frac{255^{2}}{MSE}) \\ (30) & STD = \sqrt{\sum_{i = 1}^{NI} \frac{{(b f_{i} - μ)}^{2}}{NI}} \end{array}$ where $NI$ is the number of independent executions; $b f_{i}$ is the best fitness value for the i execution; μ is the average value of the best fitness values; MSE is the mean square error, the functional equation is given as follow: $\begin{matrix} (31) & MSE = \frac{\sum_{i = 1}^{m} \sum_{j = 1}^{n} {(f (i, j) - g (i, j))}^{2}}{m n} \end{matrix}$

The experimental results of IMATSA are depicted as Table 7 and Table 8. It can be seen from Table 7 and Table 8 that in the threshold segmentation experiments of different images, with the increase of the number of thresholds, the PSNR and MEAN become better and better, and the performance is directly proportional to the number of thresholds.

Table 7
Experimental results based on Otsu

Image Threshold number Threshold PSNR MEAN STD

Baboon 2 98 155 20.3732 1539.3 3.3677

3 92 124 164 22.1472 1627.2 2.1704

4 91 139 163 185 23.5057 1694.9 2.0692

5 77 106 129 159 167 24.4709 1725.1 3.1634

Barbara 2 86 138 20.1881 1922.1 0.5480

3 76 120 171 23.8665 2038.0 3.6120

4 72 109 129 168 24.8888 2109.6 2.6600

5 50 80 116 140 163 25.7588 2145.5 3.5489

Goldhill 2 96 162 20.8618 2067.9 1.0586

3 87 125 185 22.7148 2214.8 1.9259

4 88 114 139 184 24.6551 2286.7 1.2865

5 50 70 106 138 181 26.8994 2327.2 2.6671

Peppers 2 48 115 20.8618 2866.1 0.0323

3 44 100 152 21.4611 3065.8 4.0962

4 46 73 123 159 24.6541 3181.0 2.5737

5 1 31 83 136 187 26.2798 3196.2 0.9846

Image	Threshold number	Threshold	PSNR	MEAN	STD
Baboon	2	98 155	20.3732	1539.3	3.3677
3	92 124 164	22.1472	1627.2	2.1704
4	91 139 163 185	23.5057	1694.9	2.0692
5	77 106 129 159 167	24.4709	1725.1	3.1634
Barbara	2	86 138	20.1881	1922.1	0.5480
3	76 120 171	23.8665	2038.0	3.6120
4	72 109 129 168	24.8888	2109.6	2.6600
5	50 80 116 140 163	25.7588	2145.5	3.5489
Goldhill	2	96 162	20.8618	2067.9	1.0586
3	87 125 185	22.7148	2214.8	1.9259
4	88 114 139 184	24.6551	2286.7	1.2865
5	50 70 106 138 181	26.8994	2327.2	2.6671
Peppers	2	48 115	20.8618	2866.1	0.0323
3	44 100 152	21.4611	3065.8	4.0962
4	46 73 123 159	24.6541	3181.0	2.5737
5	1 31 83 136 187	26.2798	3196.2	0.9846

Table 8

Experimental results based on Renyi entropy

Image	Threshold number	Threshold	PSNR	MEAN	STD
Baboon	2	87 145	20.3938	5.4089	0.0019
	3	52 99 153	21.6093	6.8877	0.0097
	4	38 85 108 158	22.3425	7.9453	0.0192
	5	33 60 109 131 169	24.5765	8.9822	0.0194
Barbara	2	86 161	20.8280	5.3942	0.0149
	3	72 125 168	23.8621	6.6915	0.0099
	4	85 123 155 185	24.8035	7.8016	0.0051
	5	44 75 120 152 178	27.0054	8.9229	0.0174
Goldhill	2	89 155	21.2009	5.5629	0.0010
	3	86 136 188	23.4529	6.6334	0.0079
	4	75 108 152 188	24.5245	8.0031	0.0285
	5	57 98 131 160 200	26.8371	9.1751	0.0258
Peppers	2	86 179	20.7868	5.6338	0.0052
	3	81 132 177	22.7632	6.9189	0.0326
	4	43 80 118 187	25.4732	8.1836	0.0541
	5	28 82 108 146 186	26.5449	8.97796	0.0618

Table 9 depict the experimental results of IMATSA, TSA-LEO and GWO based on Otsu. Table 10 depict the experimental results of SFO, BOA and WOA based on Otsu. Compared with TSA-LEO, GWO, SFO and BOA, IMATSA has better convergence stability and PSNR value. The overall performance of IMATSA is better.

Table 9

Experimental results based on Otsu of IMATSA, TSA-LEO and GWO

Image	Threshold number	IMATSA			TSA-LEO			GWO

		PSNR	MEAN	STD	PSNR	MEAN	STD	PSNR	MEAN	STD
Baboon	2	20.3732	1539.3	3.3677	20.2672	1546.8	1.3763	19.1677	1532.7	4.5804
Baboon	3	22.1472	1627.2	2.1704	21.5655	1622.1	3.6441	20.4411	1623.8	2.3645
Baboon	4	23.5057	1694.9	2.0692	23.1428	1685.8	3.0859	22.3685	1665.0	2.8010
Baboon	5	24.4709	1725.1	3.1634	23.7970	1705.2	1.3416	23.2318	1707.3	3.2337
Barbara	2	20.1881	1922.1	0.5480	20.0447	1917.6	0.5667	19.0034	1917.2	1.4713
Barbara	3	23.8665	2038.0	3.6120	23.8130	2056.1	1.2028	22.5791	2025.4	3.7633
Barbara	4	24.8888	2109.6	2.6600	25.0979	2105.4	3.5924	24.2586	2104.2	2.6642
Barbara	5	25.7588	2145.5	3.5489	25.4586	2141.8	3.8143	24.9751	2138.1	3.6600
Goldhill	2	20.8618	2067.9	1.0586	20.1660	2058.0	0.4691	19.8577	2063.2	1.5086
Goldhill	3	22.7148	2214.8	1.9259	22.2276	2210.7	3.2454	22.2592	2201.7	2.7854
Goldhill	4	24.6551	2286.7	1.2865	24.7314	2281.8	5.7703	23.5433	2282.0	1.6870
Goldhill	5	26.8994	2327.2	2.6671	26.1786	2316.6	3.0443	25.8779	2312.0	2.9095
Peppers	2	20.8618	2866.1	0.0323	19.0415	2861.3	0.9099	19.0818	2863.6	0.6627
Peppers	3	21.4611	3065.8	4.0962	21.2574	3061.9	2.3786	20.2925	3058.3	4.5080
Peppers	4	24.6541	3181.0	2.5737	23.4638	3143.9	7.9530	23.7163	3135.2	2.7136
Peppers	5	26.2798	3196.2	0.9846	25.5048	3186.7	2.7846	24.3948	3174.7	3.1978

Table 10

Experimental results based on Otsu of IMATSA, SFO and BOA

Image	Threshold number	SFO			BOA			WOA

		PSNR	MEAN	STD	PSNR	MEAN	STD	PSNR	MEAN	STD
Baboon	2	20.1615	1538.1	0.0469	19.2420	1537.6	3.5938	19.2573	1533.8	1.6983
Baboon	3	21.8991	1625.0	2.8692	21.0451	1623.3	3.7756	20.6814	1635.4	1.8436
Baboon	4	23.2443	1663.1	3.2419	22.4378	1660.1	4.6769	22.9135	1687.4	2.5427
Baboon	5	24.1702	1710.3	3.3113	22.9363	1699.0	5.0153	23.6814	1706.0	3.1542
Barbara	2	20.0234	1917.5	0.9045	19.0046	1909.7	3.6190	20.0302	1916.8	0.7846
Barbara	3	23.4194	2029.4	4.9169	22.4229	2022.9	4.4692	22.5525	2057.3	1.6883
Barbara	4	24.6027	2103.5	5.2654	23.7198	2085.2	3.8626	24.0911	2115.5	2.9337
Barbara	5	25.3421	2143.2	3.8835	25.3013	2134.2	5.2415	24.9184	2142.7	3.5932
Goldhill	2	20.1363	2057.7	1.2178	19.9637	2055.4	2.3392	19.9481	2068.4	1.0488
Goldhill	3	22.0165	2210.6	2.1408	22.2187	2194.1	3.8009	22.1458	2215.7	2.4157
Goldhill	4	24.3907	2276.2	4.1970	24.0631	2263.0	3.1710	24.0506	2291.2	2.7554
Goldhill	5	25.3434	2313.0	7.4193	24.8062	2298.2	4.2078	25.6350	2320.9	3.3954
Peppers	2	19.0941	2864.1	1.1196	19.0941	2861.0	3.5414	19.0821	2866.0	0.9247
Peppers	3	21.1421	3057.6	10.2922	21.2974	3045.5	4.4443	20.4827	3063.0	2.1831
Peppers	4	23.6209	3144.6	5.2335	24.3582	3108.2	4.7636	24.6072	3147.3	3.4092
Peppers	5	25.8410	3181.9	2.7068	25.5905	3168.3	5.3726	25.8654	3186.7	3.4839

Table 11 depict the experimental results of IMATSA, TSA-LEO and GWO based on Renyi entropy. Table 12 depict the experimental results of SFO, BOA and WOA based on Renyi entropy. Compared with TSA-LEO, GWO, SFO and BOA. The overall performance of IMATSA is also better.

Table 11

Experimental results based on Renyi entropy of IMATSA, TSA-LEO and GWO

Image	Threshold number	IMATSA			TSA-LEO			GWO

		PSNR	MEAN	STD	PSNR	MEAN	STD	PSNR	MEAN	STD
Baboon	2	20.3938	5.4089	0.0019	20.2251	5.2102	0.0035	19.1403	5.3011	0.0030
Baboon	3	21.6093	6.8877	0.0097	21.4362	6.4999	0.0091	21.3958	6.6149	0.0158
Baboon	4	22.3425	7.9453	0.0192	22.1316	7.7067	0.0307	21.8484	7.8201	0.0248
Baboon	5	24.5765	8.9822	0.0194	23.5546	8.8279	0.0259	23.6828	8.9076	0.0841
Barbara	2	20.8280	5.3942	0.0149	20.7765	5.3156	0.0315	19.7907	5.3721	0.0410
Barbara	3	23.8621	6.6915	0.0099	23.7863	6.5981	0.0104	22.8652	6.6631	0.0128
Barbara	4	24.8035	7.8016	0.0051	24.8444	7.7380	0.0072	24.6738	7.7960	0.0268
Barbara	5	27.0054	8.9229	0.0174	26.3162	8.8300	0.0498	25.4241	8.8920	0.0266
Goldhill	2	21.2009	5.5629	0.0010	21.0516	5.3626	0.0016	20.0677	5.4468	0.0026
Goldhill	3	23.4529	6.6334	0.0079	23.0312	6.6197	0.0059	23.2122	6.6320	0.0153
Goldhill	4	24.5245	8.0031	0.0285	24.3231	7.8281	0.0244	23.8297	7.9040	0.0354
Goldhill	5	26.8371	9.1751	0.0258	26.1748	8.9726	0.0309	25.7515	9.0464	0.0299
Peppers	2	20.7868	5.6338	0.0052	20.5997	5.3442	0.0066	20.3153	5.4419	0.0068
Peppers	3	22.7632	6.9189	0.0326	22.5103	6.7596	0.0166	22.4417	6.8403	0.0089
Peppers	4	25.4732	8.1836	0.0541	25.2825	7.9804	0.0654	24.9774	8.0755	0.0667
Peppers	5	26.5449	8.97796	0.0618	26.1325	8.9613	0.0706	25.8481	8.8758	0.0212

Table 12

Experimental results based on Renyi entropy of IMATSA, SFO and BOA

Image	Threshold number	SFO			BOA			WOA

		PSNR	MEAN	STD	PSNR	MEAN	STD	PSNR	MEAN	STD
Baboon	2	16.5259	4.9545	0.0016	20.0231	5.2982	0.0064	19.1200	5.3049	0.0197
Baboon	3	20.7684	6.2832	0.0722	20.6402	6.5883	0.0170	21.4807	6.6198	0.0107
Baboon	4	21.9843	7.6155	0.0793	21.3819	7.7694	0.0365	22.1547	7.8529	0.0215
Baboon	5	23.8616	8.8755	0.0541	22.6340	8.9233	0.0754	22.4631	8.9332	0.0542
Barbara	2	20.6616	5.3590	0.0154	19.4772	5.3657	0.0212	19.7731	5.3732	0.0322
Barbara	3	23.1641	6.6683	0.0015	23.5165	6.6525	0.0202	22.8764	6.6686	0.0121
Barbara	4	24.4704	7.7921	0.0067	24.7102	7.7317	0.0303	24.2382	7.8014	0.0156
Barbara	5	26.4964	8.8630	0.0349	25.6493	8.8026	0.1415	26.1417	8.9103	0.0261
Goldhill	2	20.2251	5.2111	0.0024	20.0901	5.4418	0.0046	19.9637	5.4469	0.0094
Goldhill	3	21.5629	6.5039	0.0021	23.0858	6.6263	0.0161	22.9837	6.7382	0.0240
Goldhill	4	22.4063	7.6863	0.0336	23.9074	7.8841	0.0462	24.7634	7.9676	0.0302
Goldhill	5	23.1026	8.7803	0.0654	25.9098	8.9518	0.0757	26.3953	9.0828	0.0486
Peppers	2	20.6385	5.3472	0.0040	20.1579	5.4387	0.0069	20.2527	5.4473	0.0248
Peppers	3	22.1454	6.7257	0.0640	22.4526	6.8133	0.0361	22.1859	6.8415	0.1131
Peppers	4	23.2766	7.9030	0.0750	25.2885	8.0318	0.0618	24.4822	8.1016	0.0971
Peppers	5	25.6468	8.8438	0.0581	25.3063	8.7958	0.0441	25.9177	8.7929	0.1132

The segmentation effect of IMATSA based on Otsu and Renyi entropy are shown in Fig. 7 and Fig. 8 respectively. As can be seen, the segmentation effect is directly proportional to the number of thresholds. With the number of thresholds increasing, the detail processing and segmentation quality are better.

Fig. 7.

Segmentation effect based on Otsu.

Fig. 8.

Segmentation effect based on Renyi entropy.

4. Discussion

In the field of parameter optimization, swarm intelligence optimization algorithm has broad prospects. At present, swarm intelligence optimization algorithm has made some achievements in the field of parameter optimization. However, the single search methods of many swarm intelligence optimization algorithms lead to lack of flexibility and mobility, and it is difficult to jump out of the local position. As a new swarm intelligence optimization algorithm, TSA also has some problems to some extent, such as lack of group diversity, local development and global search to be optimized.

This paper improves TSA in three aspects, including enhancing population diversity, local development ability and global search ability. At the same time, the algorithm proposed by this paper can adaptively coordinate and balance the local development and global search ability under a certain probability through the improved Gauss disturbance method. Based on the above improvements, this paper proposes an improved and adaptive tuner swarm algorithm (IMATSA).

A population with good distribution and randomness can be created by Tent map, and then the uniformity of the population can be further strengthened by QI. In terms of enhancing population diversity, this paper propose a Tent map and QI strategy. A population with good distribution and randomness can be created by Tent map, and then the uniformity of the population can be further strengthened by QI. Firstly, The new strategy initializes the population by Tent map, and generates new individuals according to QI. Secondly, all individuals are ordered according to the fitness values. Thirdly, the best individuals are selected as the final initialization population.

In terms of enhancing local development ability, this paper adopts Golden-Sine algorithm to reduce the search space in the local search stage to approximate the optimal solution of the algorithm and speed up the convergence speed. The principle of Golden-Sine is to use the unit circle scanning of sinusoidal function, which is similar to the spatial search of the solution of the problem to be optimized, and reduce the search space through the golden section ratio to approximate the optimal solution of the algorithm. In this paper, after updating the individual position by group behavior, IMATSA uses Golden-Sine to accelerate the convergence speed and update the individual position again.

In terms of enhancing the global search ability, this paper adopts Levi flight to improve the search range under a certain probability, and the probability value can be dynamically adjusted according to different scenes, so as to enhance the population diversity and the ability to jump out of the local optimal position. Levy flight is a random walking strategy, and its moving step size conforms to the stable distribution of a heavy tail, moving alternately for a short distance and a relatively long distance. In this paper, after each iteration, the current optimal position and fitness value are updated by Levy flight strategy, so as to improve the search range search range.

In terms of coordinating the local development and global search ability, this paper proposes an improved Gauss disturbance strategy to disturb the optimal position. In the early stage of the algorithm, the algorithm has strong global exploration ability; In the middle and later stage, gradually decreases, and the algorithm will do more sufficient local search. In this paper, after each iteration, the current optimal position and fitness value are updated by improved Gauss disturbance strategy under a certain probability, and the probability value can be dynamically adjusted according to different scenes.

In order to verify the performance of IMATSA, this paper has designed various types of experiments, including three kinds of benchmark functions experiment, ablation experiment, parameter optimization experiments of Support Vector Machine (SVM) and Gradient Boosting Decision Tree (GBDT), Wilcoxon signed rank test and image multi-threshold segmentation experiment. The benchmark functions, parameter optimization experiments of SVM and GBDT can test the general performance of the algorithm. The Wilcoxon signed rank test can verify whether there are significant differences in performance between IMATSA and other algorithms. The experiment of image multi threshold segmentation can test the image segmentation performance of the algorithm. Experimental results show that IMATSA performs better in the all kinds of benchmark functions; each component of IMATSA has a positive effect on the performance; IMATSA performs better in parameter optimization experiments of SVM experiment and GBDT; there is significant difference between IMATSA and other algorithms by Wilcoxon signed rank test; in image segmentation, the performance is directly proportional to the number of thresholds, and compared with other algorithms, IMATSA has better comprehensive performance.

There also exists a problem of a lack of application and business scenarios in IMATSA. In this paper, IMATSA has been tested and demonstrated in the scenarios of benchmark functions, machine learning and image segmentation, but it has not been verified in specific business scenarios, such as operation research optimization, task scheduling and so on. The optimization accuracy, speed and stability of imatsa under more business scenarios need to be further verified.

For future work, this team will continue to enrich the business scenarios, and investigate and compare the performance of IMATSA and other algorithms in the further studies. In addition, this team will design more types and numbers of experiments for different business scenarios, and continuously improve the algorithm according to the experimental results.

5. Conclusion

In order to improve the performance of TSA, this paper proposes IMATSA. Firstly, IMATSA adopts Tent map and QI to initialize the population. Secondly, this paper adopts Golden-Sine algorithm to accelerate the convergence of local search. Thirdly, this paper adopts Levy flight strategy to improve the search range under a certain probability. Finally, this paper adopts improved Gauss disturbance to disturb optimal solution, which can improve the population diversity, and coordinate the local development and global search ability.

In order to verify the performance of IMATSA, this paper designs various types of experiments, including benchmark functions, ablation experiment, parameter optimization experiments of SVM and GBDT, Wilcoxon signed rank test, and image multi threshold segmentation experiment.

Experiments show that IMATSA performs better in the three kinds of benchmark functions, parameter optimization experiments of SVM and GBDT; each component of IMATSA has a positive effect on the performance; there is significant difference between IMATSA and other algorithms; the performance is directly proportional to the number of thresholds, and compared with other algorithms, IMATSA has better comprehensive performance.

In the follow-up research work, the performance of IMATSA will be further optimized and applied to more research fields and business scenarios.

Footnotes

Acknowledgements

This research was funded by the National Natural Science Foundation of China, grant number 61762009 and this research was funded by the Natural Science Foundation of Guangxi Province of China, grant number 2020GXNSFAA159090.

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