A cooperative honey bee mating algorithm and its application in multi-threshold image segmentation

Information Sciences 369 (2016) 171–183

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A cooperative honey bee mating algorithm and its application in multi-threshold image segmentation Yunzhi Jiang a,b,1, Wei-Chang Yeh b,∗, Zhifeng Hao c, Zhenlun Yang c a

School of Automation, Guangdong Polytechnic Normal University, China Department of Industrial Engineering and Engineering Management, College of Engineering, National Tsing Hua University, Taiwan c School of Computer Science and Engineering, South China University of Technology, Guangzhou, China b

a r t i c l e

i n f o

Article history: Received 21 June 2014 Revised 15 June 2016 Accepted 15 June 2016 Available online 23 June 2016 Keywords: Cooperative learning Honey bee mating algorithm Multilevel thresholding Image segmentation

a b s t r a c t Multilevel thresholding has been one of the most popular image segmentation techniques; however, most of these techniques are time-consuming. This paper proposes a cooperative honey bee mating-based algorithm for natural scenery image segmentation using multilevel thresholding (CHBMA) to save computation time while conquering the curse of dimensionality. The characteristics of natural scenery images are random, imprecise, complicated, and noisy. The locations of interest points in them are not regular. Our proposed algorithm, which is based on honey bee mating algorithms (HBMA) and cooperative learning, considerably enhances the search capability of the algorithm. In the algorithm, we adopt a new population initialization strategy to make the search more efficient, and this strategy works in accordance with the characteristics of multilevel thresholding in an image; the thresholding is arranged from a low gray level to a high gray level. Extensive experiments have demonstrated that our proposed algorithm can deliver more effective and efficient results than state-of-the-art population-based thresholding methods. Thus, our algorithm can be applied in complex image processing, such as automatic target recognition. © 2016 Elsevier Inc. All rights reserved.

1. Introduction Image segmentation is an important and basic operation for the analysis of images to separate objects from their background or distinguish an object from objects that have distinct gray levels. For intensity images, there are four popular image segmentation approaches: thresholding techniques, edge-based methods, region-based techniques, and connectivitypreserving relaxation methods [3,32]. Among these four approaches, the image thresholding techniques are widely used in many image processing applications, such as optical character recognition, automatic visual inspection of defects, detection of video changes, moving object segmentation and medical image applications [19], due to their simplicity, robustness and accuracy [23,25]. The image thresholding methods can also be classified generally into parametric and nonparametric approaches. In parametric approaches, the gray-level distribution of each class has a probability density function that is usually assumed to obey a Gaussian distribution. Additionally, there is always a need to find an estimate of the parameters of the

∗

1

Corresponding author. E-mail addresses: [email protected] (Y. Jiang), [email protected] (W.-C. Yeh). Tel.: +8615217220718; fax: +862038256620.

http://dx.doi.org/10.1016/j.ins.2016.06.020 0020-0255/© 2016 Elsevier Inc. All rights reserved.

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Y. Jiang et al. / Information Sciences 369 (2016) 171–183

distribution to fit the given histogram data in parametric approaches [17,31,28]. Hence, such approaches are time-consuming and computationally expensive. Nonparametric approaches find the thresholds in an optimal manner based on some discriminating criteria such as the between-class variance, entropy, or other features. These criteria are easy to extend to multilevel thresholding. However, the amount of thresholding computation significantly increases with this extension. To overcome this problem, some multilevel thresholding techniques that are based on Intelligent Optimization Algorithms (IOA) have been proposed and proven effective in recent years. Hammouche et al. [12] proposed a fast multilevel thresholding technique based on GA and a wavelet transform-based technique to reduce the time computation. However, it is less efficient to determine the threshold number automatically. Lai et al. [18] proposed a clustering-based approach using a hierarchical evolutionary algorithm for medical image segmentation. The threshold number in the given image does not need to be known in advance, but is initialized randomly. Ye et al. [29] proposed a particle swarm optimization algorithm (PSO) to optimize Otsu’s criterion. The experimental results of this algorithm show that a PSO algorithm can yield very encouraging results in terms of the quality of solutions found and the processing time required, compared with Otsu. Yin [30] first developed a recursive programming technique that stored the results of preceding tries as the basis for the computation of succeeding ones. Then, a PSO algorithm was used for searching the optimal thresholds based on the recursive programming technique. Gao et al. [9] proposed the quantum-behaved PSO by employing a cooperative method to conquer the curse of dimensionality. It also shortens the computation time of the traditional OTSU method and has comparable or even superior search performance for many hard optimization problems, with faster and more stable convergence rates. Tao et al. [27] proposed a fuzzy entropy method incorporating the Ant Colony Optimization algorithm (ACO), which has better search performance than the one based on GA. Bhattacharyya et al. [5] proposed a multilevel method using a self-supervised multilayer self-organizing neural network and a supervised pyramidal neural network. The central idea of the method is to incorporate the contextual image information into the thresholding process. Chen et al. [26] proposed two low-complexity variants of Fuzzy C-means Clustering with spatial constraints for image segmentation by using kernel methods, and the two variants aimed at simplifying the computation of parameters. Gong [11] proposed a fuzzy energy functional based variational image segmentation model with an edge indicator integrated regularization. This method is insensitive to initialization and provides a general scheme for introducing fuzzy logic to region-based partial differential equation approaches. Horng [14] proposed a multilevel thresholding method based on the technology of the Honey Bee Mating Algorithm (HBMA) by using the maximum entropy criterion. The HBMA may also be considered as a typical swarm-based approach for optimization. In this paper, Horng also analyzed the convergence behavior and selected three other methods to be implemented on several real images for comparison. Chander et al. [7] proposed a new variant of PSO for image segmentation, which made a contribution in adapting the ‘social’ and ‘momentum’ components of the velocity equation for the particle move updates. The authors also proposed an iterative scheme based on Otsu’s method to obtain initial thresholds. This iterative scheme showed a linear growth of computational complexity. Jiang et al. [15,16] proposed an automatic parameter selection technique based on stratified sampling and Tabu Search. This technique can automatically determine the appropriate threshold number and values for image segmentation. Inspired by the balance of competition and diversity in ecology, Gao et al. [10] presented a PSO algorithm with an intermediate disturbance searching strategy (IDPSO), which can enhance the global search ability of particles and increase their convergence rates. The IDPSO algorithm was applied to a multilevel image segmentation problem, and it showed that an IDPSO-based method was more effective than some IOA-based methods, and it shortened the computation time of the traditional threshold-based segmentation methods. Cuevas et al. [8] explored the use of the Artificial Bee Colony (ABC) algorithm to compute threshold selection for image segmentation. In their algorithm, an image’s 1-D histogram is approximated through a Gaussian mixture model whose parameters are calculated by the ABC algorithm. The ABC method shows fast convergence and low sensitivity to initial conditions. Additionally, it mitigates complex time-consuming computations commonly required by gradient-based methods. Over the past decade, Swarm Intelligence, which is a sub-field of IOA, has been widely applied to engineering optimization problems and has achieved very satisfactory results. Ant Colony Algorithm [13] and Particle Swarm Optimization are the two most popular approaches in swarm intelligence. HBMA [2] is inspired by the mating process of real honey bees, and it is a typical swarm-based approach for optimization. The HBMA has been adopted to search for the optimal solution in some applications, such as clustering, market segmentation and benchmark mathematical problems. However, for the PSO, updating the position of the particle as a whole item has the problem of the curse of dimensionality. The performance of the PSO deteriorates as the dimensionality of the search space increases, as described in [4]. HBMA also has the problem of the curse of dimensionality because the broods are updated as a whole item. To solve this problem, a new HBMA with a cooperative method (called CHBMA here) is proposed in this paper. The cooperative method is specifically applied to conquer the “curse of dimensionality” by partitioning the search space of a high-dimensional problem into one-dimensional subspaces. Then, the broods in CHBMA make a contribution not only as a whole item but also in each dimension. A measure that is based on the entropy criterion is employed to evaluate the performance of CHBMA. The images that we have chosen in the experiment are images of natural scenery. The characteristics of natural scenery images are random, imprecise, complicated, and noisy. The locations of interest points in them are not regular. Combining the honey bee mating algorithm with the cooperative learning method can result in high-quality handling of imprecise and noisy image information. The experimental results indicate that CHBMA can produce effective, efficient and smoother segmentation results in a comparison with several previously developed methods.

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173

Table 1 List of symbols. Symbol

Definition and description

D Q Prob(Q,D) S(t) L floor(L/D) Xid NumberOfDrones M

A drone The queen or the number of queens The probability of a drone (D) mating with the queen (Q) The speed of the queen at time t The grayscale of the image Floor(L/D) rounds the elements of L/D to the nearest integers Towards minus infinity. The position of an arbitrary drone i in the dth dimension The number of drones The number of threshold values The decreasing factor The capacity of the spermatheca The initial speed of the mating flight The breeding ratio The generation numbers SPi is the ith individual in the queen’s spermatheca the jth individual of the next generation

α

Nsp S(0) Pc Gen SPi = (Xi 1 , Xi 2 , …, Xi 3 ) Broodj

The main contributions of this paper are as follows: (a) We introduce a new bee colony in each generation to avoid higher levels of inbreeding; (b) We also adopt a population initialization strategy to make the search more efficient and faster; (c) Compared with previous work, such as HBMA, we delete the brood mutation operator because we find from our experiments that it has low efficiency; and (d) The cooperative learning method is applied to conquer the “curse of dimensionality”. The remainder of this paper is organized as follows: Section 2 introduces the related work. Section 3 presents the proposed algorithm CHBMA. Section 4 analyzes CHBMA with respect to several aspects from the point of view of both algorithm design and experimental analysis. Performance evaluation and experimental analysis are presented in detail in Section 5. Finally, conclusions are presented in Section 6. 2. Related work 2.1. Notation In this section, we introduce the notation. Table 1 lists some basic symbols that will be used throughout the paper. Note that although we use the same symbol Q to represent the queen or the number of queens, it is always clear from the context what is intended. 2.2. Honey bee mating algorithm HBMA models the mating behavior of honey bees, one of the most studied social insects [2]. Each normal honey bee colony typically consists of one or more egg-laying queens in addition to drones, broods and workers. Queens represent the main reproductive individuals and they specialize in egg laying [21]. A queen bee can live up to 5 or 6 years, whereas worker bees and drones never live more than 6 months. After the mating process, the drones die. The drones are the sires or fathers of the colony. They are haploid and act to amplify their mother’s genomes without altering their genetic composition except through mutation. The drones are practically considered to be agents that propagate one of their mother’s gametes and function to enable females to act genetically as males. Workers specialize in brood care and sometimes in laying eggs. Broods arise either from fertilized eggs, which represent potential queens or workers, or unfertilized eggs, which represent prospective drones. In the mating process, the queens mate during their mating flight in the air. A mating flight starts with advances performed by the queens, who start a mating flight during which the drones follow the queens and mate with them. In each mating, sperm reaches the spermatheca and accumulates there to form the genetic pool of the colony. Each time, a queen lays unfertilized eggs and then randomly retrieves a mixture of the sperm that is accumulated in the spermatheca to fertilize the eggs [22]. The mating flight can be considered to be a set of transitions in a state space (the environment) in which the queen moves between the different states at some speed and mates with the drone that is encountered at each state probabilistically. The queen is initialized with some energy content at the start of the mating flight and returns to her nest when the energy is within some threshold or when her spermatheca is full. In HBMA, the functionality of the workers is confined to brood care, and thus, each worker can be regarded as a heuristic that can improve a set of broods. An annealing function [Eq. (1)] is used to describe the probability of a drone (D) mating with the queen (Q).

P rob(Q, D ) = Exp[−| f (Q ) − f (D )|/S(t )]

(1)

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Fig. 1. Pseudocode for HBMA.

where |f(Q)–f(D)| is the absolute difference between the fitness of D and the fitness of Q, and S(t) is the speed of the queen at time t. After each transition of mating, the queen’s speed and energy are decayed according to Eq. (2).

S(t + 1 ) = a × S(t ), a ∈ [0, 1]

(2)

where a is the decreasing factor. Workers adopt some heuristic mechanisms, such as crossover or mutation, to improve the brood’s genotype. The process for implementing HBMA [2] is shown in Fig. 1. 2.3. Entropy criterion-based measure Kapur et al. proposed the entropy criterion from information theory for bi-level thresholding. This method can also extend to solving multilevel thresholding problems, and it has been widely used in determining optimal threshold values in image segmentation. The entropy based image segmentation technique is dated, but new entropy-based methods have been proposed recently [30,14,6]. An image segmentation technique based on entropy is proposed and tested in this paper. The entropic method of thresholding can be described as follows. Let there be L gray levels in a given image, and these gray levels have the range [0, L–1]. Let h(i) be the observed frequency of gray-level i, and also let

N = h (0 ) + · · · + h (i ) + · · · + h (L − 1 ) pi = h(i )/N

(3)

Assume that there are M thresholds: {t1 , t2 ,…, tM }, where (1 ≤ M ≤ L – 1), which divide the original image into M + 1 classes, which are represented as follows: C0 = {0,1,…,t1 – 1}, C1 = {t1 , t1 + 2,…, t2 – 1},…, CM = {tM , tM + 2,…,L – 1}. For multilevel thresholding, the formula that is based on the entropic method is computed as follows:

f (t1 , t2 , · · · , tM ) = E0 + E1 + E2 + · · · EM t1 −1 t1 −1 pi pi ω0 = pi , E0 = − ln i=0

ω1 = ω2 =

t2 −1 i=t1

t3 −1 i=t2

.. .

pi , E1 = − pi , E2 = −

i=0

ω0

i=t1

ω1

ln

ω1

i=t2

ω2

ln

ω2

t2 −1 pi t3 −1 pi

ω0 pi pi

Y. Jiang et al. / Information Sciences 369 (2016) 171–183

ωM =

L−1 i=tM

pi , EM = −

L−1 pi i=tM

ωM

ln

pi

ωM

175

(4)

In the proposed CHBMA algorithm, we attempt to obtain the optimum thresholds {t1 , t2 ,…, tM } by maximizing Eq. (4). The objective function is also used to act as the fitness function for CHBMA. 3. Proposed algorithm 3.1. Cooperative learning The cooperative idea was presented by Potter [24] to apply to GAs successfully, and then van Den Bergh [4] applied this idea very well to the PSO. It has also been proven effective in papers [9,19]. The same concept can easily be applied to HBMA to create a cooperative HBMA. First, we introduce the weakness of the HBMA, which also exists in PSO and GA. Then, we present the solution. Each brood in HBMA represents a potential solution. Each update step is performed on a full c-dimensional particle. This arrangement leads to the possibility that some components in the brood have been moved closer to the solution, while others have been moved away from the solution. As long as the effect of the improvement outperforms the effect of the components that deteriorated, the HBMA will consider the updated brood to have an overall improvement but will neglect the deteriorated components in the brood. Therefore, it is clear that it is typically significantly more difficult to find the global optimum of a high-dimensional problem. Hence, we present a new HBMA algorithm with the cooperative learning method for solving this problem. A simple example can demonstrate the importance of the cooperative method: Consider a three-dimensional vector X = [x1 , x2 , x3 ] and the objective function f(X) = (x1 – a1 )2 + (x2 – a2 )2 + (x3 – a3 )2 , where A = (a1 , a2 , a3 ) = (10, 20, 30). This arrangement implies that the global minimum value is 0, where X = A = (10, 20, 30). Next, consider a honey bee colony that contains two broods X1 and X2 , and the Queen is (5, 12, 20) at the current time step t. We have

Queen(t ) = (5, 12, 20 ), f (Queen(t )) = 189; X1 (t ) = (8, 20, 42 ), f (X1 (t )) = 148; X2 (t ) = (6, 25, 36), f (X2 (t ) ) = 77. This construct implies that f(X1 (t)) and f(X2 (t)) are even better than the function value of the Queen. If we use HBMA without the cooperative method, then Queen(t) will be directly updated with X2 (t), and X1 (t) will be discarded. However, the second component of X1 (t) has the correct value of 20; it does not make any contribution to the Queen. The cooperative learning method can help the Queen to obtain the appropriate component by evaluating each component of the broods that are better than the Queen. Fig. 2 presents the pseudocode for the cooperative learning method. By applying this method, the Queen is updated as (8, 20, 20), f(8, 20, 20) = 104, first with the help of X1 (t), and then, it is updated as (8, 20, 36), f(8, 20, 36) = 40, with the help of X2 (t). Therefore, in the next time step t + 1, the Queen obtains a more precise result than HBMA without the cooperative method. By applying the cooperative method in the thresholding segmentation field, each brood in CHBMA contributes to the population not only as a whole item but also in each dimension. Therefore, the approach is not limited to the dimension of the optimal thresholds and is especially suitable for multilevel thresholding. Another benefit of using the cooperative method is that each brood in CHBMA can potentially make a contribution to the optimal thresholds in each dimension; this approach ensures that the search space is searched more thoroughly and that the algorithm’s chance of finding better results is improved. Additionally, the rationality of the analysis of the cooperative method is validated in Part E of Section 4. 3.2. The proposed method: CHBMA In this subsection, a cooperative honey bee mating algorithm for image segmentation using multilevel thresholding (CHBMA) is developed according to HBMA. Its specific features are presented in Section 4. 4. Analysis of our proposed CHBMA In this section, we present a detailed analysis of the proposed CHBMA from the point of view of both algorithm design and experimental analysis. In Section 4.A, we give the reason that we add a new bee colony in each generation. In Section 4.B, we present the bee colony initialization strategy of CHBMA. In Section 4.C, we illustrate the method of implementing the mating flight. In Section 4.D, we demonstrate the breeding process. In Section 4.E, we analyze the convergence behavior of the cooperative method. In Section 4.F, we present the parameters that are used in the proposed algorithm. 4.1. The introduction of a new colony In the real world, to avoid higher levels of inbreeding, a queen sometimes leaves her own colony to mate with the drones from another bee colony. Hence, we imitated this natural phenomenon in our proposed algorithm to keep from generating

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Fig. 2. The pseudocode for the cooperative learning method.

too many eccentric and homogeneous broods, one of the important factors that causes premature convergence and slow convergence. Our method is that we reduce the initial population size and introduce a new bee colony in each generation; the population size of the new bee colony is the same as the initial population size. 4.2. Initialize The positions of the drones are randomly initialized from the search space. In CHBMA, the dimensionality D of a drone is the number of thresholds. According to Bayesian inference, if the existence of the threshold value (an individual) in a closed interval is not in doubt, it has been suggested that the prior should be uniform over the interval (qualifying population). Hence, combining these with the characteristics of multilevel thresholding when arranged from a low gray level to a high gray level, the position of an arbitrary drone i in the dth dimension, i.e., Xid , should be initialized by being uniformly distributed in the interval [(d – 1) × floor(L/D), d × floor(L/D)]. This step can increase the diversity of the population and improve the global search ability. Among all of the drones, the drone that has the maximum fitness is selected as the queen Q. 4.3. Flight mating between the drones and Q In this stage, we select the set of better drones to mate with the queen Q by using the simulated annealing method. The best drone Di in the drone set D is first selected as the object of mating with Q. After the mating flight, the queen’s speed and energy decay according to Eq. (2). The mating flight continues until the number of drones that have deposited sperm in the queen’s spermatheca is more than the threshold Nsp . The value of Nsp is generally preset by the user and is less than the number of drones. A sperm can be described by SPi = (Xi 1 , Xi 2 , …, Xi 3 ), where SPi is the ith individual in the queen’s spermatheca. 4.4. Generate broods by a breeding operator Broods will be generated at this stage based upon in-flight mating between the queen and the sperm that is stored in the queen’s spermatheca. In the breeding process, the jth individual of the spermatheca is selected to breed if its corresponding random number is less than a user-defined breeding ratio Pc . The breeding process transfers the genes of the drones and the queen to the jth individual based on Eq. (5).

Brood j = Q ± rand (1 ) × abs(SPj − Q )

(5)

The parameter rand(1) is in the interval [0,1] and is randomly generated with a uniform distribution during each generation.

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177

Table 2 The parameters used in CHBMA. Parameter

Value

Q NumberOfDrones M L a Nsp S(0) Pc Gen

2 50 3, 4, 5, 6 0∼255 0.98 20 1.0 0.8 80

We find experimentally that if we use only the + sign or the – sign in Eq. (5), then the genes of the broods (Broodj ) will always keep increasing or decreasing. You also can obtain this result from a simple mathematical analysis of Eq. (5). If the + and – signs occur with equal probability, the increase and decrease of the genes will be controlled. 4.5. The analysis of the cooperative learning To analyze the convergence behavior of the cooperative learning method, Fig. 3 illustrates the changes in the queen’s fitness versus the number of generations for different threshold values. We can conclude that the cooperative learning method can not only quickly help to obtain the optimal solution but also improve the precision of the result to a certain extent. Furthermore, it is obvious that CHBMA can converge when the generation number is less than 80. Hence, the number of generations of the CHBMA algorithm can be set to 80 in the following experiments. 4.6. The parameters used in CHBMA Derived through a substantial number of experiments, the parameters that are used in the proposed algorithm are shown in Table 2. The reason that the generation numbers are set to 80 is analyzed in Section 4.5. 5. Experimental analysis A set of experiments was used to demonstrate the performance of the proposed algorithm on a personal computer with 2.40 GHz CPU, 2 GB RAM and a Windows 7 operating system. All of the programs are coded using MATLAB 7.2. The performance of CHBMA is evaluated by comparing its results with those of other algorithms that have been developed in the literature: the quantum-behaved PSO that employs the cooperative method (CQPSO) [9]; the hybrid cooperativecomprehensive learning PSO algorithm (HCPSO) [20]; and the maximum entropy-based honey bee mating optimization thresholding (HBMA) method [14].We have implemented these methods on a wide variety of images sourced from the Berkeley segmentation data set (www.eecs.berkeley.edu). We only considered scalar images because the scalar images are the basis of the segmentation technique. In the future work, we will to extend the proposed method to color images and also compare it with additional related methods. Fig. 4 presents the six original images. The parameters of CHBMA are listed in Section 4, and the parameters of the other algorithms are set as described in the original papers, except that the generation number is set to 100. In our experiments, when two thresholds satisfy the condition that the absolute value of their difference is less than 6, we unify them. Thus, the threshold number that is obtained through the experiments could be less than the initial threshold number in the algorithms. 5.1. The comparison of the segmentation results, with M = 4 For a visual interpretation of the segmentation results, we present in Fig. 5 the segmented images with M = 4. It is easy to see that the quality of the segmentation of CHBMA is better in general, especially for images a1, b1, c1, d1 and e1. When we enlarged the segmented images in Fig. 5, we observed the following: (1) our method yielded a clearer result for images (a), (b) and (c) with only 3 or 4 thresholds compared with those of the other methods; and (2) our segmented result for image (d) was absolutely clear, except that part of the horse’s body was too white. In most cases, the CHBMA segmentation results are better than those of HBMA. For example, there is often only the moon but no trees in the HBMA results of image (c); (3) The CQPSO method obtained even worse results than the other methods, and sometimes the CQPSO results were too vague, especially for images (d) and (f); and (4) the HCPSO results were also quite acceptable; however, this approach is time-consuming because it is a complicated process.

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Fig. 3. The comparison of the convergence rate between CHBMA and HBMA, with M = 4, 5 and 6.

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179

Fig. 4. The six original images.

Fig. 5. The comparison of the segmentation results: (a1)–(i1) the segmentation results of our method, CHBMA; (a2–i2) the segmentation results of HBMA; (a3–i3) the segmentation results of CQPSO; and (a4–i4) the segmentation results of HCPSO.

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5.2. The comparison of threshold values, PSNR, and computation time The quality of segmented images can be evaluated through the peak signal to noise ratio (PSNR), which is used to compare the segmentation results by using multilevel image threshold techniques [1,14,15], as follows:

P SNR = 20 × log10 (255/RMSE ) where

RMSE =

m n i=1

j=1

(6)

(I (i, j ) − I (i, j ))/(m × n )

(7)

and I and I are the original and segmented images of size m × n, respectively. A larger value of PSNR means that the quality of the segmented image is better. Furthermore, as the threshold number increases, the PSNR tends to be larger. The results from the four methods that are used for testing the images are summarized in Table 3. From Table 3, HCPSO was the slowest of the four methods. Additionally, from Table 3, the proposed CHBMA was faster than the other algorithms and its computation time was insensitive to the number of thresholds. The values of

Table 3 The comparison of threshold values, uniformity, and computation time. Method

M

Threshold values

PSNR

CPU time(s)

0–95–166–218–255 0–96–157–203–255 0–53–108–122–255 0–83–144–190–255 0–82–169–224–255 0–84–171–228–255 0–63–86–92–255 0–80–167–225–255 0–67–118–244–255 0–84–118–239–255 0–27–46–160–255 0–27–132–216–255 0–65–110–191–255 0–59–105–176–255 0–58–104–255 0–61–107–176–255 0–84–142–194–255 0–93–129–188–255 0–86–110–134–255 0–84–141–191–255 0–91–154–205–255 0–84–155–203–255 0–65–77–85–255 0–84–149–201–255

23.676 22.151 10.551 20.355 21.997 22.158 7.596 21.834 15.387 21.162 8.028 13.827 16.732 15.718 10.503 15.888 20.037 19.327 15.931 19.826 23.047 22.256 6.821 22.040

1.2947 2.0087 1.4016 1.8916 1.1008 1.6521 1.3634 2.0901 0.9853 1.4531 1.3733 1.7301 1.1358 1.6349 1.4285 2.0978 1.0905 1.7471 0.9961 2.0149 1.3245 2.3286 1.5503 2.2328

0–52–97–171–228–255 0–42–95–165–207–255 0–99–108–115–123–255 0–61–94–146–199–255 0–62–125–178–234–255 0–81–138–169–210–255 0–61–83–89–255 0–62–117–173–218–255 0–59–93–136–229–255 0–49–94–138–236–255 0–27–58–137–255 0 –55– 92–128–234–255 0–62–103–148–200–255 0–48–119–162–214–255 0–45–52–61–255 0–43–88–164–217–255 0–91–146–179–217–255 0–85–114–173–206–255 0–79–98–120–142–255 0–61–107–156–195–255

20.601 18.869 10.602 18.706 25.022 24.561 7.293 23.973 20.617 20.381 8.124 20.512 20.152 19.573 6.620 17.282 22.016 20.690 15.951 19.100

1.2962 1.6593 1.3962 2.4356 1.4498 1.5901 1.3881 2.0944 1.2297 1.4722 1.3702 2.2889 1.0489 1.6467 1.3751 2.1748 1.3455 1.5447 1.3888 2.0171

Table 3(a) M = 3 24,063.jpg

21,077.jpg

238,011.jpg

385,039.jpg

245,051.jpg

236,037.jpg

CHBMA HBMA CQPSO HCPSO CHBMA HBMA CQPSO HCPSO CHBMA HBMA CQPSO HCPSO CHBMA HBMA CQPSO HCPSO CHBMA HBMA CQPSO HCPSO CHBMA HBMA CQPSO HCPSO

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

Table 3(b) M = 4 24,063.jpg

21,077.jpg

238,011.jpg

385,039.jpg

245,051.jpg

CHBMA HBMA CQPSO HCPSO CHBMA HBMA CQPSO HCPSO CHBMA HBMA CQPSO HCPSO CHBMA HBMA CQPSO HCPSO CHBMA HBMA CQPSO HCPSO

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

(continued on next page)

Y. Jiang et al. / Information Sciences 369 (2016) 171–183

181

Table 3 (continued)

236,037.jpg

Method

M

Threshold values

PSNR

CHBMA HBMA CQPSO HCPSO

4 4 4 4

0–61–119–167–207–255 0–69–120–186–217–255 0–63–73–84–255 0–62–106–155–200–255

22.805 24.052 6.650 20.172

1.0805 1.5828 1.3974 2.4873

CPU time(s)

0–41–93–143–176–218–255 0–70–117–154–200–225–255 0–99–109–118–124–255 0–39–97–146–177–217–255 0–49–100–144–180–229–255 0–82–138–183–217–241–255 0–55–77–86–93–255 0–50–79–135–180–223–255 0–50–95–118–165–223–255 0–45–92–112–161–216–255 0–27–38–112–163–255 0–48–97–135–201–244–255 0–42–97–139–177–218–255 0–35–69–114–180–206–255 0–27–56–126–190–255 0–26–69–106–158–202–255 0–50–95–140–187–222–255 0–92–117–152–211–230–255 0–75–88–105–120–139–255 0–113–139–159–197–234–255 0–50–88–134–175–221–255 0–67–128–154–199–226–255 0–62–72–78–85–247–255 0–49–90–128–167–211–255

24.042 25.020 10.831 23.941 25.580 19.816 7.380 21.564 28.060 27.606 11.225 27.944 22.284 17.621 12.687 16.739 24.875 22.167 15.942 20.081 22.595 22.341 9.987 21.782

1.1235 1.6998 1.7656 2.4032 1.0675 1.6389 1.7829 2.1048 1.0496 1.4965 1.7734 2.1375 1.1339 1.8418 1.7571 2.2914 1.1389 1.5759 1.7544 2.1618 1.1321 1.6504 1.7534 2.3152

Table 3(c) M = 5 24,063.jpg

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CHBMA HBMA CQPSO HCPSO CHBMA HBMA CQPSO HCPSO CHBMA HBMA CQPSO HCPSO CHBMA HBMA CQPSO HCPSO CHBMA HBMA CQPSO HCPSO CHBMA HBMA CQPSO HCPSO

5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5

Table 4 The stability of CHBMA with M = 3, 4, 5, 6 (the standard deviation is in the table).

M=3 M=4 M=5 M=6

CHBMA HBMA CHBMA HBMA CHBMA HBMA CHBMA HBMA

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0.0158 0.0550 0.0359 0.1039 0.0351 0.1324 0.0474 0.1613

0.0084 0.0288 0.0346 0.0711 0.0247 0.1240 0.0424 0.1765

0.0234 0.0664 0.0363 0.1195 0.0536 0.1379 0.0828 0.2033

0.0042 0.0198 0.0095 0.0582 0.0250 0.0873 0.0271 0.1118

0.0187 0.0467 0.0416 0.0670 0.0387 0.1171 0.0727 0.1712

0.0087 0.0257 0.0202 0.0627 0.0381 0.0978 0.0260 0.1419

PSNR obtained from the proposed CHBMA on images (a), (b), (c), (d) and (e) (M = 4, as shown in Fig. 5) were the largest. Hence, the proposed CHBMA was the best or close to the best at different threshold numbers generally. Moreover, CHBMA could obtain a clearer and more complete segmentation image for video image compression. 5.3. The stability of our proposed method with M = 3, 4, 5, and 6 Because the four population-based methods in this paper are stochastic and are random search algorithms, the results of the experiments are not absolutely the same for each run and are influenced by the initial population. We analyze the stability by calculating the standard deviation (std) of the queen’s fitness, which is recorded in each independent run. In the same situation, a larger value of std proves that the results of the experiment are unstable. The standard deviation values for 100 runs of CHBMA and HBMA (with M = 3, 4, 5, 6) are presented in Table 4. From these results, we can see that CHBMA is more stable than the HBMA algorithms on each image. One reason is that a new bee colony is introduced in each generation to avoid premature convergence and slow convergence. The other reason is that the cooperative method helps CHBMA search in the feasible solution space more thoroughly. Therefore, CHBMA has a more powerful global searching ability, which ensures that it obtains more stable results in a limited amount of time. 5.4. The ability to conquer “the curse of dimensionality” with M = 5, 9, 12, and 16 To verify the search ability of our proposed method for a high dimension, we also test it on each image, with M = 5, 9, 12, 16. Table 5 shows the mean computation time (in seconds) for 100 runs. The other parameters of CHBMA are in

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Y. Jiang et al. / Information Sciences 369 (2016) 171–183 Table 5 The computation time with M = 5, 9, 12, 16. M

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6 9 12 16

1.1385 1.1623 1.2612 1.3344

1.0760 1.1416 1.2214 1.3261

0.9945 1.0683 1.1582 1.2388

1.0960 1.1779 1.2451 1.3445

1.0381 1.1156 1.1855 1.2979

1.0770 1.1453 1.2203 1.3174

Section 4, which can make our method achieve a good segmentation effect. Therefore, we can say that CHBMA is effective and efficient, and it has a powerful ability to conquer “the curse of dimensionality”. At present, it is difficult to justify the performance of the method providing an evidential basis in IOA communities. The usual method is comparing the computation times of different algorithms. The reason that the proposed algorithm can evade the curse of dimensionality is due to the characteristics of HBMA and the cooperative learning method. 6. Conclusions The problem of multilevel thresholding is regarded as an important area of research interest among research communities worldwide. In this paper, we described a cooperative honey bee mating algorithm for image segmentation using multilevel thresholding based on the cooperative learning method and HBMA, which has a powerful ability to conquer “the curse of dimensionality”. The difference between the traditional OTSU method and CHBMA is that the former finds the solution by testing thresholds in the range of the gray levels one by one, thus making it an exhaustive method; the latter obtains the solution by the cooperation of the population and, therefore, will shorten the computation time, especially in multilevel thresholding. We also adopt a new population initialization strategy that makes the search more efficient and faster. Furthermore, besides being evaluated by application to the benchmark images that were shown in this paper, the proposed method has also been tested on a wide variety of images that were provided by the Berkeley segmentation data set. The results prove that the proposed method can generally produce more favorable results than several well-known methods. 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