A Quick Artificial Bee Colony Algorithm for Image Thresholding

Abstract

The computational complexity grows exponentially for multi-level thresholding (MT) with the increase of the number of thresholds. Taking Kapur’s entropy as the optimized objective function, the paper puts forward the modified quick artificial bee colony algorithm (MQABC), which employs a new distance strategy for neighborhood searches. The experimental results show that MQABC can search out the optimal thresholds efficiently, precisely, and speedily, and the thresholds are very close to the results examined by exhaustive searches. In comparison to the EMO (Electro-Magnetism optimization), which is based on Kapur’s entropy, the classical ABC algorithm, and MDGWO (modified discrete grey wolf optimizer) respectively, the experimental results demonstrate that MQABC has exciting advantages over the latter three in terms of the running time in image thesholding, while maintaining the efficient segmentation quality.

1. Introduction

Image segmentation involves the technique of segmenting an image into several non-overlapping areas with similar features, or, in other words, it is a process of separating a digital image into multiple areas or targets [1]. These areas can provide more precise and useful information than individual pixels. Therefore, image segmentation plays an important role in image analysis and understanding, and it is also widely used in such areas as medical analysis [2], image classification [3], object recognition [4], and so on.

Thresholding is the most commonly used method in image segmentation [5]. For grayscale images, bi-level thresholds are enough to separate the objects from the background; this is, namely, bi-level thresholding. Similarly, multi-level thresholding (MT) can divide the image into several areas and produce more precise segmented areas. Numerous different thresholding approaches have been reported in the literature. Basically, thresholding methods fall into two categories; parametric and non-parametric [6,7,8]. For the parametric, it is necessary to first assume the probability density model for each segmented area and then estimate the relevant parameters for fitness features. Such methods are time-consuming and computationally expensive. On the other hand, nonparametric methods try to determine the optimal thresholds by optimizing some standards, which include between-class variance, the entropy, the error rate, and so on [6,7,8,9,10,11]. The biggest advantages of such methods lie in their robustness and accuracy [5]. Based on the above analyses, this paper takes nonparametric methods to analyze and study multi-level thresholding. After modification of the distance strategy in the neighborhood searches of the quick artificial bee colony algorithm, this paper puts forward the modified quick artificial bee colony algorithm (MQABC) by taking Kapur’s entropy as the optimized objective function. The experimental results show that the proposed method has exciting advantages in terms of the running time in image thresholding, on the premise of the efficient segmentation quality.

The rest of the paper is organized as follows. In Section 2, the previous works in image thresholding are summarized. In Section 3, the thresholding problem is formulated, and then Kapur’s entropy for image thresholding and the objective function are presented. In the next section, the standard artificial bee colony algorithm is briefly described and the proposed MQABC is described in detail. In Section 5, a comparison of experimental results is conducted, and it shows the superiority of the MQABC. The final section concludes the paper.

2. Related Works

It has proved to be feasible to determine the optimal thresholds by analyzing the histogram characteristics or optimizing objective functions. These nonparametric methods can be achieved by optimizing some objective functions. The commonly used optimization functions include maximization of the entropy [12], maximization of the between-class variance [13], the use of the fuzzy similarity measure [14], and minimization of the Bayesian error [15]. All of these techniques were originally employed in bi-level thresholding and then extended to multi-level thresholding fields. However, in multi-level thresholding, the computational complexity grows exponentially [7]. Therefore, numerical evolutionary and swarm-based intellectual computation are introduced into MT [10].

There exist two classical methods for bi-level thresholding [5]. The first is proposed by Kapur et al. [12] and uses the maximization of Shannon entropy to measure the homogeneity of the classes. The second is proposed by Otsu et al. [13] and maximizes the between-class variance. Although the two methods have proved to be highly efficient for bi-level thresholding, the computational complexity for MT increases with each new threshold [16].

As an alternative to the classical methods, MT problems have been dealt with through intelligent optimization methods. It has been proved in the literature that intelligent optimizations are able to deliver better results than classical ones in terms of precision, processing speed, and robustness [5,10]. EMO was introduced for MT by Diego Olivaa et al. [5], in which Kapur’s entropy and Otsu’s method are applied respectively. Their experimental results show that Kapur’s entropy was more efficient. Before that, they verified the same tests through the Harmony Search Optimization and obtained similar results [17]. Pedram Ghamisi et al. [18] analyzed the performances of Particle Swarm Optimization (PSO), Darwinian Particle Swarm Optimization (DPSO), and Fractional-Order Darwinian Particle Swarm Optimization (FODPSO) in MT. In comparison to the Bacteria Foraging algorithm and genetic algorithms, FODPSO shows better performance in overcoming local optimization and running time. Wasim A. Hussein et al. [19] studied the modified Bees Algorithm (BA), called the Patch-Levy-based Bees Algorithm (PLBA), to render Kapur’s and Otsu’s methods more practical. Their experimental results demonstrate that the PLBA-based thresholding algorithms are much faster than Basic BA, Bacterial Foraging Optimization (BFO), and quantum mechanisms (quantum-inspired algorithms) and perform better than the non-metaheuristic-based two-stage multi-threshold Otsu method (TSMO) in terms of the segmented image quality. In our previous work [20], we take Kapur’s entropy as the optimal objective function, with Modified Discrete Grey Wolf Optimizer (MDGWO) as the tool to achieve image segmentation. Compared with EMO, GWO, and DE (Differential Evolution), MDGWO shows better performance in segmentation quality, objective function and stability

In the multi-level thresholding field, the artificial bee colony algorithm (ABC) has become the most frequently used method [10,21,22,23,24,25]. Kurban et al. [10] had conducted comparative studies of the applications of evolutionary and swarm-based methods in MT. According to the statistical analyses, population-based methods are more precise in solving MT problems, so the authors take Otsu’s method, between-class variance, Tsallis entropy, and Kapur’s entropy for objective functions. After processing these through the ABC algorithm, they obtained better results in image segmentation. Akay [8] compared ABC with PSO by employing between-class variance and Kapur’s entropy as objective functions. The Kapur’s entropy based ABC proved to be better when the thresholds were increased, and the time complexity was also reduced. Bhandari et al. [25] conducted comparative analysis in detail between Kapur’s, Otsu, and Tsallis functions. The results show that, in remote sensing image segmentation, Kapur’s entropy-based ABC performs better than the rest generally.

The artificial bee colony algorithm [26] was first developed by Karaboga, which mimicked the foraging behavior of honey bees. It has shown superior performance on numerical optimization [27,28] and has been applied to many problems encountered in different research fields [29,30,31,32,33]. It has been proven that ABC is an easy, yet highly efficient optimal model. Furthermore, compared to other search heuristics, its iteration is much more easily implemented. More importantly, few pre-defined control parameters are required in ABC. But there are still some drawbacks in ABC regarding its solution search equation, which is fine at exploration but inferior at exploration processes [25]. Later, an advanced solution equation [34] was proposed, whereby the onlooker bee searches only around the best solution of the previous iteration to improve exploitation. Therefore, the core task is to determine the search scope. If the scope is larger than necessary, it will greatly heighten the complexity, otherwise it will be reduced to local optimization, and different problems call for different distance strategies and models. Therefore, this paper will mainly focus on the MQABC for multilevel image thresholding, and one of its purposes is to discover the search abilities of ABC in MT. In the standard ABC, the entire bee colony searches in a random way. Although QABC has improved the onlooker bee’s search models, it necessitates formulating different distance strategies in search abilities with different problems. In this paper, the MQABC is formulated in response to the problem of distance strategies for multilevel image segmentation, and it also demonstrates its superiority in convergence rate and time-efficiency. In order to evaluate the image segmentation quality, two evaluation standards, the peak-to-signal-noise (PSNR) ratio and the feature similarity index (FSIM), are adopted to give a qualitative evaluation. The experimental results show that MQABC delivers a better performance for MT.

3. Formulation of the Multilevel Thresholding

MT needs a set of thresholds. Based on that, the image can be segmented into different regions. By means of intelligent optimization to obtain the optimal thresholds, the process of image thresholding has to be formulated by taking image elements or image features as parameters to get the optimized objective function values with the purpose of getting close to the optimal thresholds.

3.1. Pixel Grouping Based on Thresholding

Assume that an image can be represented by L gray levels. The gray level for each pixel can be represented by f(x,y), where x,y stands for the pixel’s positions in Cartesian coordinates. Then the output image can be formulated by Equation (1):

$$g(x,y)=\{\begin{array}{ll}\raisebox{1ex}{$t_1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.& 0\le \mathrm{f}(\mathrm{x},\mathrm{y})<t_1\\ \raisebox{1ex}{$(t_i+t_{i+1})$}\!\left/ \!\raisebox{-1ex}{$2$}\right.& t_i\le \mathrm{f}(\mathrm{x},\mathrm{y})<t_{i+1}\\ \raisebox{1ex}{$(t_m+L)$}\!\left/ \!\raisebox{-1ex}{$2$}\right.& t_m\le \mathrm{f}(\mathrm{x},\mathrm{y})<L\end{array},$$

(1)

where $t_i$ (i = 1, 2…, m) stands for ith threshold and m is the number of thresholds. As the threshold value is optimized, the image can be segmented into m + 1 regions. Formulating the output is not the focus, the key point is to determine $t_i$ and its optimization. To realize the optimization of the thresholds, the objective function has to be initialized. The maximization or minimization of the objective function represents the optimal value and also ensures the optimization of image thresholding results.

3.2. Concept of Kapur’s Entropy for Image Thresholding

The intelligent optimization algorithm is linked with MT through objective functions so as to get better segmentation results. Based on this, the population based intelligent method using Kapur’s entropy as the objective function could get better image thesholding. Kapur’s method can be easily extended from bi-level thresholding to MT, and, with the entropy reaching maximization, the optimal thresholds are distributed naturally in the image’s histogram. Entropy of the discrete information can be obtained by the probability distribution p = $p_i$, where $p_i$, is the probability of the system in possible state i [24]. The probability for each gray level i is represented by its relative occurrence frequency, equalized by the total number of gray levels as shown in Equation (2):

$$p_i=\raisebox{1ex}{$h(i)$}\!\left/ \!\raisebox{-1ex}{${\displaystyle {\sum }_{i=0}^{L-1}h(i)}$}\right.i=0,1,\dots ,L-1$$

(2)

Kapur’s entropy is used to measure the compactness and separability of classes. For MT, Kapur’s entropy can be described as in Equation (3):

$$\begin{array}{l}{H}_0=-{\displaystyle {\sum }_{\mathrm{i}=0}^{t_1-1}\frac{p_i}{{\omega }_0}}\ln \frac{p_i}{{\omega }_0},{\omega }_0={\displaystyle {\sum }_{\mathrm{i}=0}^{t_1-1}{p}_i}\\ H_1=-{\displaystyle {\sum }_{\mathrm{i}=t_1}^{t_2-1}\frac{p_i}{{\omega }_1}}\ln \frac{p_i}{{\omega }_1},{\omega }_1={\displaystyle {\sum }_{i=t_1}^{t_2-1}{p}_i}\\ H_j=-{\displaystyle {\sum }_{\mathrm{i}=t_j}^{t_{j+1}-1}\frac{p_i}{{\omega }_j}}\ln \frac{p_i}{{\omega }_j},{\omega }_j={\displaystyle {\sum }_{\mathrm{i}=t_j}^{t_{j+1}-1}{p}_i}\\ H_m=-{\displaystyle {\sum }_{\mathrm{i}=t_m}^{L-1}\frac{p_i}{{\omega }_m}}\ln \frac{p_i}{{\omega }_m},{\omega }_m={\displaystyle {\sum }_{\mathrm{i}=t_m}^{L-1}{p}_i}\end{array}$$

(3)

Thus, the function f (T) can be obtained by Equation (4), which is used as the parameter of MQABC’s fitness function in Section 4.1.

$$f(T)={\displaystyle {\sum }_{i=0}^m-H_i}\hspace{1em}T=[t_1{t}_2\cdots t_m]$$

(4)

where, T represents a vector quantity of thresholds.

4. Brief Introduction of the Quick Qrtificial Bee Colony Algorithm

The aggregate intelligent behavior of insect or animal groups attracts the interest of more and more researchers. These behaviors include flocks of birds, colonies of ants, schools of fish, and swarms of bees. These collective behaviors are identified as swarm behavior and then abstracted as intelligent optimization methods. Compared with other methods, ABC algorithms have been widely used, as they employ fewer control parameters than other methods. This is very important because, in many cases, tuning the control parameters of the algorithm might be more difficult than the problem itself. Since the processing data are all positive real numbers, multi-level thresholding is not a large-scale problem. The ABC is advised to employ a small number of control parameters. A brief description of the ABC and the QABC algorithms are provided in the following subsections.

4.1. Standard Artificial Bee Colony Algorithm

Bees are gregarious insects. Although an individual insect’s behavior is simple, the groups formed by individuals are extremely complex in their collective behavior. Real bee colonies can, in all circumstances, collect nectar in a highly efficient way. At the same time, they are highly adaptable. Accordingly, ABC proposes the foraging behavior model of bee colonies [26].

Artificial bees can be classified into three types; employed bees, onlooker bees, and scout bees. Employed bees are related to specific food sources. Onlooker bees, by observing the dance of employed bees, decide to choose a certain food source. Scout bees will search for food randomly. The colony behavior and related simulation can be found in the literature [26].

In ABC, the solutions are represented by the positions of food source, whilst the quality of the solution is represented by the number of bees around the food source. In the foraging process, three kinds of bees adopt different strategies. The number of employed bees is equal to the number of food sources. Every employed bee is related to only one food source. It searches the area around the food source in its memory. If it finds a better food source, its memory will be updated. Otherwise, it will have to count the number of searches in its memory.

For the onlooker bee, there is no such information concerning food sources in its memory. It selects the food source by probability. The probability information is picked up from the employed bee. Therefore, the communication between the employed bee and the onlooker bee comes into being. The better the quality of the food source, the bigger the probability that the onlooker bee will select the food source. Once a better source is found, the old one will be replaced. If the number of searches associated with the food source reaches the limit, the solution is assumed to be exhausted, meaning the food source is not the optimal solution and needs to be replaced. The bee discards the food source and becomes a scout bee, which selects a random source to exploit. The main phases of the algorithm are given step-by-step in Algorithm 1.

Algorithm 1 (Main steps of the ABC algorithm)

Step 1: Initialization Phase Step 2: Repeat Step 3: Employed Bee Phase Step 4: Onlooker Bee Phase Step 5: Scout Bee Phase Step 6: Memorize the best solution achieved so far Step 7: Continue until the termination criteria is satisfied or Maximum Cycle Number has been achieved.

In the initialization phase, all the food sources are initialized by Equation (5). The number of food sources is set by pre-defined parameters:

$$t_{j,i}=l_i+random(0,1)\times (u_i-l_i),$$

(5)

where $t_{j,i}$ is the ith dimensional data of the jth food source, $l_i$ and $u_i$ stand for the lower limit and upper limit of the parameter $t_j$, and $t_j$ is the jth food source.

In the employed bee phase, the search is conducted in the bee’s memory at the specific speed ${\phi }_{j,i}$. The speed, as shown in Equation (6), determines the change rate of the food source, which affects the convergence speed. If the solution produced in Equation (6) is better than the bee’s solution, its memory is updated by a greedy selection approach:

$$v_{j,i}=t_{j,i}+{\phi }_{j,i}\times (t_{j,i}-t_{r,i}),$$

(6)

where $t_r$ represents a randomly selected food source, i is a randomly selected parameter index, and ${\phi }_{j,i}$ is a random number within the range [−1, 1].

To compare the advantages and disadvantages, the fitness of the solution is produced by Equation (7). A higher fitness value represents the better objective function value; thus maximizing the fitness function can reach the optimal thresholds:

$$fit(t_j)=\{\begin{array}{ll}\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$(1+f(t_j))$}\right.& f(t_j)\ge 0\\ 1+abs(f(t_j))& f(t_j)<0\end{array},$$

(7)

where$f(t_j)$ can be calculated by the Equation (4).

The employed bee shares information about food source fitness with the onlooker bee. The onlooker bee, by probability, selects one source to investigate. Ideally, the onlooker bee always investigates food sources with the highest-level of fitness or return. The probability is closely related to the fitness function. In standard ABC, the probability function can be represented by Equation (8):

$$pro{b}_j=\raisebox{1ex}{$fit(t_j)$}\!\left/ \!\raisebox{-1ex}{${\displaystyle {\sum }_{j=1}^{SN}fit(t_j)}$}\right.,$$

(8)

where SN represents the number of food sources. Whilst the onlooker bee selects the food source by probability, the neighboring food sources are produced by Equation (6). The fitness values are produced by Equation (7). Similar to the employed bee phase, onlooker bees greedily select better optimization solutions. When the food source cannot be improved upon with the predetermined number of trials, the food source will be abandoned. The turned scout bees will again search for an initial food source.

4.2. Modified Quick Artificial Bee Colony Algorithm for Image Thresholding

In the standard ABC algorithm, the employed bee and the onlooker bee use Equation (6) to search food sources. In other words, the employed bee and the onlooker bee adopt the same strategies in the same area to search the new and better food sources, but in real honey bee colonies, the employed bee and the onlooker bee adopt different methods to search for new food sources [34]. Therefore, when onlooker bees search for the best food, it is quite reasonable that their model differs from that of the employed bees. The modified onlooker bee behavior is adopted in this paper with the optimal fitness food source as the center.

The formulized search for new food sources can be represented by Equation (9):

$$v_{N_{j,i}}^{best}=t_{N_{j,i}}^{best}+{\phi }_{j,i}\times (t_{N_{j,i}}^{best}-t_{r,i}),$$

(9)

where $t_{N_{j,i}}^{best}$ is defined as the optimal solution of all the neighborhood food sources around the present food source $t_j$, $N_j$ represents all the neighborhood food sources including $t_j$, $v_{N_{j,i}}^{best}$ is the updated food sources for the next iteration, and ${\phi }_{j,i}$ and $t_{r,i}$ are the same as the ones in Equation (6). It can be clearly observed that the key lies in how the neighborhood food source is to be defined.

From Equation (9), the focus of the model lies in the neighboring food sources to be defined. Karaboga [34] et al. only gives the simple definition of neighboring food sources. With regard to different ways of defining different problems, it needs to define different measurements for similarity. In this paper, with similar motivation as [34] and different from its definition of neighboring food sources, the food sources in multilevel image thresholding are a two-dimensional vector of SN rows and M columns, where SN is the population size and M is the number of thresholds. Every solution corresponds to a digital sequence. The neighboring food source distance is defined as follows:

$$d_{j,i}=\raisebox{1ex}{${\displaystyle {\sum }_{x=1}^{SN}{t}_{x,i}}$}\!\left/ \!\raisebox{-1ex}{$SN$}\right.i=1,2,\dots ,M\mathrm{and}j=1,2,\dots ,SN$$

(10)

where $d_{j,i}$ is the search scope of the food source $t_j$ around the neighborhood. SN stands for the number of the food sources. M is the dimensions of a certain food source. If the Euclidean distance of a solution to $t_j$ is shorter than $d_j$, then it is regarded as the neighborhood of the present solution $t_j$, which is different from the standard ABC and QABC [34] algorithm. It is important to emphasize that all the distances of the neighborhood food sources must be within the vector $d_j$, which contains M euclidean distances respectively. When the onlooker bee reaches the food source $t_j$, firstly it investigates all the neighborhood food sources, choosing the best food source $t_{N_j}^{best}$, and improves her search by Equation (9). In $N_j$, the best food source is defined by Equation (11):

$$t_{N_j}^{best}=\arg max(fit(t_{N_j}^1),fit(t_{N_j}^2),\dots ,fit(t_{N_j}^s))$$

(11)

The improved ABC algorithm can be represented as follows:

Algorithm 2 (Main steps of the MQABC algorithm for image thresholding):

Step 1: Initialization of the population size SN, setting the number of thresholds M and maximum cycle number CN, and initialization the population of source foods by Equation (5). Step 2: Evaluate the population via the specified optimization function, while a termination criterion is not satisfied or Cycle Number < CN. Step 3: (for j = 1 to SN) Produce new solutions (food source positions) vj in the neighborhood of tj for the employed bees using Equation (6). Step 4: Apply the greedy selection process between the corresponding $t_j$ and vj. Step 5: Calculate the probability values $pro{b}_j$ for the solutions $t_j$ by means of their fitness values using Equation (8). Step 6: Calculate the Euclidean distance $d_j$ by Equation (10), search for the neighborhood food sources $N_j$ with distance less than $d_j$ in the existing population, amd then choose the best food source $t_{N_j}^{best}$ by Equation (11) in $N_j$. Step 7: Produce the new solutions (new positions) $v_{N_j}^{best}$ for the onlookers from the solutions $v_{N_j}^{best}$ using Equation (9). Then select them depending on $pro{b}_j$ and evaluate them. Step 8: Apply the greedy selection process for the onlookers between $t_{N_j}^{best}$ and $v_{N_j}^{best}$. Step 9: Determine the abandoned solution (source food), if it exists, and replace it with a new randomly produced solution tj for the scout. Step 10: Memorize the best food source position (solution) achieved so far. Step 11: end for Step 12: end while

5. Experiments and Result Discussions

On the basis of numerical comparative experiments, through comparisons and contrasts of images, data, and graph analysis, this paper has verified the superiority of the proposed algorithm. The focus of the intelligent optimization-based image thresholding algorithm is the objective function and the selection of optimized methods. From relevant literature, it can be seen that some methods are superior to others. This paper will compare the proposed algorithm with the best-so-far methods, while the proven inferior ones will be sidelined. In the following sections, the proposed algorithm will be compared to the electro-magnetism method, the standard ABC presented in literature [5], and [23] MDGWO [20], respectively. Electro-magnetism optimization and MDGWO with Kapur’s entropy as the objective function are so far the newest intelligent optimizations employed in multilevel image thresholding.

The proposed algorithm has been implemented in a set of benchmark images. Some of these images (Lena, Cameraman, Hunter, and Baboon) are widely used in multilevel image thresholding literature [5,8,17]. Others are chosen on purpose from the Berkeley Segmentation Data Set and Benchmarks 500 (BSD500 for short, see [35]), as shown in Figure 1. The experiments were carried out on a Lenovo Laptop with an Intel Core i5 processor and 4GB memory. The algorithm was developed via the signal processing toolbox, image processing toolbox, and global optimization toolbox of MatlabR2011b. The parameters used for the ABC algorithms [23] are presented in Table 1. In order to test the specific effects of these parameters in MQABC, many experiments have been conducted. Figure 2 presents the convergence of objective functions with the iterations from 50 to 500. When the iterations range from 50 to 100, the convergence is not so apparent. In the case that the iterations reach 200, the convergence is evident. However, as the iterations increase, they show little effect on convergence speed. Table 2 presents the image segmentation of Baboon with different population-sizes (detailed PSNR and FSIM evaluation is shown at Subsection 5.3). From the table, it can be seen that, when swarm size reaches 30, the quality of image segmentation is supreme. At the same time, the parameter changes of the maximum trial limit are tested. It shows that while the biggest value is 7, most of them are below 5. Therefore, the parameters presented in Table 1 can be directly applied to MQABC.

Figure 1. The original images and their histograms (a) Baboon; (b) Lena; (c) Cameraman; (d) Corn; (e) Hunter; (f) Soil; (g) Starfish; (h)and Lady, and (i–p) are their respective histograms.

Table 1. Parameters used for the modified quick artificial bee colony algorithm (MQABC).

Parameters	Population Size	No. of Iterations	Lower Bound	Upper Bound	Trial Limit
Value	30	200	1	256	10

Figure 2. The comparison of convergence with different number of iterations.

Table 2. Comparison of image segmentation quality with different population sizes.

Population Size	10	20	30	40	50
PSNR	20.3650	20.4675	20.7193	20.4243	20.4071
FSIM	0.9259	0.9192	0.9318	0.9206	0.9207

5.1. The Modified Quick Artificial Bee Colony Image Segmentation Results with Different Thresholds

Eight test images are employed, which are widely used in the literature, as shown in Figure 1. Kapur’s entropy is based on histograms, so we also present the histograms together with the test images. From Figure 1, it can be observed that each image corresponds to a different shape, which guarantees the universality and applicability of the algorithm.

Figure 3 and Figure 4 show the image segmentation results. If the thresholds are within the range 2 to 5, the quality is relatively high, but if the image size is larger or needs to be segmented into more areas, it is desirable to increase the number of thresholds, which depends on the specific application situation. Apart from the results given in Figure 3 and Figure 4, the Figures also mark the specific positions of the thresholds.

Figure 3. The segmentation results of (a–d) in Figure 1 and their thresholds in histograms.

Figure 4. The segmentation results of (e–h) in Figure 1 and their thresholds in histograms.

It is hard to compare the quality of image segmentation visually with other MT segmentation methods. As a result, comprehensive detailed evaluation systems will be given in the next sections in the form of tables to offer qualitative analysis.

5.2. Comparison of Best Objective Function Values and Their Corresponding Thresholds between EMO, ABC, MDFWO and MQABC

In this section, the results of best objective function values and their corresponding thresholds acquired by various images are discussed. Table 3 and Table 4 depict the number of thresholds, objective values, and corresponding optimal thresholds obtained by the EMO, ABC, MDGWO, and MQABC methods, respectively.

Table 3. Comparison of optimum threshold values between EMO, ABC, MDGWO and MQABC.

Image	M	Optimum Threshold Values
		EMO	ABC	MDGWO	MQABC
Baboon	2 3 4 5	78 143 46 100 153 46 100 153 235 34 75 115 160 235	80 144 46 101 153 45 100 153 236 34 75 116 161 236	80 144 46 100 153 34 89 147 235 27 65 107 154 235	80 144 46 100 153 48 100 153 235 31 73 118 165 235
Lena	2 3 4 5	97 164 81 126 177 16 82 127 177 16 64 97 137 179	97 164 83 127 178 16 87 123 174 16 65 96 133 178	97 164 81 126 177 16 82 127 177 16 64 97 138 179	97 164 81 126 176 16 83 126 174 16 64 97 137 178
Camera man	2 3 4 5	124 197 43 103 197 40 96 145 197 40 96 145 191 221	124 197 43 103 197 41 96 1 45 197 28 63 98 145 197	124 197 43 103 197 36 95 145 197 25 61 100 145 197	124 197 43 103 197 40 95 144 197 28 65 102 147 197
Corn	2 3 4 5	99 177 82 140 198 73 118 164 211 69 107 144 181 219	99 177 82 140 199 73 115 158 207 68 106 145 179 216	99 177 82 140 198 73 118 164 211 67 105 142 180 218	99 177 80 138 195 73 116 161 208 67 104 139 178 215
Hunter	2 3 4 5	90 178 58 117 178 44 89 133 180 44 89 132 176 213	90 178 60 118 178 43 87 132 180 51 95 135 179 220	90 178 60 118 178 45 90 133 180 41 86 131 178 222	90 178 59 117 178 44 90 134 180 46 91 133 178 222
Soil	2 3 4 5	104 166 95 151 213 78 124 169 218 17 82 128 172 218	104 166 94 152 212 84 129 172 218 17 64 113 159 216	104 166 95 151 213 82 128 171 218 64 103 141 180 220	103 164 94 153 212 79 125 171 219 17 78 122 167 214
Starfish	2 3 4 5	90 170 74 131 185 68 117 166 208 57 96 135 173 211	90 170 73 127 182 63 111 161 207 55 94 132 170 210	90 170 74 131 185 67 114 161 205 54 92 131 170 209	90 170 72 127 182 69 119 167 206 53 90 130 170 210
Lady	2 3 4 5	108 197 95 144 207 36 84 140 203 36 84 134 179 211	109 197 94 143 207 35 84 140 203 35 85 136 181 211	108 197 95 144 206 34 84 140 203 32 82 132 179 211	108 197 94 145 206 36 84 139 203 34 83 136 180 211

Table 4. Comparison of best objective function values between EMO (Electro-Magnetism optimization), the artificial bee colony algorithm (ABC), modified discrete grey wolf optimizer (MDGWO), and MQABC.

Image	M	Best Objective Function Values
		EMO	ABC	MDGWO	MQABC
Baboon	2 3 4 5	17.6799 22.1331 26.5254 30.6432	17.6802 22.1331 26.5249 30.4394	17.6802 22.1331 26.4671 30.5622	17.6802 22.1331 26.5248 30.6111
Lena	2 3 4 5	17.8234 22.1102 26.1107 30.0052	17.8234 22.1091 26.1064 29.9870	17.8234 22.1102 26.1107 30.0045	17.8234 22.1101 26.1062 30.0029
Cameraman	2 3 4 5	17.7870 22.3541 26.9258 30.8656	17.7870 22.3541 26.9231 30.9041	17.7870 22.3541 26.9150 30.9071	17.7870 22.3541 26.9232 30.9098
Corn	2 3 4 5	18.6341 23.2676 27.5028 31.4310	18.6341 23.2668 27.4910 31.4183	18.6341 23.2676 27.5028 31.4305	18.6341 23.2661 27.4976 31.4205
Hunter	2 3 4 5	17.9294 22.6077 26.8221 30.8369	17.9294 22.6076 26.8094 30.8709	17.9294 22.6076 26.8216 30.8744	17.9294 22.6076 26.8213 30.8784
Soil	2 3 4 5	17.7911 22.4002 26.6100 30.7030	17.7911 22.3949 26.6079 30.6860	17.7911 22.4002 26.6152 30.5266	17.7902 22.3922 26.6067 30.6872
Starfish	2 3 4 5	18.7518 23.3205 27.5691 31.5536	18.7518 23.3192 27.5609 31.5338	18.7518 23.3205 27.5696 31.5537	18.7518 23.3192 27.5675 31.5519
Lady	2 3 4 5	17.3196 21.8911 26.0155 30.1406	17.3195 21.8901 26.0140 30.1187	17.3196 21.8908 26.0137 30.1276	17.3196 21.8882 26.0147 30.1282

From the tables, it can be observed that all the images with different methods show relatively high objective function values. Notably, with the increase of the number of thresholds, the four methods can get high objective function values. Generally, MQABC and EMO show similar or higher objective function values, while ABC and MDGWO are a little inferior, but the difference range is less than 0.1.

5.3. The Multilevel Image Segmentation Quality Assessment by PSNR and FSIM

For a comparison with the most advanced MT method so far, we adopt PSNR (the peak-to-signal ratio) and FSIM (the feature similarity index). One of the most popular performance indicators, peak signal to noise ratio (PSNR), is used to compare the segmentation results between the original image and the segmented image. It is defined by Equation (12):

$$PSNR(i,j)=20{\log }_{10}(\raisebox{1ex}{$255$}\!\left/ \!\raisebox{-1ex}{$RMSE(i,j)$}\right.)$$

(12)

where RMSE is the root mean-squared error, present in the literature [5]. By contrast, the feature similarity index (FSIM) was used to determine the similarity of an image (image segmented) to the reference image (original image) based on the image quality assessment. It can be defined as Equation (13):

$$FSIM(X)=\raisebox{1ex}{${\displaystyle {\sum }_{x\in \mathsf{\Omega }}{S}_L(X)\times P{C}_m(X)}$}\!\left/ \!\raisebox{-1ex}{${\displaystyle {\sum }_{x\in \mathsf{\Omega }}P{C}_m(X)}$}\right.$$

(13)

The detailed definitions of $S_L(X)$ and $P{C}_m(X)$ can be found in [36].

Table 5 and Table 6 demonstrate the PSNR and FSIM metrics of the test images segmented with different thresholds by using EMO, ABC, MDGWO, and QMABC. The results show MQABC achieves the highest assessment, testifying to the superiority of MQABC. In addition, as the number of thresholds increases, the superior PSNR value stands out. In terms of FSIM, the MQABC algorithm produces higher FSIM values on all items in Figure 1 except for the Soil, the Lady images with M = 1, the Hunter, and the Starfish images with M = 2, on which the values yielded are only a little less than those yielded by other algorithms. It needs to be emphasized that, since the use of the same objective function gets similar thresholds (as shown in Table 3 and Table 4), the values of PSNR and FSIM are also very close in the experiments. Similar results can be found in the literature [5,6,7,8,9,10], in which the same objective function is used. The main contribution of this paper is to ensure high objective function values and better time efficiency.

Table 5. Peak-to-signal-noise ratio (PSNR) metrics of the test images segmented with different threshold.

Image	M	PSNR
		EMO	ABC	MDGWO	MQABC
Baboon	2 3 4 5	15.9947 18.5921 18.5921 20.5234	16.0070 18.5921 18.5435 20.5058	16.0070 18.5921 17.7340 19.9261	16.0070 18.5921 18.6718 20.7193
Lena	2 3 4 5	14.5901 17.2122 18.5488 20.3250	14.5901 17.1178 18.3798 20.2440	14.5901 17.2122 18.5488 20.3152	14.5901 17.2328 18.6011 20.3360
Cameraman	2 3 4 5	13.9202 14.4620 20.1078 20.2439	13.9202 14.4620 20.1172 20.6463	13.9202 14.4620 20.0183 20.7726	13.9202 14.4620 20.1187 21.0713
Corn	2 3 4 5	13.5901 15.1819 16.3460 17.1081	13.5901 15.1772 16.3688 17.2131	13.5901 15.1819 16.3460 17.3272	13.5901 15.2913 16.7947 17.3402
Hunter	2 3 4 5	15.1897 18.5073 21.0490 21.1563	15.1897 18.5091 21.0204 21.0040	15.1897 18.5091 21.0584 21.0582	15.1897 18.5116 21.0668 21.1618
Soil	2 3 4 5	15.0106 15.9665 18.5457 19.2186	15.0106 16.0513 18.2730 18.9467	15.0106 15.9665 18.3966 18.6843	15.0813 16.0565 18.6280 19.3609
Starfish	2 3 4 5	14.3952 16.9727 18.1811 20.0649	14.3952 17.1199 18.4836 20.2304	14.3952 16.9727 18.3818 20.2925	14.3952 17.1430 18.5054 20.3400
Lady	2 3 4 5	13.5137 18.0864 18.8255 19.7622	13.5108 18.1279 18.8180 19.7680	13.5137 18.1007 18.8099 19.4766	13.5137 18.1559 18.9093 19.8081

Table 6. Feature similarity index (FSIM) metrics of the test images segmented with different thresholds.

Image	M	FSIM
		EMO	ABC	MDGWO	MQABC
Baboon	2 3 4 5	0.8586 0.9004 0.9004 0.9218	0.8601 0.9004 0.8997 0.9218	0.8601 0.9004 0.8863 0.9131	0.8601 0.9004 0.9012 0.9318
Lena	2 3 4 5	0.7262 0.7911 0.8018 0.8457	0.7262 0.7886 0.7952 0.8415	0.7262 0.7911 0.8018 0.8455	0.7262 0.7919 0.8036 0.8455
Cameraman	2 3 4 5	0.6776 0.7893 0.8433 0.8470	0.6776 0.7893 0.8431 0.8636	0.6776 0.7893 0.8426 0.8689	0.6776 0.7893 0.8439 0.8743
Corn	2 3 4 5	0.6902 0.7661 0.8092 0.8364	0.6902 0.7653 0.8096 0.8363	0.6902 0.7661 0.8092 0.8370	0.6902 0.7676 0.8107 0.8373
Hunter	2 3 4 5	0.7117 0.8175 0.8815 0.8851	0.7117 0.8156 0.8828 0.8741	0.7117 0.8156 0.8801 0.8871	0.7117 0.8168 0.8804 0.8880
Soil	2 3 4 5	0.7805 0.8182 0.8761 0.8977	0.7805 0.8216 0.8649 0.9074	0.7805 0.8182 0.8694 0.9014	0.7779 0.8229 0.8789 0.9100
Starfish	2 3 4 5	0.6058 0.6870 0.7336 0.7913	0.6058 0.6924 0.7417 0.7956	0.6058 0.6870 0.7417 0.7951	0.6058 0.6922 0.7450 0.7957
Lady	2 3 4 5	0.5895 0.7205 0.7645 0.8261	0.5900 0.7222 0.7646 0.8230	0.5895 0.7223 0.7645 0.8220	0.5895 0.7254 0.7667 0.8277

5.4. Comparison of the Characteristics of Running Time and Convergence

On the prior condition that segmentation quality is guaranteed, we examine the running time of image thresholding and iterations to convergence to test performance. As shown in Table 1, experiments are set to the same number of iterations for testing. However, in order to obtain the operating time for convergence as described in Table 7 and Table 8, when the fitness function remains unchanged for 20 consecutive times, the experimental iteration is forced to be terminated. Table 7 and Table 8 present data concerning the running time of image thresholding and convergence. EMO enjoys certain advantages in terms of iterations, but its running time is much longer than the other three. First, compared with EMO, the ABC method gets the highest increase of 368% for the Baboon image with M = 5; the lowest increase is still 70%, for the Corn image with M = 4. Then, by comparing MQABC to EMO, we find MQABC to get more significant time efficiency. It is close to the maximum 461% and minimum 76% of the time efficiency on the same images and thresholds as ABC. On the whole, when compared with EMO, the running time of ABC and MQABC increase by an average of 165% and 205%, respectively, for the eight pictures shown in Figure 1. At the same time, even if we reduce the running time by decreasing the iterations of EMO, the number of iterations for convergence will be improved accordingly, so it is still inefficient. Take Baboon for example; under the condition of the biggest iterations reaching 100 and with the threshold = 5, the running time of EMO is 18.007129 and the iterations to convergence are 47. Its running time is still longer.

Table 7. The running time of image thresholding for EMO, ABC, MDGWO, and MQABC.

Image	M	Running Time (S)
		EMO	ABC	MDGWO	MQABC
Baboon	4 5	2.887928 8.747538	1.369329 1.870215	1.616275 1.676552	1.269490 1.559394
Lena	4 5	3.923755 5.297557	1.537726 1.997193	1.571869 1.703536	1.180458 1.471279
Cameraman	4 5	4.171417 6.263097	1.375140 1.699723	1.637016 1.751039	1.246388 1.494485
Corn	4 5	4.221177 9.523326	2.481717 2.989686	2.707189 2.958905	2.402332 2.702894
Hunter	4 5	3.187744 6.207372	1.572262 2.044084	1.600798 1.654322	1.479363 1.596739
Soil	4 5	4.546173 8.693048	2.359536 2.822248	2.748231 2.853070	2.122324 2.481809
Starfish	4 5	4.651453 5.812047	2.537318 2.972738	2.790188 2.954895	2.308098 2.647345
Lady	4 5	4.621457 9.362831	2.524725 3.055700	2.736448 2.947035	2.437168 2.837646

Table 8. The number of iterations to convergence for EMO, ABC, MDGWO, and MQABC.

Image	M	Iterations to Convergence
		EMO	ABC	MDGWO	MQABC
Baboon	4 5	15 53	55 140	145 141	46 81
Lena	4 5	29 42	127 194	149 148	20 78
Cameraman	4 5	19 49	82 140	149 149	49 82
Corn	4 5	33 60	80 193	148 150	54 100
Hunter	4 5	17 26	97 183	145 147	64 89
Soil	4 5	23 50	113 115	58 147	17 61
Starfish	4 5	30 35	131 194	149 150	59 117
Lady	4 5	29 61	100 188	147 150	33 145

The comparison of ABC and MQABC with the same parameters as shown in Table 1 shows that the latter enjoys significantly better time efficiency, especially when the threshold number increases. The MQABC method gets the highest increase of 36% for the Lena image when M = 5; the lowest increase is 3.3% for the Corn image when M = 4. From the perspective of the threshold number, the maximum and minimum run times increase by 30% and 3.3% for the Lena and Corn images, respectively, when M = 4. Further, when M = 5, the maximum and minimum run times increase by 36% and 7.7% for the Lena and Lady images, respectively. On average, the MQABC method shows more than 18% of the time improvement for the eight figures when M = 5 and more than 10% when M = 4. Therefore, the MQABC method ensures the optimal threshold values so as to obtain the best time efficiency.

Compared with ABC and MDGWO, MQABC also enjoys better time efficiency. The MQABC method gets the highest increase of 33% for the Lena image when M = 4; the lowest increase is by 3.6% for the Hunter image when M = 5. From the perspective of the threshold number, the maximum and minimum run times increase by 33% and 8.2% for the Lena and Hunter images, respectively, when M = 4. Furhter, when M = 5, the maximum and minimum run times increase by 17% and 3.6% for the Cameraman and Hunter images, respectively. On average, the MQABC method shows 22% of the time improvement for the eight figures when M = 4 and more than 10% when M = 5. Therefore, the MQABC method ensures the optimal threshold value so as to obtain the best time efficiency

In order to compare the efficiency of MQABC and ABC in convergence rates, Figure 5 demonstrates the convergence curving lines of images in Figure 1 when the thresholds are 5. Combined with Table 7 and Table 8, it can be easily found that MQABC shows obvious superiority not only in terms of iterations to convergence, but also in its speed towards the optimal objective function. Obviously, the convergence rate of MQABC is better than that of ABC.

Figure 5. The comparison of convergence of images (a–h) in Figure 1 with 5 thresholds.

From Table 3, Table 4, Table 5, Table 6, Table 7 and Table 8 and Figure 3, Figure 4 and Figure 5, together with visual effect analysis, it can be observed that MQABC demonstrates better time efficiency and excellent segmentation results and has obvious advantages, especially over ABC, in convergence performance. Therefore, it can be safely assumed that MQABC is a desirable image segmentation method with high efficiency and of high quality.

6. Conclusions

In this paper, MQABC has been employed to optimize histogram-based Kapur’s entropy in order to realize MT image thresholding. The experimental results demonstrated that MQABC is highly efficient in running time, convergence, and image segmentation quality. By improving on the distance strategies in the onlooker bee phase, MQABC can be successfully applied to multilevel image thresholding. Through numerical quantitative and visual experimental comparisons, it can be observed that MQABC is able to obtain better convergence speeds and shows obvious superiority over EMO, ABC, and MDGWO in terms of the running time of image thresholding and image segmentation quality.