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.
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 Section 2. 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 and it is shown as . is the pixels number of the image and is the pixel number of gray value . is described as the probability distribution of each gray value , where .
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 is , is the pixels number of gray-level , and the total pixels number of the image is . If the probability of a given gray-level is given as , obviously, , . Assume the image segmentation threshold is , shows the average gray-level value of all pixels less than or equal to , and shows the average gray-level value of all pixels larger than in the image. Then variance between binary image classes is described as
where , . can be defined as a reparability criterion. So can be computed by maximizing .
The formula (1) can be extended to multithreshold segmentation (assuming M level). Threshold group can divide one image into k-classes objectives. And the variance between classes can be described as follows:
The threshold group of multiobjective image segmentation can be given by maximizing the between-classes variance .
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 . shows the pixels number of the image with gray-value . shows the total pixels number of the image and . The probability density is and the entropy of the image is defined as .
For image multilevel segmentation (assuming M level), the image can be divided into M subclasses: , , , and the entropy is defined as
where .
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 population is generated randomly. Choose solutions with smallest cost as imperialists, and assign 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 to tep 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 string . The decoding process is a series of integers arranged in ascending order , in which is the order 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 country , in which 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 former countries as imperialists and the rest as the colony, so the number of colonies can be described as .
Step 3. Calculate the power of each imperialist , , and determine the colony number allocated to each imperialist , where . 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 that is the set of all colonies in empire .
The detailed steps of ICA are shown below.
Step 1. Suppose colony , and compute the consistency between colony and their imperialist . If the two solutions are consistent, judge whether the reserve country set is empty. If it is empty, the colony will not be assimilated. Otherwise, randomly choose a reserve country instead of .
Step 2. If colony is inconsistent with the imperialist or has been replaced by the reserve country , the colony will move close to imperialist through the global search operation.
The global search in tep 2 is divided into two steps. First, suppose the imperialist is and a colony is ; then the new colony can be obtained according to formula (4).
Among them, is a random number between 0 and 2. And the colony can be amended and a feasible solution can be adjusted depending on formula (5).
The revised colony is assimilated colony and replaced colony with colony .
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 imperialist , new individuals are obtained from the imperialist self-learning.
The specific imperialist self-learning process is shown as follows:
Step 1. Let , generate a random integer , and .
Step 2. Take , and is the random number between 0 and 1.
Step 3. Use the formula (6) to modify , to get the new individual , and to calculate its cost value.
Step 4. If the cost value is better than , will be put into the reserve country set and replaced ; otherwise, will be put into the reserve country set.
Among them, the size of the reserve country set is , and if the number of reserve countries is larger than , 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 is described as follows.
Step 1. Generate a random number , if continues; otherwise, end the revolution.
Step 2. If the cost value is in the first 80% of all colonies, then the new solution will directly replace the original colony 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 from will be chosen randomly to replace the original colony .
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 from 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 cost is calculated according to formula (7), which is related to the cost value of the imperialist and all the empire colonies.Among them, is the cost value of the colony , and is the cost value of the imperialist . Weight coefficient is less than 1, and it is 0.1 according to Reference [21]. Calculate the empire power . And the probability vector can be constructed as = . There, is a random number vector between 0 and 1 and is the number of imperialist. Suppose the value of element in is the largest; then the empire is the winning empire. A reserve country is randomly selected from and compared with the weakest colony in the weakest empire. If is better than , is replaced by and is allocated to the winning empire. Otherwise, will be allocated directly to the winning empire.
3.6. Algorithm Flow
The flow chart of the hybrid imperialist competition algorithm is described in Figure 1. The termination condition is that the evaluation number of the objective function reaches a set value .
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 size . Test5-Test10 is from BSD and the size of picture Test5 is . The size of Test6 and Test9-10 is . The size of Test7-8 is . They are shown in Figure 2.
(a) Test1
(b) Test2
(c) Test3
(d) Test4
(e) Test5
(f) Test6
(g) Test7
(h) Test8
(i) Test9
(j) Test10
In 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 size , the ratio of number of imperialists to size , revolutionary probability , and the maximum objective functions evaluation number .
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 Table 1. 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.
From Table 3, 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: , , , and .
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. means the results of 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 , the difference between algorithms is significant. Table 11 gives the average CPU time of all algorithms.
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 Table 10 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.
Data Availability
The data we used to support the findings of this study is available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is financially supported by Shandong Province Science and Technology Development Plan Project Foundation (No. 2014GGX101030).