Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2015 / Article

Research Article | Open Access

Volume 2015 |Article ID 272947 | 23 pages | https://doi.org/10.1155/2015/272947

Artificial Bee Colony Optimizer with Bee-to-Bee Communication and Multipopulation Coevolution for Multilevel Threshold Image Segmentation

Academic Editor: Sishaj P. Simon
Received29 Sep 2014
Accepted28 Jan 2015
Published15 Apr 2015

Abstract

This paper proposes a modified artificial bee colony optimizer (MABC) by combining bee-to-bee communication pattern and multipopulation cooperative mechanism. In the bee-to-bee communication model, with the enhanced information exchange strategy, individuals can share more information from the elites through the Von Neumann topology. With the multipopulation cooperative mechanism, the hierarchical colony with different topologies can be structured, which can maintain diversity of the whole community. The experimental results on comparing the MABC to several successful EA and SI algorithms on a set of benchmarks demonstrated the advantage of the MABC algorithm. Furthermore, we employed the MABC algorithm to resolve the multilevel image segmentation problem. Experimental results of the new method on a variety of images demonstrated the performance superiority of the proposed algorithm.

1. Introduction

Multithreshold image segmentation technique is considered as a simple but effective method to extract useful objects from the digital image background, which required multiple threshold values to accomplish segmentation. The image segmentation can be roughly grouped into optimal threshold [14] and property-based threshold [57]. The purpose of the first category is to find the proper thresholds for classifying the histogram at a desired level by optimizing an objective function. The second category searches the optimal thresholds by measuring some property of the histogram, which is fast and suitable for the case of multilevel threshold but hard to determine the number of thresholds. Such methods include entropy-based threshold methods, minimization of Bayesian error, and clustering-based methods.

Many existing algorithms utilize the exhaustive search approach to pursue the desired thresholded classes [812]. References [8, 9] proposed some novel methods derived from optimizing an objective function. Several algorithms originally proposed for bilevel threshold have been later extended to multilevel threshold [3, 4, 11, 12]. However, due to the exhaustive search employed, the computational complexity of these methods will rise exponentially when the number of thresholds increases, which limits the multilevel threshold applications.

Recently, owing to their predominant abilities of coping with complex nonlinear optimizations, several swarm intelligence (SI) algorithms as the powerful optimization tools have been applied to address this concerning issue [1215]. Among them, the artificial bee colony algorithm (ABC) is a popular SI algorithm, which simulates the social foraging behavior of a honeybee swarm [16]. Due to its simple arithmetic and good robustness, the ABC algorithm has been widely used in solving many numerical optimizations and engineering optimization problems [1720]. However, facing up complex problems, similar to other SIs, ABC algorithm suffers from the following drawbacks [17]. The solution search equation of ABC works well in global exploration but is poor in the exploitation process. With the dimension increasing, the information exchange of each individual is limited in a random dimension, resulting in a slow convergence rate. Due to the random selection of the neighbor bee and dimensions, food sources with higher fitness are not utilized, which influences the ability of global search.

Thus, this paper presents a modified artificial bee colony algorithm for multilevel threshold image segmentation, namely, MABC, to conquer the original ABC drawbacks to some extent, which synergizes the idea of multipopulation cooperative model with the bee-to-bee communication strategy. In the bee-to-bee communication mechanism, with the enhanced information change strategy, individuals can share more information from the elites in the early exploration stage of the algorithm. With the multipopulation cooperative approach, the hierarchical colony with different topologies can be structured, which can maintain diversity of the whole community. By incorporating this hybrid strategy, the proposed MABC can be claimed very powerful, because the exploitation and exploration can be elaborately balanced, which guarantees finding the optimal thresholds at a more reasonable time.

The rest of the paper is organized as follows. Section 2 describes the canonical ABC algorithm in detail. In Section 3, the proposed modified artificial bee colony (MABC) algorithm is given. Section 4 presents the experimental studies of the proposed MABC and the other algorithms with descriptions of the involved benchmark functions, experimental settings, and experimental results. And its application to image segmentation has been presented in Section 5. Finally, Section 6 outlines the conclusion.

2. Canonical ABC Algorithm

The artificial bee colony (ABC) algorithm was proposed by Karaboga in 2005 [16], simulating the intelligent foraging behavior of a honeybee swarm. The entire bee colony contains three groups of bees: employed bees, onlookers, and scouts [17, 18]. Employed bees are responsible for exploring the specific food sources; meanwhile they pass the food information to onlooker bees. The number of employed bees is equal to that of food sources. Then onlooker bees choose good food sources based on the received information, and then they further exploit the food near their selected food source. The food source with higher quality would have a larger opportunity to be selected by onlookers. There is a control parameter called ‘‘limit’’ in the canonical ABC algorithm. If a food source is not improved anymore when limit is exceeded, it is assumed to be abandoned by its employed bee and the employed bee associated with that food source becomes a scout to search for a new food source randomly. The fundamental mathematic representations are listed as follows.

Step 1 (initialization phase). In initialization phase, a group of food sources are generated randomly in the search space using the following equation:where , . is the number of food sources. is the number of variables, that is, problem dimension. and are the lower upper and upper bounds of the th variable, respectively.

Step 2 (employed bees’ phase). In the employed bees’ phase, the neighbor food source (candidate solution) can be generated from the old food source of each employed bee in its memory using the following expression:where is a randomly selected food source and must be different from ; is a randomly chosen index; is a random number in range [−1, 1].

Step 3 (onlooker bees’ phase). In the onlooker bees’ phase, an onlooker bee selects a food source depending on the probability value associated with that food source; can be calculated as follows:where is the fitness value of the th solution.

Step 4 (scout bees’ phase). In scout bees’ phase, if a food source cannot be improved further through a predetermined cycle (called “limit” in ABC), the food source is supposed to be abandoned. The employed bee subsequently becomes a scout. A new food source will be produced randomly in the search space using (1).
The employed, onlooker, and scout bees’ phase will recycle until the termination condition is met.

3. Modified Artificial Bee Colony Algorithm

3.1. Bee-to-Bee Communication Model
3.1.1. Von Neumann Topology Structure

The inherent property of swarm intelligence is the “population” where unsophisticated individuals share global information with their neighborhoods to create collective intelligence in some specific topologies [21, 22]. The spatial topology structure significantly affects the spreading path of information flow across the structured population and also determines the social interaction between individuals.

Recently, using population topology technique to improve the performances of evolutionary algorithms has been strongly suggested [2325]. In [23], after deeply investigating the relationship between various topologies and performances, Kennedy drew an interesting conclusion that Von Neumann topology demonstrates superior performances on a wide range of testing functions, as shown in Figure 1(a).

In ABC, from (1), it is clearly visible that any candidate neighborhood of individual can be selected randomly from the entire population, which means every individual is connected to individual . In other words, the population topology of ABC belongs to the Star type, as shown in Figure 1(b). As depicted in [23], it was theorized that although the connectivity of the Von Neumann structure is lower than the Star type, the Von Neumann type can enable individuals of population to cover a larger search space and to maintain diversity of population, avoiding falling into the deep local optimum. The detailed steps of constructing the Von Neumann structure are given as shown in Algorithm 1.

Step  1. Split the population of individuals into rows and cols, and .
Step  2.  For .
Step  3.   1 =
        if 1
        1.
Step  4.   2,
        if ,
        2.
Step  5.   3,
        if ,
        .
Step  6.   4,
        if 4,
        4
Step  7.   nhood() = 1234.
     End For

In the MABC algorithm, the Von Neumann model is utilized in the employed bee phase where each individual has three neighborhoods, namely, the above node, the left node, and the below node. And in the onlooker bee phase and the scout bee phase, the original topology is still employed. This can maintain an appropriate balance between improving convergence rate and enhancing diversity of population.

3.1.2. Enhanced Information Exchange Based on Crossover

In canonical ABC, from (2), we can see that the information exchange between individuals is restricted to a narrow scope, which relies on a single random dimension of a single random bee. To address this, the crossover-based information sharing technique is employed in MABC. To benefit from the global best information derived from the entire population, a given number of elites from excellent individuals are subjected to crossover operation. The underlying idea behind this method is to facilitate better search in complex higher dimensional search space as opposed to classical ABC that perturbs a single parameter. The steps for crossover operation phase are given as follows and the schematic diagram is illustrated in Figure 2.

Algorithm 2 (crossover operation). Consider the following.

Step 1 (select elites to the best-performing list (BPL)). A set of individuals with higher fitness from current food sources are chosen to construct the best-performing list (BPL), whose size is equal to the current population size.

Step 2 (crossover operation). Parents are chosen from the BPL’s elites only for crossover by the tournament selection scheme. Then, two offspring are generated by the arithmetic crossover operation:where is new offspring, Parent 1 and Parent 2 are randomly chosen from BPL, and is a random coefficient in range of .

Step 3 (update population). A selecting rate is used to determine the replaced individuals in the population. If population size is S, then the replaced individuals size is , and they are chosen randomly from the population. Through the greedy selection between the replaced individual and the new offspring , the better one remains in the population.

3.2. Multipopulation Cooperative Mechanism

The multipopulation cooperative concept has been widely adopted to improve the search ability of algorithms via decomposing complex problems into simple tasks tackled in parallel. With the multipopulation cooperative mechanism as shown in Figure 3, the single population in ABC can be structured into two levels with different topologies and information exchange strategies. The new characteristics can be abstracted as follows.

In each foraging process, this two-level multipopulation evolution can be formulated as follows:where is the global best individual within species and is the best individual found so far among all species. From (5), we can see that the term represents the information exchange within species (individual level) in a specific topology, and denotes the cooperation of agents between species (species level). This will help to amplify the diversity of population. Furthermore, the information exchange strategy significantly improves the search ability of the proposed method.

3.3. The Proposed Algorithm

By incorporating the aforementioned evolution model and strategies, namely, multipopulation cooperative model and bee-to-bee communication strategy, each individual can regulate its trajectory according to the effective topology and information exchange strategy. In particular, the population evolution process can make use of the historical experience of individual level and the global best information of the species level, which can maintain the diversity of population. The main steps of the proposed algorithm are given as the following processes.

Algorithm 3 (the proposed MABC algorithm). Consider the following.

Step 1 (initialization). Consider the following.

Step 1.1. Randomly generate species, each possessing individuals.

Step 1.2. Initialize the population topology structure according to Algorithm 1.

Step 1.3. Evaluate the fitness of each bee.

Step 1.4. Set maximum cycles (LimitC).

Step 2. Consider iteration = 0.

Step 3 (employ bee phase). Loop over each food source.

Step 3.1. Generate a candidate solution by (4) and evaluate .

Step 3.2. Greedy selection and memorize the better solution.

Step 4. Calculate the probability value by (3).

Step 5 (onlooker bee phase). Consider the following.

Step 5.1. Select a food source by and generate a candidate solution by (4).

Step 5.2. Greedy selection and memorize the better solution.

Step 6 (crossover phase). Consider the following.

Step 6.1. Select elites to the best-performing list (BPL) as Algorithm 2.

Step 6.2. Crossover and mutation operation as Algorithm 2.

Step 6.3. Update with greedy selection strategy as Algorithm 2.

Step 7. Consider iteration = iteration + 1.

Step 8. If the iteration is greater than LimitC, stop the procedure; otherwise, go to Step 3.

Step 9. Output the best solution achieved.

In summary, in order to facilitate the below presentation and test formulation, we define unified parameters for MABC model in Table 1 and the flowchart of MABC algorithm is summarized in Figure 4.


MABC = ()

Population size
Dimension of problem
Species number
Selection rate

4. Benchmark Test

In the experimental studies, 15 benchmark functions are employed to test the proposed MABC algorithm [2630]. The involved benchmarks can be classified as basic benchmarks () and CEC 2005 benchmarks (). The formula for each basic benchmark and CEC 2005 test functions is shown in Tables 2 and 3. In our test, the function evaluations (FEs) criterion is used as a time measure substituting the number of iterations, due to the fact that the algorithms do differing amounts of work in their inner loops.


NameFunctionLimits

Sphere ()

Rosenbrock ()

Quadric ()

Sin ()

Rastrigin ()

Schwefel ()

Ackley ()

Griewank ()


NameFunctionLimits

Shifted Sphere (), 

Shifted Schwefel’s Problem 1.2 (),  

Shifted Rosenbrock’s function ()

Shifted rotated Griewank without bounds ()No bounds

Shifted rotated Ackley (), 

Shifted Rastrigin (), 

Shifted rotated Rastrigin ()

4.1. Parameters Settings for the Involved Algorithms

Experiment is conducted to compare with successful EA and SI algorithms, including original artificial bee colony algorithm (ABC) [17], canonical PSO with constriction factor (PSO) [31], genetic algorithm with elitism (EGA) [32], and covariance matrix adaptation evolution strategy (CMA-ES) [33]. The independent run time is 30, and the maximum evaluation number is 100000. For the involved benchmarks, the dimensions are all set as 30.

All related parameters of the involved compared algorithms are set as their original literatures suggest: initialization conditions of CMA-ES are referred to in [33], and the candidate offspring number per time step is set as . For ABC, the limit parameter is , where is employed bee number and is the benchmark dimension; the learning rates and of PSO are both 2.05 and the constriction factor ; the crossover rate and mutation rate of EGA are 0.8 and 0.01, respectively, and the global elite with a rate of 0.06 is used [31]. The parameter setting for all algorithms is listed in Table 4. For the proposed MABC, the species number and selection rate should be tuned firstly in the next section.


TypeMABCABCPSOEGACMA-ES

550505050
10NANANANA
1NANANANA
NANA0.729NANA
NANA2NANA
NANA2NANA
NANANANA12/0.1/20/0.12/0.08
Crossover/mutation rateNANANA0.8.0.01NA

4.2. Sensitivity in Relation to Parameters of MABC

(1)  Effect of Agent Number in Species Level. In order to study the sensitivity in relation to the parameter , six classical 30-dimensional benchmark functions were employed, namely, unimodal sphere (), unimodal nonseparable Rosenbrock (), multimodal Schwefel (), unimodal shifted sphere (), multimodal rotated Rastrigin (), and multimodal nonseparable rotated Ackley () functions, as defined in Table 2.

In this test, the values of parameter were changed from 1 to 15, while the selection rate is preset as 1. Results of this experiment are illustrated in Figure 5. From Figure 5, it is clearly visible that when is near 10, MABC performed superior in terms of mean values and convergence rate on most test functions. However, the effect of this parameter is not very obvious in the test.

(2) Effect of Selection Rate CR. Five basic benchmark functions () are employed to evaluate the performance of MABC variants with different s. All the functions are implemented for 30 run times, whose dimension is 30. From Table 5, we can observe that MABC variant with equal to 1 performed best on four functions among all five functions while the variant with equal to 0.05 obtained best results on function . Therefore, the value of selection rate in each crossover operation can be set to 1 as an optimal value in all the following experiments.


Func.0.050.1 0.2.0.40.61

Mean
Std.

Mean
Std.

Mean
Std.

Mean
Std.

Mean
Std.

4.3. Numerical Results and Comparison
4.3.1. Results for the Classical Benchmarks

The means and standard deviations of the 30 run times of the nine involved algorithms on the classical test suite are listed in Table 6. The best results among those algorithms are shown in bold in Table 6. Figures 6(a)6(g) present the average convergence rates of each algorithm for each benchmark.


FunctionsMABCABCPSOCMA-ESGA

Mean
Std.
Rank

Mean
Std.
Rank

Mean
Std.
Rank

Mean