# A Novel Imperialist Competitive Algorithm for Multithreshold Image Segmentation

Authors: Mei Wang; Guohua Pan; Yan Liu
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

Multithreshold image segmentation plays a very important role in computer vision and pattern recognition. However, the computational complexity of multithreshold image segmentation increases exponentially with the increasing number of thresholds. Thus, in this paper, a novel imperialist competitive algorithm is proposed to solve the multithreshold image segmentation problem. Firstly, a new strategy of revolution and assimilation is adopted to improve the search efficiency of the algorithm. Secondly, imperialist self-learning and reserve country set are introduced to enhance the search of outstanding individuals in the population. Combining with the reserve country set, a novel imperialist competition strategy is proposed to remove the poorer individuals and improve the overall quality of the population. Finally, the sensitivity of the algorithm parameters is analyzed. Ten standard test pictures are selected to test. The experimental results show that the novel imperialist competitive algorithm has faster convergence speed, higher quality, and higher stability in solving multithreshold segmentation problems than methods from literature.


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## Body

## 1. Introduction

Image segmentation is the technology of dividing an image into several specific or unique regions and extracting the interesting objects. It plays very important role in computer vision and pattern recognition. At present, thousands of image segmentation algorithms have been proposed [1–10], among which threshold segmentation is a better method and is used widely. Traditional threshold segmentation algorithm is very effective for single threshold segmentation, but with the increase of the thresholds number, the amount of computation will increase dramatically [3]. For a given image, the process of searching the optimal threshold can be regarded as a constrained optimization problem. In order to solve the problem of large amount of computation in multithreshold segmentation, many intelligent swarm optimization algorithms are used for image segmentation [11, 12], such as particle swarm optimization (PSO [13]), Gray Wolf algorithm (GWO [3]), teaching optimization algorithm (TLBO [14]), Cuckoo search algorithm [15], bee colony algorithm [16–18], and dragonfly based algorithm [19]. However, due to the limitations of the algorithm itself, the search ability is not ideal, and it is easy to premature convergence, fall into local optimum, and the solution stability is insufficient. [20]ICA is a metaheuristic based on social and political behavior [21]. At present, the algorithm has been successfully applied to power grid energy management [22], nuclear reactor fuel loading [23], parameter estimation [24], production scheduling [25–29], and so on. According to literature [30], ICA not only is an effective global optimization method, but also has a strong neighborhood search capability. Its structure is flexible and easy to be fused with other algorithms. Thus, ICA is suitable for solving different problems with high stability. However, at present, ICA has not been applied to solve multithreshold segmentation problem. Therefore, a new imperialist competition algorithm in this paper has been given to improve the quality and stability of the multithreshold segmentation.In this paper, a new imperialist competition algorithm is proposed to solve the multithreshold image segmentation problem. Firstly, a new strategy of revolution and assimilation is adopted to improve the search efficiency of the algorithm. Then, in order to enhance the search for excellent individuals in the population, imperialist self-learning and reserve country set are introduced. Thirdly, a new imperialist competition strategy combining with the reserve country set is proposed to weed out the poorer individuals and improve the overall quality of the population. Finally, the sensitivity of the algorithm parameters is analyzed. Ten standard test pictures are selected to test the performance comparing with three efficient metaheuristics, and the experimental results show that the novel imperialist competitive algorithm has faster convergence speed, higher quality, and higher stability in solving multithreshold segmentation problems.The remainder of the paper is organized as follows. Problem under study is described in Section2. ICA for the problem is reported in Section 3. Numerical test experiments on ICA are shown in Section 4 and the conclusions are summarized in the final section and some topics of the future research are provided.


## 2. Problem Description

Threshold segmentation method is used commonly for image segmentation and the traditional threshold selection method is described as Otsu [1] and Kapur’s [2] method. Assume an image gray level is L and it is shown as 0,1,2,…,L-1. N is the pixels number of the image and ni is the pixel number of gray value i.p=pi∣i=0,1,2,…,L-1 is described as the probability distribution of each gray value i, where pi=ni/N.


### 2.1. Otsu Method

The classical Otsu method [1] is known as the optimal threshold selection method. And Otsu for multiobjective image segmentation is the method to determine the optimal segmentation threshold group based on the maximum between-class variance criterion. The traditional Otsu’s method is described as follows.Assume the image gray level isL, ni is the pixels number of gray-level i, and the total pixels number of the image is N=∑i=0L-1ni. If the probability of a given gray-level i is given as pi=ni/N, obviously, pi≥0, ∑i=0L-1pi=1. Assume the image segmentation threshold is θt, g0 shows the average gray-level value of all pixels less than or equal to θt, and g1 shows the average gray-level value of all pixels larger than θt in the image. Then variance between binary image classes is described as (1)σB2t=ω0σ02+ω1σ12=ω0g0-θt2+ω1g1-θt2whereω0=∑i=0t-1pi, ω1=∑i=tL-1pi=1-ω0. σB2t can be defined as a reparability criterion. So θt can be computed by maximizing σB2t.The formula (1) can be extended to multithreshold segmentation (assuming M level). Threshold group t1,t2,…,tM-1 can divide one image into k-classes objectives. And the variance between classes can be described as follows:(2)σB2t1,t2,…,tM-1=∑j=0M-1ωjgj-θj2The threshold groupt1,t2,…,tM-1 of multiobjective image segmentation can be given by maximizing the between-classes variance σB2t1,t2,…,tM-1.


### 2.2. Kapur’s Method

Optimum entropy threshold method presented by Kapur [2] can be described as follows. For gray images, the grayscale also can be expressed as 0,1,2,…,L-1L=256. ni shows the pixels number of the image with gray-value i. N shows the total pixels number of the image and N=∑i=0L-1ni. The probability density is pi=ni/N and the entropy of the image is defined as Hn=-∑i=0L-1piln⁡pi.For image multilevel segmentation (assuming M level), the image can be divided into M subclasses:C0=0,1,…,t1, C1=t1+1,t1+2,…,t2,…, CM-1=tM-1+1,tM-1+2,…,L-1, and the entropy is defined as(3)φt1,t2,…,tM-1=-∑i=0t1piP0ln⁡piP0-∑i=t1+1t2piP1ln⁡piP1-⋯-∑i=tM-1L-1piPM-1ln⁡piPM-1wherePj=∑i∈Cjpi,j=0,1,…M-1.The optimum entropy threshold selection method proposed by Kapur does not need prior knowledge. It can be used to segment nonideal bimodal histogram, but it takes a lot of computation to determine threshold, especially multithreshold [3].


## 2.1. Otsu Method

The classical Otsu method [1] is known as the optimal threshold selection method. And Otsu for multiobjective image segmentation is the method to determine the optimal segmentation threshold group based on the maximum between-class variance criterion. The traditional Otsu’s method is described as follows.Assume the image gray level isL, ni is the pixels number of gray-level i, and the total pixels number of the image is N=∑i=0L-1ni. If the probability of a given gray-level i is given as pi=ni/N, obviously, pi≥0, ∑i=0L-1pi=1. Assume the image segmentation threshold is θt, g0 shows the average gray-level value of all pixels less than or equal to θt, and g1 shows the average gray-level value of all pixels larger than θt in the image. Then variance between binary image classes is described as (1)σB2t=ω0σ02+ω1σ12=ω0g0-θt2+ω1g1-θt2whereω0=∑i=0t-1pi, ω1=∑i=tL-1pi=1-ω0. σB2t can be defined as a reparability criterion. So θt can be computed by maximizing σB2t.The formula (1) can be extended to multithreshold segmentation (assuming M level). Threshold group t1,t2,…,tM-1 can divide one image into k-classes objectives. And the variance between classes can be described as follows:(2)σB2t1,t2,…,tM-1=∑j=0M-1ωjgj-θj2The threshold groupt1,t2,…,tM-1 of multiobjective image segmentation can be given by maximizing the between-classes variance σB2t1,t2,…,tM-1.


## 2.2. Kapur’s Method

Optimum entropy threshold method presented by Kapur [2] can be described as follows. For gray images, the grayscale also can be expressed as 0,1,2,…,L-1L=256. ni shows the pixels number of the image with gray-value i. N shows the total pixels number of the image and N=∑i=0L-1ni. The probability density is pi=ni/N and the entropy of the image is defined as Hn=-∑i=0L-1piln⁡pi.For image multilevel segmentation (assuming M level), the image can be divided into M subclasses:C0=0,1,…,t1, C1=t1+1,t1+2,…,t2,…, CM-1=tM-1+1,tM-1+2,…,L-1, and the entropy is defined as(3)φt1,t2,…,tM-1=-∑i=0t1piP0ln⁡piP0-∑i=t1+1t2piP1ln⁡piP1-⋯-∑i=tM-1L-1piPM-1ln⁡piPM-1wherePj=∑i∈Cjpi,j=0,1,…M-1.The optimum entropy threshold selection method proposed by Kapur does not need prior knowledge. It can be used to segment nonideal bimodal histogram, but it takes a lot of computation to determine threshold, especially multithreshold [3].


## 3. A Novel Imperialist Competitive Algorithm for Multithreshold Image Segmentation

ICA is a population-based metaheuristic. Each individual of population represents a country and some best countries are selected as imperialists in the initialization. Its main process is described as follows:Step 1 (initialize the empire).
An initial populationP is generated randomly. Choose Nimp solutions with smallest cost as imperialists, and assign Ncol remaining countries to the imperialists.Step 2 (assimilation and revolution).
In every empire, assimilation is carried out on every colony and executes revolution of some colonies.Step 3.
Exchange position of colony and imperialist if possible.Step 4 (imperialist competition).
According to the total cost of the empire, the weakest colonies of the worst empire are redistributed to the winning empire.Step 5.
If there are no members in an Empire, the Empire is eliminated directly.Step 6.
If the termination condition holds, the search ends; otherwise, go toStep 2.The cost is related to the objective function. The better the objective function, the better the country. The specific process of the initial empire, assimilation, revolution, and imperialist empire competition of basic ICA can be found in Reference [21].In the traditional imperialist competition, the new colony solution comes from the global search and local search. The global search or local search of imperialists is seldom considered. Imperialists are often excellent individuals in the population. The quality of solutions is higher than other colonies. Using these excellent individuals to search will help to improve the search efficiency and get better quality solutions more easily. For this reason, imperialists self-learning and reserve country set are introduced. In addition, in order to improve the search efficiency of the algorithm, a new strategy of revolution and assimilation is adopted. Above all, a novel ICA is proposed to solve multithreshold image segmentation. The steps of ICA are described in detail below.Multithreshold image segmentation (assuming M level) is represented by an integer stringϕ=ϕ1,ϕ2,…,ϕM-1. The decoding process is a series of integers arranged M-1 in ascending order ϕ¯=ϕ1¯,ϕ2¯,…,ϕM-1¯, in which ϕi¯ is the order i segmentation threshold.


### 3.1. Initial Empire

The establishment of the initial empire is a key step for ICA. The specific steps are described as follows:Step 1.
Calculate the cost of countrylcostl=f, in which f is shown as the objective function value according to formula (2) or formula (3).Step 2.
Descending order for all solutions according to the cost of the country, choose the formerNimp countries as imperialists and the rest as the colony, so the number of colonies can be described as Ncol=N-Nimp.Step 3.
Calculate the power of each imperialistk, Pk=costk/∑i=1Nimpcosti, and determine the colony number NCl allocated to each imperialist k, where NCk=roundPk×Ncol. The specific allocation steps are shown in Reference [21].


### 3.2. Assimilation

The assimilation is a process that the colonies in the empire gradually move close to their imperialists. It will make the colonies similar to their imperialists. When the colony is the same as their imperialist, the assimilation operation will lose its effect and reduce the diversity of the population. Therefore, in order to improve the efficiency of assimilation and the diversity of the population, a new strategy of assimilation is given.Suppose thatWk is the set of all colonies in empire k.The detailed steps of ICA are shown below.Step 1.
Suppose colonyτ∈Wk, and compute the consistency between colony τ and their imperialist k. If the two solutions are consistent, judge whether the reserve country set Ω is empty. If it is empty, the colony will not be assimilated. Otherwise, randomly choose a reserve country λ instead of τ.Step 2.
If colonyτ is inconsistent with the imperialist k or has been replaced by the reserve country λ, the colony will move close to imperialist k through the global search operation.The global search inStep 2 is divided into two steps. First, suppose the imperialist k is ϕk=ϕ1k,ϕ2k,…,ϕM-1k and a colony τ∗ is ϕτ=ϕ1τ,ϕ2τ,…,ϕM-1τ; then the new colony ϕτ∗=ϕ1τ∗,ϕ2τ∗,…,ϕM-1τ∗ can be obtained according to formula (4).(4)ϕiτ∗=roundϕiτ+β×ϕik-ϕiτAmong them,β is a random number between 0 and 2. And the colony τ∗ can be amended and a feasible solution can be adjusted depending on formula (5).(5)ϕiτ∗=0ϕiτ∗<0L-1ϕiτ∗≥Lϕiτ∗0≤ϕiτ∗<LThe revised colonyϕτ∗ is assimilated colony and replaced colony τ with colony ϕτ∗.


### 3.3. Imperialist Self-Learning

Imperialist self-learning and reserve country set are introduced to strengthen the development of imperialists and retain some suboptimal individuals. The algorithm search efficiency can be improved by the local search of imperialist. The suboptimal individuals obtained in the process of imperialist self-learning can be retained in the reserve country set. Therefore, the excellent individuals can be made better use.For each imperialistk, ϕk=ϕ1k,ϕ2k,…,ϕKk new individuals ϕk∗ are obtained from the imperialist self-learning.The specific imperialist self-learning process is shown as follows:Step 1.
Letϕk∗=ϕk, generate a random integer α, and α∈1,M-1.Step 2.
Takeϕαk∗=ϕαk+rand×L, and rand is the random number between 0 and 1.Step 3.
Use the formula (6) to modify ϕαk∗, to get the new individual ϕk∗, and to calculate its cost value.Step 4.
If the cost valueϕk∗ is better than ϕk, ϕk will be put into the reserve country set Ω and replaced ϕk; otherwise, ϕk∗ will be put into the reserve country set. (6)ϕαk∗=0ϕαk∗<0L-1ϕαk∗≥Lϕiτ∗0≤ϕαk∗<LAmong them, the size of the reserve country setΩ is 3Nimp, and if the number of reserve countries is larger than 3Nimp, the weaker individuals will be eliminated according to the cost value.


### 3.4. Revolution

Revolution is another way for ICA to generate new solutions. A new way of revolution is proposed in this paper to balance the local search ability and global search ability and adopted different search strategies according to the problem characteristics.Neighborhood search for the better individuals in the population can often improve the quality of the solution [26]. However, for the weaker individuals in the population, it is often difficult to obtain the better individuals by neighborhood search. Therefore, it is considered to select a reserve country randomly to eliminate the weaker individuals in the population in order to improve the overall quality of the population. The process of the revolution in each colony τ is described as follows.Step 1.
Generate a random numberrand, if rand<pr continues; otherwise, end the revolution.Step 2.
If the cost valueτ is in the first 80% of all colonies, then the new solution τ∗ will directly replace the original colony τ after carrying out the consistent operation of the imperialist self-learning.Step 3.
If the cost value is in the worst 20% of all colonies, a reserve country fromΩ will be chosen randomly to replace the original colony τ.


### 3.5. Imperialist Competition

Imperialist competition is the process of colonies redistribution. The weakest empire needs to hand over the weakest colonies to the winning empire, but the weakest colonies of the weakest empire often contribute less to the evolution of the winning empire and will weaken the competitiveness of the winning empire. Therefore, this paper considers randomly selecting a reserve country fromΩ and compares with the weakest colonies of the weakest empire. Then, the better individual will be handed to the winning empire and the other is eliminated.The specific operation of imperialist competition is described below.Firstly, the empire total costTCk is calculated according to formula (7), which is related to the cost value of the imperialist k and all the empire colonies.(7)TCk=costk+ξ∑k=1NClcostlNCkAmong them, costl is the cost value of the colony l, and costk is the cost value of the imperialist k. Weight coefficient ξ is less than 1, and it is 0.1 according to Reference [21]. Calculate the empire power TPk=TCk/∑i=1NimpTCi. And the probability vector can be constructed as D=TP-R = TP1-r1,TP2-r2,…,TPNimp-rNimp. There, R is a random number vector between 0 and 1 and Nim is the number of imperialist. Suppose the value of element k in D is the largest; then the empire k is the winning empire. A reserve country x is randomly selected from Ω and compared x with the weakest colony x∗ in the weakest empire. If x is better than x∗, x∗ is replaced by x and x∗ is allocated to the winning empire. Otherwise, x∗ will be allocated directly to the winning empire.


### 3.6. Algorithm Flow

The flow chart of the hybrid imperialist competition algorithm is described in Figure1. The termination condition is that the evaluation number of the objective function reaches a set value max_it.Figure 1
Algorithm flow chart.


## 3.1. Initial Empire

The establishment of the initial empire is a key step for ICA. The specific steps are described as follows:Step 1.
Calculate the cost of countrylcostl=f, in which f is shown as the objective function value according to formula (2) or formula (3).Step 2.
Descending order for all solutions according to the cost of the country, choose the formerNimp countries as imperialists and the rest as the colony, so the number of colonies can be described as Ncol=N-Nimp.Step 3.
Calculate the power of each imperialistk, Pk=costk/∑i=1Nimpcosti, and determine the colony number NCl allocated to each imperialist k, where NCk=roundPk×Ncol. The specific allocation steps are shown in Reference [21].


## 3.2. Assimilation

The assimilation is a process that the colonies in the empire gradually move close to their imperialists. It will make the colonies similar to their imperialists. When the colony is the same as their imperialist, the assimilation operation will lose its effect and reduce the diversity of the population. Therefore, in order to improve the efficiency of assimilation and the diversity of the population, a new strategy of assimilation is given.Suppose thatWk is the set of all colonies in empire k.The detailed steps of ICA are shown below.Step 1.
Suppose colonyτ∈Wk, and compute the consistency between colony τ and their imperialist k. If the two solutions are consistent, judge whether the reserve country set Ω is empty. If it is empty, the colony will not be assimilated. Otherwise, randomly choose a reserve country λ instead of τ.Step 2.
If colonyτ is inconsistent with the imperialist k or has been replaced by the reserve country λ, the colony will move close to imperialist k through the global search operation.The global search inStep 2 is divided into two steps. First, suppose the imperialist k is ϕk=ϕ1k,ϕ2k,…,ϕM-1k and a colony τ∗ is ϕτ=ϕ1τ,ϕ2τ,…,ϕM-1τ; then the new colony ϕτ∗=ϕ1τ∗,ϕ2τ∗,…,ϕM-1τ∗ can be obtained according to formula (4).(4)ϕiτ∗=roundϕiτ+β×ϕik-ϕiτAmong them,β is a random number between 0 and 2. And the colony τ∗ can be amended and a feasible solution can be adjusted depending on formula (5).(5)ϕiτ∗=0ϕiτ∗<0L-1ϕiτ∗≥Lϕiτ∗0≤ϕiτ∗<LThe revised colonyϕτ∗ is assimilated colony and replaced colony τ with colony ϕτ∗.


## 3.3. Imperialist Self-Learning

Imperialist self-learning and reserve country set are introduced to strengthen the development of imperialists and retain some suboptimal individuals. The algorithm search efficiency can be improved by the local search of imperialist. The suboptimal individuals obtained in the process of imperialist self-learning can be retained in the reserve country set. Therefore, the excellent individuals can be made better use.For each imperialistk, ϕk=ϕ1k,ϕ2k,…,ϕKk new individuals ϕk∗ are obtained from the imperialist self-learning.The specific imperialist self-learning process is shown as follows:Step 1.
Letϕk∗=ϕk, generate a random integer α, and α∈1,M-1.Step 2.
Takeϕαk∗=ϕαk+rand×L, and rand is the random number between 0 and 1.Step 3.
Use the formula (6) to modify ϕαk∗, to get the new individual ϕk∗, and to calculate its cost value.Step 4.
If the cost valueϕk∗ is better than ϕk, ϕk will be put into the reserve country set Ω and replaced ϕk; otherwise, ϕk∗ will be put into the reserve country set. (6)ϕαk∗=0ϕαk∗<0L-1ϕαk∗≥Lϕiτ∗0≤ϕαk∗<LAmong them, the size of the reserve country setΩ is 3Nimp, and if the number of reserve countries is larger than 3Nimp, the weaker individuals will be eliminated according to the cost value.


## 3.4. Revolution

Revolution is another way for ICA to generate new solutions. A new way of revolution is proposed in this paper to balance the local search ability and global search ability and adopted different search strategies according to the problem characteristics.Neighborhood search for the better individuals in the population can often improve the quality of the solution [26]. However, for the weaker individuals in the population, it is often difficult to obtain the better individuals by neighborhood search. Therefore, it is considered to select a reserve country randomly to eliminate the weaker individuals in the population in order to improve the overall quality of the population. The process of the revolution in each colony τ is described as follows.Step 1.
Generate a random numberrand, if rand<pr continues; otherwise, end the revolution.Step 2.
If the cost valueτ is in the first 80% of all colonies, then the new solution τ∗ will directly replace the original colony τ after carrying out the consistent operation of the imperialist self-learning.Step 3.
If the cost value is in the worst 20% of all colonies, a reserve country fromΩ will be chosen randomly to replace the original colony τ.


## 3.5. Imperialist Competition

Imperialist competition is the process of colonies redistribution. The weakest empire needs to hand over the weakest colonies to the winning empire, but the weakest colonies of the weakest empire often contribute less to the evolution of the winning empire and will weaken the competitiveness of the winning empire. Therefore, this paper considers randomly selecting a reserve country fromΩ and compares with the weakest colonies of the weakest empire. Then, the better individual will be handed to the winning empire and the other is eliminated.The specific operation of imperialist competition is described below.Firstly, the empire total costTCk is calculated according to formula (7), which is related to the cost value of the imperialist k and all the empire colonies.(7)TCk=costk+ξ∑k=1NClcostlNCkAmong them, costl is the cost value of the colony l, and costk is the cost value of the imperialist k. Weight coefficient ξ is less than 1, and it is 0.1 according to Reference [21]. Calculate the empire power TPk=TCk/∑i=1NimpTCi. And the probability vector can be constructed as D=TP-R = TP1-r1,TP2-r2,…,TPNimp-rNimp. There, R is a random number vector between 0 and 1 and Nim is the number of imperialist. Suppose the value of element k in D is the largest; then the empire k is the winning empire. A reserve country x is randomly selected from Ω and compared x with the weakest colony x∗ in the weakest empire. If x is better than x∗, x∗ is replaced by x and x∗ is allocated to the winning empire. Otherwise, x∗ will be allocated directly to the winning empire.


## 3.6. Algorithm Flow

The flow chart of the hybrid imperialist competition algorithm is described in Figure1. The termination condition is that the evaluation number of the objective function reaches a set value max_it.Figure 1
Algorithm flow chart.


## 4. Computational Experiments

In order to verify the superiority of the novel imperialist competition algorithm in solving multithreshold segmentation, a lot of computational experiments have been carried out. These experiments' programming has been implemented and run on a computer with 16.0G RAM and 2.80 GHz by using MATLAB 2015b.


### 4.1. Selection of Test Examples and Comparison Algorithm

In this paper, 10 standard test pictures are selected for the experiment. Ten pictures are named Test1-Test10. Test1-Test4 is from USC-SIPI with the picture size512×512. Test5-Test10 is from BSD and the size of picture Test5 is 1024×1024. The size of Test6 and Test9-10 is 321×481. The size of Test7-8 is 481×321. They are shown in Figure 2.Figure 2
Test example.
(a)
Test1 (b)
Test2 (c)
Test3 (d)
Test4 (e)
Test5 (f)
Test6  (g)
Test7 (h)
Test8 (i)
Test9 (j)
Test10In order to verify the effectiveness of the new ICA, we compare ICA with particle swarm optimization (PSO [13]), Gray Wolf algorithm (GWO [3]), and teaching optimization algorithm (TLBO [14]), which is proposed in recent years to solve multithreshold segmentation problems.


### 4.2. Parameters Settings

Four important parameters must be tested in new ICA: population sizeN, the ratio of number of imperialists to size Nimp/N, revolutionary probability pm, and the maximum objective functions evaluation number max_it.In order to study the influence of parameter selection on the algorithm performance and to find the best combination of parameters, a four-level DOE experiment is designed using Test1 test picture. The parameters of each level are shown in Table1. According to the orthogonal Table 2, each group's parameters are run independently 10 times and the whole objective value is calculated. The results are shown in Table 2. According to the results obtained in Table 2, the influence of each parameter on the algorithm performance is ranked and it is shown in Table 3. The influence trend of each parameter on the performance of the algorithm is shown in Figure 3.Table 1
Parameter values at various levels.
parameters level 1 2 3 4 population sizeN 20 40 60 80 N i m p / N 5% 10% 15% 20% revolutionary probabilitypm 0.2 0.4 0.6 0.8 m a x _ i t 2000 4000 8000 10000Table 2
Parameter orthogonal table and object function values.
Combination number level object function values N N i m p / N p r m a x _ i t 1 1 1 1 1 17.9845 2 1 2 2 2 18.0123 3 1 3 3 3 18.0123 4 1 4 4 4 17.9946 5 2 1 2 3 17.9985 6 2 2 1 4 17.9985 7 2 3 4 1 17.9765 8 2 4 3 2 17.9865 9 3 1 3 4 18.0118 10 3 2 4 3 18.0083 11 3 3 1 2 18.0123 12 3 4 2 1 18.0006 13 4 1 4 2 17.9900 14 4 2 3 1 17.9908 15 4 3 2 4 18.0123 16 4 4 1 3 18.0118Table 3
Average of parameters.
Level N N i m p / N p r m a x _ i t 1 18.00 18.00 18.00 17.99 2 17.99 18.00 18.01 18.00 3 18.01 18.00 18.00 18.01 4 18.00 18.00 17.99 18.00 Delta 0.02 0.01 0.01 0.02 Rank 2 4 3 1Figure 3
Main effect diagram of mean value.From Table3, it can be seen that the ICA is mainly affected by the objective function evaluations maximum number and the population size. Thus, four parameters of the new imperial competition are as follows: N=60, Nimp/N=15%, Pr=0.4, and max_it=8000.


### 4.3. Comparative Analysis with Other Algorithms

In order to analyze the superiority of the new ICA in solving the multithreshold segmentation problem, three algorithms, PSO [13], GWO [3], TLBO [14], are selected as the comparison algorithms. The termination conditions of the three algorithms are set to 8000, and the other parameters are set according to References [3, 13, 14]. Tables 4 and 5 give the average values and maximum values of Kapur’s objective values obtained by the new ICA and three contrast algorithms on 40 groups of images with four kinds of levels 3-6. Tables 6 and 7 show the average values and maximum values of Ostu’s objective values obtained by four algorithms on 40 sets of examples. Tables 8 and 9 give the threshold values obtained by ICA, PSO, GWO, and TLBO based on Kapur’s entropy methods and Ostu’s between-class variance methods. Testi_M-1 means the results of Testi in M level. Figures 4 and 5 show the segmented images obtained by ICA-Kaur multilevel thresholding method and ICA-Ostu multilevel thresholding method. Table 10 shows the results of statistical analysis. We set confidence level as 95%. When P_value≤0.05, the difference between algorithms is significant. Table 11 gives the average CPU time of all algorithms.Table 4
The maximum values of Kapur’s objective values.
Example ICA PSO GWO TLBO Example ICA PSO GWO TLBO Test1_2 12.3466 12.3466 12.3463 12.3466 Test6_2 12.6616 12.6616 12.6616 12.6616 3 15.3183 15.3183 15.3183 15.3183 3 15.8436 15.8436 15.8431 15.8436 4 18.0123 18.0123 17.9996 18.0109 4 18.6570 18.6570 18.6481 18.6545 5 20.6095 20.6095 20.6008 20.6078 5 21.4315 21.4315 21.3996 21.4295 Test2_2 12.2115 12.2115 12.2115 12.2115 Test7_2 12.9606 12.9606 12.9605 12.9606 3 15.5039 15.5039 15.4997 15.5039 3 16.2204 16.2204 16.2198 16.2204 4 18.3121 18.3121 18.3072 18.3110 4 19.3650 19.3650 19.3181 19.3645 5 20.9088 20.9088 20.4932 20.9088 5 22.1924 22.1924 21.9295 22.1899 Test3_2 12.2178 12.2178 12.2178 12.2178 Test8_2 11.2873 11.2873 11.2873 11.2873 3 15.2792 15.2792 15.2750 15.2792 3 14.1185 14.1185 14.1148 14.1185 4 18.1267 18.1267 18.1216 18.1267 4 16.7098 16.7098 16.7087 16.7090 5 20.7896 20.7896 20.7226 20.7890 5 19.7066 19.7066 19.5136 19.7055 Test4_2 12.6346 12.6346 12.6346 12.6346 Test9_2 12.6390 12.6390 12.6390 12.6390 3 15.6887 15.6887 15.6887 15.6887 3 15.5952 15.5952 15.5952 15.5950 4 18.5394 18.5394 18.4917 18.5387 4 18.5295 18.5295 18.5183 18.5292 5 21.2818 21.2818 21.0860 21.2746 5 21.1922 21.1922 21.1899 21.1853 Test5_2 12.6357 12.6357 12.6355 12.6357 Test10_2 12.9682 12.9682 12.9682 12.9682 3 15.8115 15.8115 15.8075 15.8115 3 16.1254 16.1254 16.1145 16.1254 4 18.7441 18.7441 18.6589 18.7362 4 19.0579 19.0579 19.0300 19.0576 5 21.6195 21.6195 21.5462 21.6057 5 21.8090 21.8090 21.8043 21.8050Table 5
The average values of Kapur’s objective values.
Example ICA PSO GWO TLBO Example ICA PSO GWO TLBO Test1_2 12.3466 12.3466 12.2945 12.3466 Test6_2 12.6616 12.6616 12.6167 12.6616 3 15.3183 15.3183 15.2436 15.3181 3 15.8436 15.8436 15.7336 15.8434 4 18.0095 18.0057 17.8894 18.0023 4 18.6512 18.6436 18.3885 18.6363 5 20.6095 20.6095 20.1170 20.5508 5 21.4315 21.4315 20.8351 21.4053 Test2_2 12.2115 12.2115 12.1910 12.2114 Test7_2 12.9606 12.9606 12.9265 12.9606 3 15.5039 15.5039 15.2467 15.5013 3 16.2204 16.2204 16.0922 16.2178 4 18.3121 18.3121 18.0079 18.3087 4 19.3650 19.3650 19.0597 19.3487 5 20.9087 20.9087 20.0152 20.8759 5 22.1924 22.1393 21.5222 22.1388 Test3_2 12.2178 12.2178 12.1838 12.2178 Test8_2 11.2873 11.2873 11.2329 11.2873 3 15.2792 15.2792 15.1968 15.2790 3 14.1184 14.1185 13.9991 14.1177 4 18.1267 18.1264 17.9916 18.1130 4 16.6985 16.6646 16.5734 16.6814 5 20.7896 20.7895 20.3838 20.7483 5 19.7066 19.6472 19.1285 19.6403 Test4_2 12.6346 12.6346 12.5996 12.6346 Test9_2 12.6390 12.6390 12.6015 12.6390 3 15.6887 15.6887 15.6841 15.6877 3 15.5952 15.5952 15.5491 15.5936 4 18.5221 18.5066 18.3674 18.5258 4 18.5295 18.5295 18.3060 18.5248 5 21.2818 21.2626 20.6440 21.2553 5 21.1794 21.1922 20.8378 21.1554 Test5_2 12.6357 12.6357 12.5795 12.6357 Test10_2 12.9682 12.9682 12.9528 12.9682 3 15.8115 15.8115 15.6541 15.8107 3 16.1254 16.1254 15.9787 16.1251 4 18.7347 18.6983 18.4361 18.7110 4 19.0579 19.0579 18.7716 19.0463 5 21.6195 21.5351 21.1728 21.5724 5 21.8082 21.8088 21.3168 21.7935Table 6
The maximum values of Ostu’s objective values.
Example ICA PSO GWO TLBO Example ICA PSO GWO TLBO Test1_2 1961.817 1961.817 1961.790 1961.817 Test6_2 2976.191 2976.191 2976.191 2976.191 3 2128.308 2128.308 2128.308 2128.308 3 3145.463 3145.463 3145.463 3145.463 4 2191.870 2191.870 2190.602 2191.870 4 3216.403 3216.403 3215.399 3216.271 5 2217.799 2217.799 2209.967 2217.726 5 3255.995 3255.995 3238.241 3255.908 Test2_2 1948.719 1948.719 1948.719 1948.719 Test7_2 7209.906 7209.906 7208.633 7209.906 3 2024.834 2024.834 2024.834 2024.834 3 7531.419 7531.419 7531.235 7531.419 4 2070.077 2070.077 2066.278 2070.057 4 7617.718 7617.718 7602.299 7617.682 5 2096.139 2096.139 2075.997 2095.610 5 7666.598 7666.598 7662.395 7666.079 Test3_2 1549.083 1549.083 1549.052 1549.083 Test8_2 5068.434 5068.434 5068.434 5068.434 3 1639.532 1639.532 1639.532 1639.532 3 5232.919 5232.919 5232.833 5232.919 4 1693.195 1693.195 1692.378 1693.195 4 5310.356 5310.356 5309.691 5310.265 5 1719.057 1719.057 1718.145 1718.883 5 5349.222 5349.222 5346.560 5349.043 Test4_2 2532.321 2532.321 2532.321 2532.321 Test9_2 1665.411 1665.411 1665.411 1665.411 3 2703.572 2703.572 2703.374 2703.572 3 1746.197 1746.197 1745.604 1746.197 4 2766.459 2766.459 2765.768 2766.371 4 1797.124 1797.124 1797.098 1797.015 5 2810.842 2810.842 2809.046 2810.694 5 1823.625 1823.625 1821.739 1823.002 Test5_2 2947.502 2947.502 2947.502 2947.502 Test10_2 2551.975 2551.975 2551.975 2551.975 3 3126.855 3126.855 3126.467 3126.855 3 2784.227 2784.227 2784.227 2784.227 4 3208.810 3208.810 3208.694 3208.694 4 2869.431 2869.431 2869.415 2869.431 5 3254.797 3254.797 3248.700 3254.562 5 2916.273 2916.273 2915.191 2916.087Table 7
The average values of Ostu’s objective values.
Example ICA PSO GWO TLBO Example ICA PSO GWO TLBO Test1_2 1961.817 1961.817 1938.178 1961.812 Test6_2 2976.191 2976.191 2953.020 2976.191 3 2128.308 2128.308 2090.090 2128.195 3 3145.463 3145.463 3095.006 3145.392 4 2191.870 2191.870 2154.799 2191.482 4 3216.403 3216.403 3163.828 3215.248 5 2217.231 2217.516 2178.598 2216.291 5 3255.995 3253.210 3209.790 3253.837 Test2_2 1948.719 1948.719 1943.368 1948.719 Test7_2 7209.906 7209.906 7181.118 7209.906 3 2024.831 2024.834 2015.234 2024.772 3 7531.419 7531.419 7414.411 7531.202 4 2070.077 2070.076 2045.085 2069.334 4 7617.718 7617.718 7556.210 7614.405 5 2096.139 2096.135 2062.247 2094.870 5 7665.778 7664.147 7621.531 7662.482 Test3_2 1549.083 1549.083 1518.499 1549.083 Test8_2 5068.434 5068.434 5033.695 5068.434 3 1639.532 1639.532 1620.771 1639.385 3 5232.919 5232.919 5209.283 5232.787 4 1693.195 1693.190 1647.427 1692.546 4 5310.356 5310.356 5273.633 5309.334 5 1719.041 1719.044 1694.932 1717.250 5 5349.222 5349.221 5317.495 5347.035 Test4_2 2532.321 2532.321 2519.423 2532.312 Test9_2 1665.411 1665.411 1657.895 1665.411 3 2703.572 2703.572 2667.182 2703.274 3 1746.197 1746.197 1739.602 1746.118 4 2766.459 2766.459 2704.149 2765.460 4 1797.117 1797.124 1778.585 1796.539 5 2810.842 2810.838 2767.802 2809.198 5 1823.590 1823.618 1796.961 1821.105 Test5_2 2947.502 2947.502 2905.698 2947.500 Test10_2 2551.975 2551.975 2512.428 2551.975 3 3126.855 3126.855 3093.664 3126.821 3 2784.227 2784.227 2749.713 2783.919 4 3208.810 3208.810 3177.217 3207.818 4 2869.428 2869.431 2827.326 2868.631 5 3254.775 3254.797 3223.476 3251.292 5 2916.271 2916.270 2870.529 2913.415Table 8
The threshold values obtained by ICA, PSO, GWO, and TLBO based on Kapur’s entropy methods.
Example ICA PSO GWO TLBO Test1_2 97 164 97 164 98 164 97 164 3 82 126 175 82 126 175 82 126 175 82 126 175 4 64 97 137 179 64 97 137 179 62 97 140 178 64 97 138 180 5 63 94 128 163 194 63 94 128 163 194 62 94 129 161 194 62 93 126 162 193 Test2_2 71 173 71 173 71 173 71 173 3 69 127 183 69 127 183 67 127 183 69 127 183 4 67 106 145 185 67 106 145 185 67 104 146 185 66 105 144 185 5 60 89 123 155 187 60 89 123 155 187 44 90 125 142 183 60 89 123 155 187 Test3_2 78 143 78 143 18 143 78 143 3 45 98 152 45 98 152 52 103 154 45 98 152 4 33 73 114 159 33 73 114 159 38 78 118 161 33 73 114 159 5 33 68 103 138 172 33 68 103 138 172 28 54 94 134 171 33 68 103 137 171 Test4_2 75 147 75 147 75 147 75 147 3 61 113 165 61 113 165 61 113 165 61 113 165 4 58 105 148 194 58 105 148 194 62 100 149 196 57 105 149 194 5 42 77 114 154 195 42 77 114 154 195 35 83 133 164 194 40 76 116 155 195 Test5_2 91 172 91 172 90 172 91 172 3 62 117 174 62 117 174 64 115 174 62 117 174 4 62 117 174 230 62 117 174 230 50 100 136 176 59 115 173 230 5 46 89 131 174 230 46 89 131 174 230 36 88 131 172 231 42 90 134 175 230 Test6_2 99 172 99 172 99 172 99 172 3 88 138 194 88 138 194 87 138 194 88 138 194 4 87 135 174 214 87 135 174 214 89 132 172 214 88 134 173 214 5 63 98 136 174 214 63 98 136 174 214 62 95 139 172 212 63 98 137 174 215 Test7_2 94 168 94 168 94 171 94 168 3 53 110 164 53 110 164 52 110 164 53 110 164 4 53 110 163 212 53 110 163 212 54 104 165 211 55 110 163 212 5 38 78 118 163 212 38 78 118 163 212 46 99 138 175 207 38 76 118 163 213 Test8_2 48 104 48 104 48 104 48 104 3 36 72 115 36 72 115 41 75 116 36 72 115 4 35 69 102 135 35 69 102 135 36 69 102 135 35 69 102 134 5 46 101 153 201 243 46 101 153 201 243 36 72 147 192 244 46 102 153 202 242 Test9_2 115 181 115 181 115 181 115 181 3 112 158 202 112 158 202 112 158 202 74 119 183 4 73 116 161 205 73 116 161 205 75 116 156 200 74 116 161 205 5 73 114 150 185 220 73 114 150 185 220 71 112 149 183 220 71 115 150 184 221 Test10_2 91 170 91 170 91 170 91 170 3 75 130 183 75 130 183 80 130 183 75 130 183 4 68 116 164 206 68 116 164 206 73 113 166 208 67 116 164 206 5 56 93 132 170 209 56 93 132 170 209 55 95 134 174 212 54 90 129 170 209Table 9
The threshold values obtained by ICA, PSO, GWO, and TLBO based on Ostu’s between-class variance methods.
Example ICA PSO GWO TLBO Test1_2 93 151 93 151 92 151 93 151 3 81 127 171 81 127 171 81 127 171 81 127 171 4 75 114 145 180 75 114 145 180 75 111 142 181 75 113 145 180 5 73 109 136 160 188 73 109 136 160 188 70 86 119 150 180 74 110 137 160 188 Test2_2 113 173 113 173 113 173 113 173 3 93 145 191 93 145 191 93 145 191 93 145 191 4 84 129 172 203 84 129 172 203 93 128 170 203 85 129 173 203 5 69 107 143 180 205 69 107 143 180 205 90 124 162 171 204 70 104 142 178 205 Test3_2 98 150 98 150 97 150 98 150 3 86 125 161 86 125 161 86 125 161 86 125 161 4 72 106 137 168 72 106 137 168 68 105 136 167 72 106 137 168 5 68 100 126 150 175 68 100 126 150 175 63 95 123 149 175 66 97 123 148 174 Test4_2 68 135 68 135 68 135 68 135 3 63 119 166 63 119 166 61 118 166 63 119 166 4 47 86 126 169 47 86 126 169 49 88 129 170 47 85 125 169 5 43 79 113 146 177 43 79 113 146 177 48 82 111 144 176 42 78 112 147 178 Test5_2 57 124 57 124 57 124 57 124 3 39 92 142 39 92 142 40 93 141 39 92 142 4 35 82 124 164 35 82 124 164 35 81 123 164 35 81 123 164 5 28 65 100 134 172 28 65 100 134 172 37 75 102 134 172 28 64 100 133 172 Test6_2 87 162 87 162 87 162 87 162 3 82 136 190 82 136 190 82 136 190 82 136 190 4 74 112 149 197 74 112 149 197 76 110 147 196 74 112 149 198 5 73 109 143 183 221 73 109 143 183 221 71 88 116 150 197 72 108 143 183 220 Test7_2 60 157 60 157 62 155 60 157 3 54 131 206 54 131 206 53 131 206 54 131 206 4 40 84 143 208 40 84 143 208 49 96 135 206 40 84 142 208 5 39 81 136 191 227 39 81 136 191 227 45 81 134 191 229 38 81 138 194 228 Test8_2 108 196 108 196 108 196 108 196 3 89 135 206 89 135 206 90 135 206 89 135 206 4 68 106 145 210 68 106 145 210 71 108 145 209 69 106 145 210 5 63 97 126 157 215 63 97 126 157 215 69 97 124 158 214 63 98 127 158 218 Test9_2 78 144 78 144 78 144 78 144 3 75 119 177 75 119 177 75 122 177 75119 177 4 66 90 131 186 66 90 131 186 65 89 131 186 66 90 132 188 5 59 79 101 140 191 59 79 101 140 191 61 82 106 151 192 59 78 100 141 195 Test10_2 85 157 85 157 85 157 85 157 3 69 120 178 69 120 178 69 120 178 69 120 178 4 60 101 138 187 60 101 138 187 61 101 138 187 60 101 138 187 5 52 86 117 150 194 52 86 117 150 194 53 86 114 147 194 52 87 118 150 195Table 10
Statistical analysis results.
t-test P _ v a l u e D I R P _ v a l u e ρ t-test(ICA,PSO) 0.001 0.041 t-test(ICA,GWO) ≤ 0.001 ≤ 0.001 t-test(ICA,TLBO) ≤ 0.001 ≤ 0.001Table 11
The average CPU time of all algorithms (in seconds).
Example ICA PSO GWO TLBO Example ICA PSO GWO TLBO Test1_2 1.0148 1.8466 1.0573 1.2695 Test6_2 0.9013 1.8463 0.9634 1.1757 3 0.9968 1.9547 1.0824 1.2020 3 1.0516 2.0183 1.0631 1.4172 4 1.0817 2.1041 1.0777 1.2727 4 1.0725 2.1534 1.0789 1.3710 5 1.1677 2.2427 1.1135 1.3929 5 1.0992 2.2310 1.1540 1.4416 Test2_2 0.9160 2.1234 0.9569 1.1933 Test7_2 0.9479 1.9786 1.0088 1.2306 3 1.0238 2.1349 1.0966 1.2583 3 1.0616 2.0336 1.0571 1.3009 4 1.0121 2.2104 1.1203 1.4549 4 1.0868 2.2236 1.1175 1.3781 5 1.0935 2.2470 1.1195 1.4557 5 1.1588 2.3927 1.1682 1.4598 Test3_2 0.9684 1.8421 0.9593 1.2024 Test8_2 0.9654 1.9061 1.0042 1.2234 3 0.9690 1.9660 1.0271 1.2812 3 1.0587 2.0592 1.0648 1.2818 4 1.0981 2.1097 1.0739 1.3281 4 1.1558 2.2072 1.1166 1.3552 5 1.0956 2.2191 1.1637 1.4219 5 1.1965 2.3133 1.1706 1.4322 Test4_2 0.9394 1.8492 0.9677 1.2796 Test9_2 0.9599 1.8732 0.9750 1.2245 3 1.0026 1.9714 1.0213 1.2786 3 1.0476 1.9952 1.0556 1.2709 4 1.0562 2.1719 1.1014 1.3773 4 1.0656 2.1281 1.0893 1.3672 5 1.1098 2.2701 1.1188 1.3913 5 1.1195 2.2444 1.1403 1.4266 Test5_2 0.9514 2.3564 0.9939 1.2206 Test10_2 0.9563 1.9158 0.9922 1.3281 3 1.0401 2.0438 1.0557 1.2901 3 1.0342 2.0040 1.0496 1.3778 4 1.0839 2.1590 1.1043 1.3911 4 1.0759 2.2321 1.0948 1.3701 5 1.1561 2.3302 1.1637 1.4313 5 1.1585 2.3338 1.1481 1.4289Figure 4
Segmented images obtained by ICA-Kapur multilevel thresholding method (a) represent 3-level thresholding, (b) represent 3-level thresholding, (c) represent 4-level thresholding, and (d) represent 6-level thresholding.Figure 5
Segmented images obtained by ICA-Ostu multilevel thresholding method (a) represent 3-level thresholding, (b) represent 3-level thresholding, (c) represent 4-level thresholding, and (d) represent 6-level thresholding.Firstly, compared with the four algorithms in solving Kapur’s objective value, ICA has obtained the optimal solution in all 40 groups of examples, of which 30 groups are better than GWO and 19 groups are better than TLBO. Aiming at the average solution of 20 runs, among 40 groups of instances, the average solution of ICA has achieved better or equal to the results of the other four algorithms. Among them, 11 groups of instances are better than PSO, 30 groups are better than TLBO, and all instances are better than GWO. This also proves the stability of ICA algorithm. Thus ICA can get better results than other three algorithms on most of instances in similar computation times and has advantages in solving multithreshold image segmentation.For Ostu’s objective value, ICA also obtained all the best solutions in 40 groups of examples, and 28 groups of examples were better than GWO and 17 groups of examples were better than TLBO, which further verified the high quality of ICA solution. In terms of the average solution of 20 runs, ICA achieved better or equal results to the other four algorithms in 33 groups of instances, and 8 groups of instances were better than PSO, 33 groups were better than TLBO, and 40 groups were better than GWO. Furthermore, the stability of ICA solution is verified again. The statistical results in Table10 also validate this conclusion, which proves the superiority of ICA in solving multithreshold segmentation.


## 4.1. Selection of Test Examples and Comparison Algorithm

In this paper, 10 standard test pictures are selected for the experiment. Ten pictures are named Test1-Test10. Test1-Test4 is from USC-SIPI with the picture size512×512. Test5-Test10 is from BSD and the size of picture Test5 is 1024×1024. The size of Test6 and Test9-10 is 321×481. The size of Test7-8 is 481×321. They are shown in Figure 2.Figure 2
Test example.
(a)
Test1 (b)
Test2 (c)
Test3 (d)
Test4 (e)
Test5 (f)
Test6  (g)
Test7 (h)
Test8 (i)
Test9 (j)
Test10In order to verify the effectiveness of the new ICA, we compare ICA with particle swarm optimization (PSO [13]), Gray Wolf algorithm (GWO [3]), and teaching optimization algorithm (TLBO [14]), which is proposed in recent years to solve multithreshold segmentation problems.


## 4.2. Parameters Settings

Four important parameters must be tested in new ICA: population sizeN, the ratio of number of imperialists to size Nimp/N, revolutionary probability pm, and the maximum objective functions evaluation number max_it.In order to study the influence of parameter selection on the algorithm performance and to find the best combination of parameters, a four-level DOE experiment is designed using Test1 test picture. The parameters of each level are shown in Table1. According to the orthogonal Table 2, each group's parameters are run independently 10 times and the whole objective value is calculated. The results are shown in Table 2. According to the results obtained in Table 2, the influence of each parameter on the algorithm performance is ranked and it is shown in Table 3. The influence trend of each parameter on the performance of the algorithm is shown in Figure 3.Table 1
Parameter values at various levels.
parameters level 1 2 3 4 population sizeN 20 40 60 80 N i m p / N 5% 10% 15% 20% revolutionary probabilitypm 0.2 0.4 0.6 0.8 m a x _ i t 2000 4000 8000 10000Table 2
Parameter orthogonal table and object function values.
Combination number level object function values N N i m p / N p r m a x _ i t 1 1 1 1 1 17.9845 2 1 2 2 2 18.0123 3 1 3 3 3 18.0123 4 1 4 4 4 17.9946 5 2 1 2 3 17.9985 6 2 2 1 4 17.9985 7 2 3 4 1 17.9765 8 2 4 3 2 17.9865 9 3 1 3 4 18.0118 10 3 2 4 3 18.0083 11 3 3 1 2 18.0123 12 3 4 2 1 18.0006 13 4 1 4 2 17.9900 14 4 2 3 1 17.9908 15 4 3 2 4 18.0123 16 4 4 1 3 18.0118Table 3
Average of parameters.
Level N N i m p / N p r m a x _ i t 1 18.00 18.00 18.00 17.99 2 17.99 18.00 18.01 18.00 3 18.01 18.00 18.00 18.01 4 18.00 18.00 17.99 18.00 Delta 0.02 0.01 0.01 0.02 Rank 2 4 3 1Figure 3
Main effect diagram of mean value.From Table3, it can be seen that the ICA is mainly affected by the objective function evaluations maximum number and the population size. Thus, four parameters of the new imperial competition are as follows: N=60, Nimp/N=15%, Pr=0.4, and max_it=8000.


## 4.3. Comparative Analysis with Other Algorithms

In order to analyze the superiority of the new ICA in solving the multithreshold segmentation problem, three algorithms, PSO [13], GWO [3], TLBO [14], are selected as the comparison algorithms. The termination conditions of the three algorithms are set to 8000, and the other parameters are set according to References [3, 13, 14]. Tables 4 and 5 give the average values and maximum values of Kapur’s objective values obtained by the new ICA and three contrast algorithms on 40 groups of images with four kinds of levels 3-6. Tables 6 and 7 show the average values and maximum values of Ostu’s objective values obtained by four algorithms on 40 sets of examples. Tables 8 and 9 give the threshold values obtained by ICA, PSO, GWO, and TLBO based on Kapur’s entropy methods and Ostu’s between-class variance methods. Testi_M-1 means the results of Testi in M level. Figures 4 and 5 show the segmented images obtained by ICA-Kaur multilevel thresholding method and ICA-Ostu multilevel thresholding method. Table 10 shows the results of statistical analysis. We set confidence level as 95%. When P_value≤0.05, the difference between algorithms is significant. Table 11 gives the average CPU time of all algorithms.Table 4
The maximum values of Kapur’s objective values.
Example ICA PSO GWO TLBO Example ICA PSO GWO TLBO Test1_2 12.3466 12.3466 12.3463 12.3466 Test6_2 12.6616 12.6616 12.6616 12.6616 3 15.3183 15.3183 15.3183 15.3183 3 15.8436 15.8436 15.8431 15.8436 4 18.0123 18.0123 17.9996 18.0109 4 18.6570 18.6570 18.6481 18.6545 5 20.6095 20.6095 20.6008 20.6078 5 21.4315 21.4315 21.3996 21.4295 Test2_2 12.2115 12.2115 12.2115 12.2115 Test7_2 12.9606 12.9606 12.9605 12.9606 3 15.5039 15.5039 15.4997 15.5039 3 16.2204 16.2204 16.2198 16.2204 4 18.3121 18.3121 18.3072 18.3110 4 19.3650 19.3650 19.3181 19.3645 5 20.9088 20.9088 20.4932 20.9088 5 22.1924 22.1924 21.9295 22.1899 Test3_2 12.2178 12.2178 12.2178 12.2178 Test8_2 11.2873 11.2873 11.2873 11.2873 3 15.2792 15.2792 15.2750 15.2792 3 14.1185 14.1185 14.1148 14.1185 4 18.1267 18.1267 18.1216 18.1267 4 16.7098 16.7098 16.7087 16.7090 5 20.7896 20.7896 20.7226 20.7890 5 19.7066 19.7066 19.5136 19.7055 Test4_2 12.6346 12.6346 12.6346 12.6346 Test9_2 12.6390 12.6390 12.6390 12.6390 3 15.6887 15.6887 15.6887 15.6887 3 15.5952 15.5952 15.5952 15.5950 4 18.5394 18.5394 18.4917 18.5387 4 18.5295 18.5295 18.5183 18.5292 5 21.2818 21.2818 21.0860 21.2746 5 21.1922 21.1922 21.1899 21.1853 Test5_2 12.6357 12.6357 12.6355 12.6357 Test10_2 12.9682 12.9682 12.9682 12.9682 3 15.8115 15.8115 15.8075 15.8115 3 16.1254 16.1254 16.1145 16.1254 4 18.7441 18.7441 18.6589 18.7362 4 19.0579 19.0579 19.0300 19.0576 5 21.6195 21.6195 21.5462 21.6057 5 21.8090 21.8090 21.8043 21.8050Table 5
The average values of Kapur’s objective values.
Example ICA PSO GWO TLBO Example ICA PSO GWO TLBO Test1_2 12.3466 12.3466 12.2945 12.3466 Test6_2 12.6616 12.6616 12.6167 12.6616 3 15.3183 15.3183 15.2436 15.3181 3 15.8436 15.8436 15.7336 15.8434 4 18.0095 18.0057 17.8894 18.0023 4 18.6512 18.6436 18.3885 18.6363 5 20.6095 20.6095 20.1170 20.5508 5 21.4315 21.4315 20.8351 21.4053 Test2_2 12.2115 12.2115 12.1910 12.2114 Test7_2 12.9606 12.9606 12.9265 12.9606 3 15.5039 15.5039 15.2467 15.5013 3 16.2204 16.2204 16.0922 16.2178 4 18.3121 18.3121 18.0079 18.3087 4 19.3650 19.3650 19.0597 19.3487 5 20.9087 20.9087 20.0152 20.8759 5 22.1924 22.1393 21.5222 22.1388 Test3_2 12.2178 12.2178 12.1838 12.2178 Test8_2 11.2873 11.2873 11.2329 11.2873 3 15.2792 15.2792 15.1968 15.2790 3 14.1184 14.1185 13.9991 14.1177 4 18.1267 18.1264 17.9916 18.1130 4 16.6985 16.6646 16.5734 16.6814 5 20.7896 20.7895 20.3838 20.7483 5 19.7066 19.6472 19.1285 19.6403 Test4_2 12.6346 12.6346 12.5996 12.6346 Test9_2 12.6390 12.6390 12.6015 12.6390 3 15.6887 15.6887 15.6841 15.6877 3 15.5952 15.5952 15.5491 15.5936 4 18.5221 18.5066 18.3674 18.5258 4 18.5295 18.5295 18.3060 18.5248 5 21.2818 21.2626 20.6440 21.2553 5 21.1794 21.1922 20.8378 21.1554 Test5_2 12.6357 12.6357 12.5795 12.6357 Test10_2 12.9682 12.9682 12.9528 12.9682 3 15.8115 15.8115 15.6541 15.8107 3 16.1254 16.1254 15.9787 16.1251 4 18.7347 18.6983 18.4361 18.7110 4 19.0579 19.0579 18.7716 19.0463 5 21.6195 21.5351 21.1728 21.5724 5 21.8082 21.8088 21.3168 21.7935Table 6
The maximum values of Ostu’s objective values.
Example ICA PSO GWO TLBO Example ICA PSO GWO TLBO Test1_2 1961.817 1961.817 1961.790 1961.817 Test6_2 2976.191 2976.191 2976.191 2976.191 3 2128.308 2128.308 2128.308 2128.308 3 3145.463 3145.463 3145.463 3145.463 4 2191.870 2191.870 2190.602 2191.870 4 3216.403 3216.403 3215.399 3216.271 5 2217.799 2217.799 2209.967 2217.726 5 3255.995 3255.995 3238.241 3255.908 Test2_2 1948.719 1948.719 1948.719 1948.719 Test7_2 7209.906 7209.906 7208.633 7209.906 3 2024.834 2024.834 2024.834 2024.834 3 7531.419 7531.419 7531.235 7531.419 4 2070.077 2070.077 2066.278 2070.057 4 7617.718 7617.718 7602.299 7617.682 5 2096.139 2096.139 2075.997 2095.610 5 7666.598 7666.598 7662.395 7666.079 Test3_2 1549.083 1549.083 1549.052 1549.083 Test8_2 5068.434 5068.434 5068.434 5068.434 3 1639.532 1639.532 1639.532 1639.532 3 5232.919 5232.919 5232.833 5232.919 4 1693.195 1693.195 1692.378 1693.195 4 5310.356 5310.356 5309.691 5310.265 5 1719.057 1719.057 1718.145 1718.883 5 5349.222 5349.222 5346.560 5349.043 Test4_2 2532.321 2532.321 2532.321 2532.321 Test9_2 1665.411 1665.411 1665.411 1665.411 3 2703.572 2703.572 2703.374 2703.572 3 1746.197 1746.197 1745.604 1746.197 4 2766.459 2766.459 2765.768 2766.371 4 1797.124 1797.124 1797.098 1797.015 5 2810.842 2810.842 2809.046 2810.694 5 1823.625 1823.625 1821.739 1823.002 Test5_2 2947.502 2947.502 2947.502 2947.502 Test10_2 2551.975 2551.975 2551.975 2551.975 3 3126.855 3126.855 3126.467 3126.855 3 2784.227 2784.227 2784.227 2784.227 4 3208.810 3208.810 3208.694 3208.694 4 2869.431 2869.431 2869.415 2869.431 5 3254.797 3254.797 3248.700 3254.562 5 2916.273 2916.273 2915.191 2916.087Table 7
The average values of Ostu’s objective values.
Example ICA PSO GWO TLBO Example ICA PSO GWO TLBO Test1_2 1961.817 1961.817 1938.178 1961.812 Test6_2 2976.191 2976.191 2953.020 2976.191 3 2128.308 2128.308 2090.090 2128.195 3 3145.463 3145.463 3095.006 3145.392 4 2191.870 2191.870 2154.799 2191.482 4 3216.403 3216.403 3163.828 3215.248 5 2217.231 2217.516 2178.598 2216.291 5 3255.995 3253.210 3209.790 3253.837 Test2_2 1948.719 1948.719 1943.368 1948.719 Test7_2 7209.906 7209.906 7181.118 7209.906 3 2024.831 2024.834 2015.234 2024.772 3 7531.419 7531.419 7414.411 7531.202 4 2070.077 2070.076 2045.085 2069.334 4 7617.718 7617.718 7556.210 7614.405 5 2096.139 2096.135 2062.247 2094.870 5 7665.778 7664.147 7621.531 7662.482 Test3_2 1549.083 1549.083 1518.499 1549.083 Test8_2 5068.434 5068.434 5033.695 5068.434 3 1639.532 1639.532 1620.771 1639.385 3 5232.919 5232.919 5209.283 5232.787 4 1693.195 1693.190 1647.427 1692.546 4 5310.356 5310.356 5273.633 5309.334 5 1719.041 1719.044 1694.932 1717.250 5 5349.222 5349.221 5317.495 5347.035 Test4_2 2532.321 2532.321 2519.423 2532.312 Test9_2 1665.411 1665.411 1657.895 1665.411 3 2703.572 2703.572 2667.182 2703.274 3 1746.197 1746.197 1739.602 1746.118 4 2766.459 2766.459 2704.149 2765.460 4 1797.117 1797.124 1778.585 1796.539 5 2810.842 2810.838 2767.802 2809.198 5 1823.590 1823.618 1796.961 1821.105 Test5_2 2947.502 2947.502 2905.698 2947.500 Test10_2 2551.975 2551.975 2512.428 2551.975 3 3126.855 3126.855 3093.664 3126.821 3 2784.227 2784.227 2749.713 2783.919 4 3208.810 3208.810 3177.217 3207.818 4 2869.428 2869.431 2827.326 2868.631 5 3254.775 3254.797 3223.476 3251.292 5 2916.271 2916.270 2870.529 2913.415Table 8
The threshold values obtained by ICA, PSO, GWO, and TLBO based on Kapur’s entropy methods.
Example ICA PSO GWO TLBO Test1_2 97 164 97 164 98 164 97 164 3 82 126 175 82 126 175 82 126 175 82 126 175 4 64 97 137 179 64 97 137 179 62 97 140 178 64 97 138 180 5 63 94 128 163 194 63 94 128 163 194 62 94 129 161 194 62 93 126 162 193 Test2_2 71 173 71 173 71 173 71 173 3 69 127 183 69 127 183 67 127 183 69 127 183 4 67 106 145 185 67 106 145 185 67 104 146 185 66 105 144 185 5 60 89 123 155 187 60 89 123 155 187 44 90 125 142 183 60 89 123 155 187 Test3_2 78 143 78 143 18 143 78 143 3 45 98 152 45 98 152 52 103 154 45 98 152 4 33 73 114 159 33 73 114 159 38 78 118 161 33 73 114 159 5 33 68 103 138 172 33 68 103 138 172 28 54 94 134 171 33 68 103 137 171 Test4_2 75 147 75 147 75 147 75 147 3 61 113 165 61 113 165 61 113 165 61 113 165 4 58 105 148 194 58 105 148 194 62 100 149 196 57 105 149 194 5 42 77 114 154 195 42 77 114 154 195 35 83 133 164 194 40 76 116 155 195 Test5_2 91 172 91 172 90 172 91 172 3 62 117 174 62 117 174 64 115 174 62 117 174 4 62 117 174 230 62 117 174 230 50 100 136 176 59 115 173 230 5 46 89 131 174 230 46 89 131 174 230 36 88 131 172 231 42 90 134 175 230 Test6_2 99 172 99 172 99 172 99 172 3 88 138 194 88 138 194 87 138 194 88 138 194 4 87 135 174 214 87 135 174 214 89 132 172 214 88 134 173 214 5 63 98 136 174 214 63 98 136 174 214 62 95 139 172 212 63 98 137 174 215 Test7_2 94 168 94 168 94 171 94 168 3 53 110 164 53 110 164 52 110 164 53 110 164 4 53 110 163 212 53 110 163 212 54 104 165 211 55 110 163 212 5 38 78 118 163 212 38 78 118 163 212 46 99 138 175 207 38 76 118 163 213 Test8_2 48 104 48 104 48 104 48 104 3 36 72 115 36 72 115 41 75 116 36 72 115 4 35 69 102 135 35 69 102 135 36 69 102 135 35 69 102 134 5 46 101 153 201 243 46 101 153 201 243 36 72 147 192 244 46 102 153 202 242 Test9_2 115 181 115 181 115 181 115 181 3 112 158 202 112 158 202 112 158 202 74 119 183 4 73 116 161 205 73 116 161 205 75 116 156 200 74 116 161 205 5 73 114 150 185 220 73 114 150 185 220 71 112 149 183 220 71 115 150 184 221 Test10_2 91 170 91 170 91 170 91 170 3 75 130 183 75 130 183 80 130 183 75 130 183 4 68 116 164 206 68 116 164 206 73 113 166 208 67 116 164 206 5 56 93 132 170 209 56 93 132 170 209 55 95 134 174 212 54 90 129 170 209Table 9
The threshold values obtained by ICA, PSO, GWO, and TLBO based on Ostu’s between-class variance methods.
Example ICA PSO GWO TLBO Test1_2 93 151 93 151 92 151 93 151 3 81 127 171 81 127 171 81 127 171 81 127 171 4 75 114 145 180 75 114 145 180 75 111 142 181 75 113 145 180 5 73 109 136 160 188 73 109 136 160 188 70 86 119 150 180 74 110 137 160 188 Test2_2 113 173 113 173 113 173 113 173 3 93 145 191 93 145 191 93 145 191 93 145 191 4 84 129 172 203 84 129 172 203 93 128 170 203 85 129 173 203 5 69 107 143 180 205 69 107 143 180 205 90 124 162 171 204 70 104 142 178 205 Test3_2 98 150 98 150 97 150 98 150 3 86 125 161 86 125 161 86 125 161 86 125 161 4 72 106 137 168 72 106 137 168 68 105 136 167 72 106 137 168 5 68 100 126 150 175 68 100 126 150 175 63 95 123 149 175 66 97 123 148 174 Test4_2 68 135 68 135 68 135 68 135 3 63 119 166 63 119 166 61 118 166 63 119 166 4 47 86 126 169 47 86 126 169 49 88 129 170 47 85 125 169 5 43 79 113 146 177 43 79 113 146 177 48 82 111 144 176 42 78 112 147 178 Test5_2 57 124 57 124 57 124 57 124 3 39 92 142 39 92 142 40 93 141 39 92 142 4 35 82 124 164 35 82 124 164 35 81 123 164 35 81 123 164 5 28 65 100 134 172 28 65 100 134 172 37 75 102 134 172 28 64 100 133 172 Test6_2 87 162 87 162 87 162 87 162 3 82 136 190 82 136 190 82 136 190 82 136 190 4 74 112 149 197 74 112 149 197 76 110 147 196 74 112 149 198 5 73 109 143 183 221 73 109 143 183 221 71 88 116 150 197 72 108 143 183 220 Test7_2 60 157 60 157 62 155 60 157 3 54 131 206 54 131 206 53 131 206 54 131 206 4 40 84 143 208 40 84 143 208 49 96 135 206 40 84 142 208 5 39 81 136 191 227 39 81 136 191 227 45 81 134 191 229 38 81 138 194 228 Test8_2 108 196 108 196 108 196 108 196 3 89 135 206 89 135 206 90 135 206 89 135 206 4 68 106 145 210 68 106 145 210 71 108 145 209 69 106 145 210 5 63 97 126 157 215 63 97 126 157 215 69 97 124 158 214 63 98 127 158 218 Test9_2 78 144 78 144 78 144 78 144 3 75 119 177 75 119 177 75 122 177 75119 177 4 66 90 131 186 66 90 131 186 65 89 131 186 66 90 132 188 5 59 79 101 140 191 59 79 101 140 191 61 82 106 151 192 59 78 100 141 195 Test10_2 85 157 85 157 85 157 85 157 3 69 120 178 69 120 178 69 120 178 69 120 178 4 60 101 138 187 60 101 138 187 61 101 138 187 60 101 138 187 5 52 86 117 150 194 52 86 117 150 194 53 86 114 147 194 52 87 118 150 195Table 10
Statistical analysis results.
t-test P _ v a l u e D I R P _ v a l u e ρ t-test(ICA,PSO) 0.001 0.041 t-test(ICA,GWO) ≤ 0.001 ≤ 0.001 t-test(ICA,TLBO) ≤ 0.001 ≤ 0.001Table 11
The average CPU time of all algorithms (in seconds).
Example ICA PSO GWO TLBO Example ICA PSO GWO TLBO Test1_2 1.0148 1.8466 1.0573 1.2695 Test6_2 0.9013 1.8463 0.9634 1.1757 3 0.9968 1.9547 1.0824 1.2020 3 1.0516 2.0183 1.0631 1.4172 4 1.0817 2.1041 1.0777 1.2727 4 1.0725 2.1534 1.0789 1.3710 5 1.1677 2.2427 1.1135 1.3929 5 1.0992 2.2310 1.1540 1.4416 Test2_2 0.9160 2.1234 0.9569 1.1933 Test7_2 0.9479 1.9786 1.0088 1.2306 3 1.0238 2.1349 1.0966 1.2583 3 1.0616 2.0336 1.0571 1.3009 4 1.0121 2.2104 1.1203 1.4549 4 1.0868 2.2236 1.1175 1.3781 5 1.0935 2.2470 1.1195 1.4557 5 1.1588 2.3927 1.1682 1.4598 Test3_2 0.9684 1.8421 0.9593 1.2024 Test8_2 0.9654 1.9061 1.0042 1.2234 3 0.9690 1.9660 1.0271 1.2812 3 1.0587 2.0592 1.0648 1.2818 4 1.0981 2.1097 1.0739 1.3281 4 1.1558 2.2072 1.1166 1.3552 5 1.0956 2.2191 1.1637 1.4219 5 1.1965 2.3133 1.1706 1.4322 Test4_2 0.9394 1.8492 0.9677 1.2796 Test9_2 0.9599 1.8732 0.9750 1.2245 3 1.0026 1.9714 1.0213 1.2786 3 1.0476 1.9952 1.0556 1.2709 4 1.0562 2.1719 1.1014 1.3773 4 1.0656 2.1281 1.0893 1.3672 5 1.1098 2.2701 1.1188 1.3913 5 1.1195 2.2444 1.1403 1.4266 Test5_2 0.9514 2.3564 0.9939 1.2206 Test10_2 0.9563 1.9158 0.9922 1.3281 3 1.0401 2.0438 1.0557 1.2901 3 1.0342 2.0040 1.0496 1.3778 4 1.0839 2.1590 1.1043 1.3911 4 1.0759 2.2321 1.0948 1.3701 5 1.1561 2.3302 1.1637 1.4313 5 1.1585 2.3338 1.1481 1.4289Figure 4
Segmented images obtained by ICA-Kapur multilevel thresholding method (a) represent 3-level thresholding, (b) represent 3-level thresholding, (c) represent 4-level thresholding, and (d) represent 6-level thresholding.Figure 5
Segmented images obtained by ICA-Ostu multilevel thresholding method (a) represent 3-level thresholding, (b) represent 3-level thresholding, (c) represent 4-level thresholding, and (d) represent 6-level thresholding.Firstly, compared with the four algorithms in solving Kapur’s objective value, ICA has obtained the optimal solution in all 40 groups of examples, of which 30 groups are better than GWO and 19 groups are better than TLBO. Aiming at the average solution of 20 runs, among 40 groups of instances, the average solution of ICA has achieved better or equal to the results of the other four algorithms. Among them, 11 groups of instances are better than PSO, 30 groups are better than TLBO, and all instances are better than GWO. This also proves the stability of ICA algorithm. Thus ICA can get better results than other three algorithms on most of instances in similar computation times and has advantages in solving multithreshold image segmentation.For Ostu’s objective value, ICA also obtained all the best solutions in 40 groups of examples, and 28 groups of examples were better than GWO and 17 groups of examples were better than TLBO, which further verified the high quality of ICA solution. In terms of the average solution of 20 runs, ICA achieved better or equal results to the other four algorithms in 33 groups of instances, and 8 groups of instances were better than PSO, 33 groups were better than TLBO, and 40 groups were better than GWO. Furthermore, the stability of ICA solution is verified again. The statistical results in Table10 also validate this conclusion, which proves the superiority of ICA in solving multithreshold segmentation.


## 5. Conclusion

A new imperialist competition algorithm is proposed in order to solve the multithreshold segmentation problem in this paper. A new strategy of revolution, assimilation, imperialist self-learning, and a novel imperialist competition can be designed to improve the algorithm performance. Ten standard test pictures are selected to test and compared with three new optimization algorithms. The experimental results show that the proposed imperialist competition algorithm has fast convergence speed, high quality, and high stability in solving multithreshold segmentation problems.Meanwhile, the algorithm in this paper will be less affected by noise, because the algorithm in this paper only pursues the optimization of the objective function. The goal of multithreshold image segmentation is to separate multiple objects of interest. The image noise is usually an isolated point. Thus, we can remove the noise points by mathematical morphological small targets after threshold target segmentation.Although Kapur and Otsu methods are two widely used image threshold methods, they still cannot satisfy all kinds of images. Therefore, simply seeking the optimal value of one of the indicators is easy to make the segmentation result appear over and under segmentations, which is also the limitation of our algorithm in this paper. In the future, we will continue to consider adopting multiobjective evolutionary algorithm to achieve better optimization effect by balancing different indexes at the same time. This problem will be our next research step.



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*Source: 5982410-2019-06-04.xml*