# Fuzzy Multilevel Image Thresholding Based on Modified Quick Artificial Bee Colony Algorithm and Local Information Aggregation

Authors: Linguo Li; Lijuan Sun; Jian Guo; Chong Han; Shujing Li
Journal: Mathematical Problems in Engineering
 (2016)

---

## Abstract

Thresholding segmentation based on fuzzy entropy and intelligent optimization is one of the most commonly used and direct methods. This paper takes fuzzy Kapur’s entropy as the best optimal objective function, with modified quick artificial bee colony algorithm (MQABC) as the tool, performs fuzzy membership initialization operations through Pseudo Trapezoid-Shaped (PTS) membership function, and finally, according to the image’s spacial location information, conducts local information aggregation by way of median, average, and iterative average so as to achieve the final segmentation. The experimental results show that the proposed FMQABC (fuzzy based modified quick artificial bee colony algorithm) and FMQABCA (fuzzy based modified quick artificial bee colony and aggregation algorithm) can search out the best optimal threshold very effectively, precisely, and speedily and in particular show exciting efficiency in running time. This paper experimentally compares the proposed method with Kapur’s entropy-based Electromagnetism Optimization (EMO) method, standard ABC, and FDE (fuzzy entropy based differential evolution algorithm), respectively, and concludes that MQABCA is far more superior to the rest in terms of segmentation quality, iterations to convergence, and running time.


---

## Body

## 1. Introduction

Thresholding is one of the most direct, simple, and efficient approaches to image segmentation, which can discriminate objects from the background through a set of thresholds at pixel levels. The automatic separation between objects and background remains the most difficult yet intriguing domain for image processing and pattern recognition [1]. It not only can show well defined regions with the minimum overlap and effectiveness of aggregation, but also can provide initial prediction or preprocessing for more complicated segmentation methods (such as level-sets or active contours) [2]. Consequently, thresholding segmentation is regarded as the key link in image analysis and image understanding and therefore is widely used in many areas like medical analysis [3], image classification [4], object recognition [5] and image copy detection [6], and so forth.Thresholding segmentation is the most commonly used method for image segmentation [7]. Bilevel thresholds are mainly used to separate the object from the background, namely, bilevel thresholding. Multilevel thresholding (MT) can classify the pixels into different groups; hence multiple regions are created, which can lead to finer segmentation results. The thresholding methods proposed in the literature can be divided into two categories, parametric and nonparametric [8–10]. The parametric are usually time consuming and computationally expensive, whereas the nonparametric are more robust and precise and as a result get more attention, as they can determine the optimal threshold by optimizing some standards like between-class variance, the entropy, the error rate, and so forth [8–11]. This paper from the perspective of nonparametric methods will be focused on the study of MT methods through intelligent optimization and fuzzy theory.Of nonparametric methods, gray level global thresholding adopting information entropy is the most popular one. Such method takes the intelligent optimization as the tool and by optimizing some information entropy to get the maximum (or minimum) objective function to achieve the final segmentation. The most commonly used objective functions include the maximization of the entropy (e.g., Kapur’s entropy) [12], maximization of the between-class variance (e.g., Otsu’s method) [13], the use of the fuzzy similarity measure [14], and minimization of the Bayesian error [15]. All of these techniques have been extended into the area of multilevel image segmentation. But the computational complexity for the application of such methods in MT grows dramatically. As a consequence, scholars in recent years have extensively used evolutionary and swarm-based intelligent computational techniques to promote multilevel computational efficiency for multilevel image segmentation and precision of threshold [11]. But the global multilevel thresholds generally get the final segmentation results by maximizing the separation between classes. Yet it neglects the complexity in uneven illumination and soft transitions between gray levels and fails to consider the spatial relationships between pixels, which lead to the imperfections of segmentation results [1]. The biggest problem with the global thresholding lies in the fact that it is pixel oriented rather than region oriented. Therefore, pixels that have the same gray levels are segmented into the same regions if such factors as connectivity of pixels and closeness of objects are not considered. As a result, such methods are prone to create isolated pixels.Of all the research findings for the various thresholding methods, each one of them is designed specifically to deal with one particular kind of images. Sezgin and Sankur [16] were the first to classify the image thresholding methods into six categories. Aja-Fernández et al. [1] further grouped them into three kinds based on the six classifications. (a) Methods that calculate a global threshold for the whole image. (b) Methods that use an adaptive local threshold. (c) Methods that use spatial local information for classifying the pixels. The first kind was the earliest to be employed. But due to the drawbacks inherent in it, currently, scholars have directed their focus on the latter two. The second kind is an improvement on and complement to the first kind which selects the threshold locally through adaptability so as to avoid local nonadaptability, but the computational complexity will grow accordingly. The third kind is mainly employed to make up for the deficiency in spatial relationship and introduces the local spatial information to heighten the effects of segmented results. In this paper, we introduce local spatial information to solve segmentation problems and at the same time explore the applications of fuzzy logic and fuzzy theory in image thresholding. The starting point of introducing local spatial information comes from the definition of one class or target, where the membership of a pixel is highly correlated with the membership of the pixels of surrounding regions [1]. In order to promote the quality of image segmentation, fuzzy membership function is used to assign each pixel with fuzzy degrees. And according to fuzzy theory, the traditional hard assigning regions or classes will be replaced with soft assigning pattern. Detailedly, the fuzzy Kapur’s entropy which is modified by Kapur’s entropy is adopted in this method as the objective function. It combines with the MQABC algorithm to generate the optimal thresholds which are used as the centroids for fuzzy membership assignment and local information aggregation.The rest of the paper is organized as follows. Section2 introduces related work of fuzzy entropy, intelligent optimization, and local information aggregation in the field of MT. In Section 3, the nonfuzzy hard thresholding and fuzzy soft thresholding are formulated, and then the corresponding Kapur’s entropy for image segmentation and the ultimate objective function are presented. In Section 4, the standard bee colony algorithm is briefly described and the proposed MQABC is dealt with in detail. Section 5 analyzes the multiple thresholds-based fuzzy membership initialization and fuzzy aggregation methods. The experiment results and discussions are presented in Section 6. The last section presents the conclusions.


## 2. Related Works

Fuzzy theory and fuzzy logic are well-known to be flexible tools where imprecise knowledge or not-well defined features have to be used [17]. It has been extensively used in system control domain and applied in image processing to some extent [18–21]. In the past 10 years, thresholding segmentation approaches have been proposed based on fuzzy logic and fuzzy calculations. In such approaches, fuzzy logic is combined with clustering method [22], fuzzy compactness [23], the interpretation of thresholds as type 2 fuzzy logic [24], and the soft calculations (such as ant colonies, bee colonies, and bacteria migration heuristics) [11], respectively, mainly used to search for the optimal thresholds, but they failed to take the pixel spatial relationship segmentation methods into full account. In these fuzzy based thresholding methods, clustering methods are among the most popular, but they are highly complicated where computational time is concerned. With the progress and upgradation of optimization methods, fuzzy optimization in thresholding segmentation gets more and more attention and it has been proved feasible to employ intelligent optimization to solve thresholding problems [10]. A large number of literature show that intelligent optimizations are far superior to traditional methods in processing speed, precision and robustness [5, 10]. Oliva et al. [7] introduced Electromagnetism Optimization into thresholding segmentation and compared it with Kapur’s entropy and Otsu’s method. The experiments show that Kapur’s entropy is more efficient. Ghamisi et al. [25] have fully analyzed the performance in thresholding of such methods as Particle swarm optimization (PSO), Darwinian Particle Swarm Optimization (DPSO), and Fractional-order Darwinian Particle Swarm Optimization (FODPSO). In comparison with Bacteria Foraging algorithm (BF) and genetic algorithms (GA), FOPDSO delivers better performance in overcoming local optimum trap and computational speed. Kurban et al., [10] have conducted a comparative study of the applications of the computational techniques involved in MT exhaustively. According to the statistical analysis of the objective value, the swarm-based method is more able to precisely solve segmentation problems. In a further analysis, swarm-based thresholding segmentation mainly takes Otsu’s method, between-class variance, Tsallis entropy, and Kapur’s entropy as the objective functions; the experimental results show that the optimized processing, by way of artificial bee colony, could obtain better segmentation results. Akay [8] took between-class variance and Kapur’s entropy as the objective functions, respectively, and compared artificial bee colony method with the particle swarms one; it shows that Kapur’s entropy-based artificial bee colony method shows better performance when the number of the thresholds increases, and the complexity of computational time is also reduced. Bhandari et al. [26] conducted detailed comparative analysis between Kapur’s Otsu and Tsallis functions through experiments, which shows that in the remote sensing satellite image segmentation, Kapur’s entropy-based artificial bee colony method is far superior.The above-mentioned optimized methods have made great progress in precision, efficiency, and segmentation quality in solving thresholding problems. But classifying regions by hard thresholding will easily lead to edge fuzziness and even isolated pixels [27]. In more recent years, fuzzy theory based thresholding method has become the focus of study, as the proposed soft thresholding strategy can effectively solve the problem. Zhao et al. [28] were the earliest to define the three-membership functions and then applied them to three-level thresholding problems. Based on that, Tao et al., [29] proposed three-level fuzzy entropy based thresholding methods; the three functions, namely, I-function, F-function, and S-function, are defined in detail. The method maximizes global fuzzy entropy through Genetic Algorithm (GA) to get the optimal thresholds. These two methods only show the visual effects, failing to present a qualitative analysis. Pratamasunu et al. [30] assigned fuzzy coefficient to each I-function and then surveyed the function to achieve the automation of thresholding by defining the distance between the fuzzy set and its closest crisp set. Although the method shows a comparative advantage over the Lopes method and Otsu thresholding, the experimental results are only applicable to bilevel thresholds and fail to show the effects of MT. Muppidi et al. [31] used Triangular membership function, Trapezoidal membership function, and Bell-shaped membership function to define three kinds of fuzzy entropies, respectively, and search out the optimal parameters set through GA. Such methods are only compared with Otsu thresholding and failed to give adequate quantitative analysis. Sarkar et al., [27] based on image histogram and fuzzy entropy theory, used Different Evolution method to get the optimal threshold to obtain multilevel thresholding segmentation. Compared with PSO and GA, such method delivers better performance in speed and precision and at the same time the author offers a large number of numerical experimental comparisons and data contrast. Aja-Fernández et al. [1] have come up with the solution scheme for multiregion thresholding segmentation by using fuzzy entropy and aggregation methods. The scheme also provides solutions for histogram and fuzzy entropy. More importantly, the scheme noted the drawbacks of pixel spatial relationship which the thresholding segmentation failed to take into account. Based on this, by local information aggregation, it improves the segmentation results and removes the isolated pixels. In local information aggregation methods, the author presents median aggregation, average aggregation, and iterative average aggregation. The experimental results testified to the effectiveness of these aggregation methods. Taken together, the fuzzy and nonfuzzy multilevel thresholding methods, it can be concluded that the proposed method based on fuzzy entropy and intelligent optimization, in combination with local spatial information aggregation, is a feasible solution scheme for multilevel thresholding segmentation.Artificial bee colony method is a simple yet highly efficient optimized pattern, and its search iteration is much easier to be implemented compared with other heuristic search methods [32]. More importantly, artificial bee colony employs few predefined control parameters, which reduces the interference of artificiality set parameters. But there still exists some drawbacks in its solution search equation, which is fine at exploration but inferior at exploitation process [26]. In quick artificial bee colony methods [33], an advanced solution equation was proposed. In the process of searching out the optimal equation, the onlooker bee searches only around the best solution of previous iterations to improve exploitation. Therefore, the core is to determine the search scope. The scope, if larger than necessary, will greatly heighten the complexity, otherwise it will be reduced to local optimization. However, different problems call for different distance strategies and patterns; it is also true with MT. This paper will be focused on the study of the multilevel thresholding methods which are based on Kapur’s entropy, modified quick artificial bee colony algorithm for image segmentation, and local information aggregation.


## 3. Formulation of Multilevel Fuzzy Kapur’s Entropy

MT needs to set a set of thresholdsti and the image can be segmented into different regions based on different gray levels. Let image I is be segmented; in order to achieve the final segmentation, the proposed method assumes that every pixel in the image I will have a degree of membership in each of the N regions. That membership will be modeled using fuzzy membership functions. In what follows, we will give the optimal threshold obtained through fuzzy entropy and artificial bee colony and then assign each pixel with a fuzzy degree in different regions.


### 3.1. Concept of Kapur’s Entropy for Hard Thresholding

Based on the above analyses, the intelligent optimization which takes Kapur’s entropy as the objective function gets the better segmentation results. Kapur’s method can be easily extended from bilevel thresholding to multilevel thresholding. Entropy-based thresholding takes the gray level histogram as the basis. When the entropy reaches the maximization, the optimal threshold will be distributed normally.Entropy of a discrete information source can be obtained by the probability distributionp=pi, where pi is the probability of the system in possible state i [7]. The probability for each gray level i is its relative occurrence frequency. Let the number of gray levels for image I be h(i); the probability distribution can be described as follows:(1)pi=hi∑i=0L-1hii=0,1,…,L-1.Kapur’s entropy is used to measure the compactness and separability of classes. For MT, Kapur’s entropy can be described in(2)H0=-∑i=0t1-1piω0ln⁡piω0,ω0=∑i=0t1-1pi,H1=-∑i=t1t2-1piω1ln⁡piω1,ω1=∑i=t1t2-1pi,Hj=-∑i=tjtj+1-1piωjln⁡piωj,ωj=∑i=tjtj+1-1pi,Hm=-∑i=tmL-1piωmln⁡piωm,ωm=∑i=tmL-1pi.Thus, the function of (T) can be obtained by (3), used as a parameter of MQABC’s fitness function in Section 4.(3)fT=∑i=0m-HiT=t0t1⋯tm,where T represents a vector quantity of thresholds and m represents the stop value of the number of thresholds.


### 3.2. Concept of Fuzzy Kapur’s Entropy for Soft Thresholding

Let the setR be a collection of elements that can either belong to or not belong to region r; according to the fuzzy theory, it can be defined as follows:(4)R=i,μri∣i∈I,0≤μri≤1,r∈1,N,where ur(i) is called membership function, which is used to measure the closeness between pixel i and region r. The higher the value is, the bigger the probability that the pixel will be distributed into region r is, and N represents the number of regions.Option of membership function is the key factor for fuzzy thresholding segmentation. A large number of literature adopts different membership functions to test its effectiveness; trapezoidal membership function is the most commonly used and also the most efficient method [31]. Just as in literature [27], in the case that there are four fuzzy parameters, the curve of the fuzzy degrees can be shown in Figure 1.Figure 1
Fuzzy trapezoidal membership function.The fuzzy trapezoidal membership function with four parameters can be easily extended to the scenario ofn(n=2∗m+1), which can be formulated as in(5)μ0k=1,k≤a1,k-a2a1-a2,a1<k≤a2,0,k>a2,μ1k=0,k≤a1,k-a1a2-a1,a1<k≤a2,1a2<k≤a3,k-a4a3-a4,a3<k≤a4,0,k>a4⋮μmk=0,k≤an-1,k-an-1an-an-1,an-1<k≤an,1,k>an,where a1,a2,…,an represent the fuzzy parameters of the trapezoidal membership function and k is the gray value of a single pixel. Based on the trapezoidal membership function, the fuzzification of Kapur’s entropy in Section 3.1 can be described as in(6)H0=-∑i=0L-1μ0i∗piω0ln⁡μ0i∗piω0,ω0=∑i=0L-1μ0i∗pi,H1=-∑i=0L-1μ1i∗piω1ln⁡μ1i∗piω1,ω1=∑i=0L-1μ1i∗pi,Hj=-∑i=0L-1μji∗piωjln⁡μji∗piωj,ωj=∑i=0L-1μji∗pi,Hm=-∑i=0L-1μmi∗piωmln⁡μmi∗piωm,ωm=∑i=0L-1μmi∗pi.In order to get the specificN parameters in (5), this paper will adopt MQABC to minimize the total Kapur’s entropy, as shown in (7). By comparing (2) and (6), we can clearly see that the fuzzy entropy involves more complicated computational procedures. In such a case, it is of necessity to adopt intelligent optimization.(7)ϕa1,a2,…,an=fT=∑i-0m-Hi,T=t0t1⋯tm.AfterN parameters are obtained through the solution, the corresponding thresholds can be represented as in(8)t0=a1+a22,t1=a3+a42⋮tm=an-1+an2.


## 3.1. Concept of Kapur’s Entropy for Hard Thresholding

Based on the above analyses, the intelligent optimization which takes Kapur’s entropy as the objective function gets the better segmentation results. Kapur’s method can be easily extended from bilevel thresholding to multilevel thresholding. Entropy-based thresholding takes the gray level histogram as the basis. When the entropy reaches the maximization, the optimal threshold will be distributed normally.Entropy of a discrete information source can be obtained by the probability distributionp=pi, where pi is the probability of the system in possible state i [7]. The probability for each gray level i is its relative occurrence frequency. Let the number of gray levels for image I be h(i); the probability distribution can be described as follows:(1)pi=hi∑i=0L-1hii=0,1,…,L-1.Kapur’s entropy is used to measure the compactness and separability of classes. For MT, Kapur’s entropy can be described in(2)H0=-∑i=0t1-1piω0ln⁡piω0,ω0=∑i=0t1-1pi,H1=-∑i=t1t2-1piω1ln⁡piω1,ω1=∑i=t1t2-1pi,Hj=-∑i=tjtj+1-1piωjln⁡piωj,ωj=∑i=tjtj+1-1pi,Hm=-∑i=tmL-1piωmln⁡piωm,ωm=∑i=tmL-1pi.Thus, the function of (T) can be obtained by (3), used as a parameter of MQABC’s fitness function in Section 4.(3)fT=∑i=0m-HiT=t0t1⋯tm,where T represents a vector quantity of thresholds and m represents the stop value of the number of thresholds.


## 3.2. Concept of Fuzzy Kapur’s Entropy for Soft Thresholding

Let the setR be a collection of elements that can either belong to or not belong to region r; according to the fuzzy theory, it can be defined as follows:(4)R=i,μri∣i∈I,0≤μri≤1,r∈1,N,where ur(i) is called membership function, which is used to measure the closeness between pixel i and region r. The higher the value is, the bigger the probability that the pixel will be distributed into region r is, and N represents the number of regions.Option of membership function is the key factor for fuzzy thresholding segmentation. A large number of literature adopts different membership functions to test its effectiveness; trapezoidal membership function is the most commonly used and also the most efficient method [31]. Just as in literature [27], in the case that there are four fuzzy parameters, the curve of the fuzzy degrees can be shown in Figure 1.Figure 1
Fuzzy trapezoidal membership function.The fuzzy trapezoidal membership function with four parameters can be easily extended to the scenario ofn(n=2∗m+1), which can be formulated as in(5)μ0k=1,k≤a1,k-a2a1-a2,a1<k≤a2,0,k>a2,μ1k=0,k≤a1,k-a1a2-a1,a1<k≤a2,1a2<k≤a3,k-a4a3-a4,a3<k≤a4,0,k>a4⋮μmk=0,k≤an-1,k-an-1an-an-1,an-1<k≤an,1,k>an,where a1,a2,…,an represent the fuzzy parameters of the trapezoidal membership function and k is the gray value of a single pixel. Based on the trapezoidal membership function, the fuzzification of Kapur’s entropy in Section 3.1 can be described as in(6)H0=-∑i=0L-1μ0i∗piω0ln⁡μ0i∗piω0,ω0=∑i=0L-1μ0i∗pi,H1=-∑i=0L-1μ1i∗piω1ln⁡μ1i∗piω1,ω1=∑i=0L-1μ1i∗pi,Hj=-∑i=0L-1μji∗piωjln⁡μji∗piωj,ωj=∑i=0L-1μji∗pi,Hm=-∑i=0L-1μmi∗piωmln⁡μmi∗piωm,ωm=∑i=0L-1μmi∗pi.In order to get the specificN parameters in (5), this paper will adopt MQABC to minimize the total Kapur’s entropy, as shown in (7). By comparing (2) and (6), we can clearly see that the fuzzy entropy involves more complicated computational procedures. In such a case, it is of necessity to adopt intelligent optimization.(7)ϕa1,a2,…,an=fT=∑i-0m-Hi,T=t0t1⋯tm.AfterN parameters are obtained through the solution, the corresponding thresholds can be represented as in(8)t0=a1+a22,t1=a3+a42⋮tm=an-1+an2.


## 4. Brief Explanation of Quick Artificial Bee Colony Algorithm

ABC algorithms have been widely used as they show high efficiency and better convergence performance and employ fewer control parameters, in particular when the tuning of control parameters is more difficult than the problem itself. In fuzzy MT, the fuzzy entropy involves much more procedures of computations; the algorithm which employs a small number of control parameters is advised to be applied. A brief description of the ABC and the QABC algorithms will be given in the following subsections.


### 4.1. Standard Artificial Bee Colony Algorithm

In standard ABC, according to the migration behavior, artificial bees can be classified into three types, namely, employed bees, onlooker bees, and scout bees. Employed bees are related to specific food sources; onlooker bees decide to choose a certain food source by observing the dance of employed bees, while scout bees will search food randomly. The specific bee colony behavior and related simulation can be found in literature [32].In the initialization phase, all the food sources are initialized by (9). The number of food sources is set by predefined parameters.(9)tj,i=li+random0,1∗ui-li,where tj,i is the ith dimensional data of the jth food source. li and ui stand for the lower limit and upper limit of the parameter tj. tj is the jth food source.In the employed bee phase, a search is conducted in her memory at a specific speedφj,i. The speed (as shown in (10)) determines the change rate of the tj,i food source, which, to a certain degree, affects the convergence speed. If the solution produced by (10) is better than the bee’s solution, then the memory is updated by a greedy selection approach.(10)vj,i=tj,i+φj,i∗tj,i-tr,i,where tr represents a randomly selected food source, i is a randomly selected parameter index, and φj,i is a random number within the range [-1,1].To compare the advantages and disadvantages, the fitness of the solution is produced by (11). The higher fitness value represents a better objective function value, thus maximizing the fitness function which can reach the optimal thresholds.(11)fittj=11+ftj,ftj≥0,1+absftj,ftj<0,where f(tj) can be calculated by (4).An employed bee shares information about food source fitness with an onlooker bee. The onlooker bee by probability selects one source to investigate. The probability function can be represented by(12)probj=fittj∑j=1SNfittj,where SN represents the number of food sources.


### 4.2. Modified Quick Artificial Bee Colony Algorithm (MQABC) for Image Thresholding

In standard ABC, the employed bee and the onlooker bee use (10) to search for food sources. In other words, the employed bee and the onlooker bee adopt the same strategies in the same area to search for a new and better food source. But in real honey bee colonies, the employed bee and the onlooker bee adopt different methods to search for new food sources [33]. Therefore, for onlooker bees, when they search for the best food, it is quite reasonable that the pattern differs from the search method. A brand new modified onlooker bee behavior is adopted in the literature with the optimal fitness food source as the center.The formulated search for new food source can be represented by(13)vNj,ibest=tNj,ibest+φt,i∗tNt,ibest-tr,i,where tNj,ibest is defined as the optimal solution of all the neighborhood food sources around the present food source and tj,Nj represents all the neighborhood food sources including tj; the rest of the parameters are the same as the ones in (10). It can be clearly observed that the key is how the neighborhood food source is defined.From (13), the core of this pattern lies in how the neighboring food source is defined. Karaboga and Gorkemli [33] only give a simple definition of neighboring food sources. With regard to different ways of defining problem solution, it needs to define different measurement for similarity. For MT, every solution corresponds to a digital sequence. In this paper, the neighboring food source distance is defined as follows.(14)dj,i=∑x=1SNtx,iSNi=1,2,…,SL,where dj,i is the search scope of the food source tj around the neighborhood. SN stands for the number of the food sources and SL is the dimensions of a certain food source. If the Euclidean distance of a solution to tj is shorter than dj,i, then it is regarded as the neighborhood of the present solution tj, which is different from the standard ABC algorithm. When an onlooker bee reaches the food source tj, firstly it will investigate all the neighborhood food sources, choosing the best food source tNjbest, and improving her search by (13). In Nj, the best food source is defined by(15)fittNjbest=max⁡fittNj1,fittNj2,…,fittNjs.The improved ABC algorithm can be represented as follows.Algorithm 1 (main steps of the MQABC algorithm for image thresholding).
Step  1. Initialization of the population size SN, provide the number of thresholds M and maximum cycle number CN, initialization of the population of source foods:(16)tj,i=li+random0,1∗ui-liThen, evaluate the population via the specified optimization function;
Step  2. While (a termination criterion is not satisfied or Cycle Number < CN) do
Step  3. For j=1 to SN do
Produce new solutions (food source positions)vj in the neighborhood of tj for the employed bees using the formula: (17)vj,i=tj,i+φj,i∗tj,i-tr,iStep 4. Apply the greedy selection process between the corresponding tj and vj;
Step  5. Calculate the probability values probj for the solutions tj by means of their fitness values using the formula:(18)probj=fittj∑i=1SNfittjStep 6. Calculate the Euclidean distance dj, chose the best food source tNjbest in Nj;(19)dj,i=∑x=1SNtx,iSNi=1,2,…,SL,fittNjbest=max⁡fittNj1,fittNj2,…,fittNjsStep7. Produce the new solutions (new positions) vNjbest for the onlookers from the solutions tNjbest using the formula: (20)vNj,ibest=tNj,ibest+φt,i∗tNt,ibest-tr,iAnd then, selected depending on probi, and evaluate them;
Step  8. Apply the greedy selection process for the onlookers between tNjbest and vNjbest;
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


## 4.1. Standard Artificial Bee Colony Algorithm

In standard ABC, according to the migration behavior, artificial bees can be classified into three types, namely, employed bees, onlooker bees, and scout bees. Employed bees are related to specific food sources; onlooker bees decide to choose a certain food source by observing the dance of employed bees, while scout bees will search food randomly. The specific bee colony behavior and related simulation can be found in literature [32].In the initialization phase, all the food sources are initialized by (9). The number of food sources is set by predefined parameters.(9)tj,i=li+random0,1∗ui-li,where tj,i is the ith dimensional data of the jth food source. li and ui stand for the lower limit and upper limit of the parameter tj. tj is the jth food source.In the employed bee phase, a search is conducted in her memory at a specific speedφj,i. The speed (as shown in (10)) determines the change rate of the tj,i food source, which, to a certain degree, affects the convergence speed. If the solution produced by (10) is better than the bee’s solution, then the memory is updated by a greedy selection approach.(10)vj,i=tj,i+φj,i∗tj,i-tr,i,where tr represents a randomly selected food source, i is a randomly selected parameter index, and φj,i is a random number within the range [-1,1].To compare the advantages and disadvantages, the fitness of the solution is produced by (11). The higher fitness value represents a better objective function value, thus maximizing the fitness function which can reach the optimal thresholds.(11)fittj=11+ftj,ftj≥0,1+absftj,ftj<0,where f(tj) can be calculated by (4).An employed bee shares information about food source fitness with an onlooker bee. The onlooker bee by probability selects one source to investigate. The probability function can be represented by(12)probj=fittj∑j=1SNfittj,where SN represents the number of food sources.


## 4.2. Modified Quick Artificial Bee Colony Algorithm (MQABC) for Image Thresholding

In standard ABC, the employed bee and the onlooker bee use (10) to search for food sources. In other words, the employed bee and the onlooker bee adopt the same strategies in the same area to search for a new and better food source. But in real honey bee colonies, the employed bee and the onlooker bee adopt different methods to search for new food sources [33]. Therefore, for onlooker bees, when they search for the best food, it is quite reasonable that the pattern differs from the search method. A brand new modified onlooker bee behavior is adopted in the literature with the optimal fitness food source as the center.The formulated search for new food source can be represented by(13)vNj,ibest=tNj,ibest+φt,i∗tNt,ibest-tr,i,where tNj,ibest is defined as the optimal solution of all the neighborhood food sources around the present food source and tj,Nj represents all the neighborhood food sources including tj; the rest of the parameters are the same as the ones in (10). It can be clearly observed that the key is how the neighborhood food source is defined.From (13), the core of this pattern lies in how the neighboring food source is defined. Karaboga and Gorkemli [33] only give a simple definition of neighboring food sources. With regard to different ways of defining problem solution, it needs to define different measurement for similarity. For MT, every solution corresponds to a digital sequence. In this paper, the neighboring food source distance is defined as follows.(14)dj,i=∑x=1SNtx,iSNi=1,2,…,SL,where dj,i is the search scope of the food source tj around the neighborhood. SN stands for the number of the food sources and SL is the dimensions of a certain food source. If the Euclidean distance of a solution to tj is shorter than dj,i, then it is regarded as the neighborhood of the present solution tj, which is different from the standard ABC algorithm. When an onlooker bee reaches the food source tj, firstly it will investigate all the neighborhood food sources, choosing the best food source tNjbest, and improving her search by (13). In Nj, the best food source is defined by(15)fittNjbest=max⁡fittNj1,fittNj2,…,fittNjs.The improved ABC algorithm can be represented as follows.Algorithm 1 (main steps of the MQABC algorithm for image thresholding).
Step  1. Initialization of the population size SN, provide the number of thresholds M and maximum cycle number CN, initialization of the population of source foods:(16)tj,i=li+random0,1∗ui-liThen, evaluate the population via the specified optimization function;
Step  2. While (a termination criterion is not satisfied or Cycle Number < CN) do
Step  3. For j=1 to SN do
Produce new solutions (food source positions)vj in the neighborhood of tj for the employed bees using the formula: (17)vj,i=tj,i+φj,i∗tj,i-tr,iStep 4. Apply the greedy selection process between the corresponding tj and vj;
Step  5. Calculate the probability values probj for the solutions tj by means of their fitness values using the formula:(18)probj=fittj∑i=1SNfittjStep 6. Calculate the Euclidean distance dj, chose the best food source tNjbest in Nj;(19)dj,i=∑x=1SNtx,iSNi=1,2,…,SL,fittNjbest=max⁡fittNj1,fittNj2,…,fittNjsStep7. Produce the new solutions (new positions) vNjbest for the onlookers from the solutions tNjbest using the formula: (20)vNj,ibest=tNj,ibest+φt,i∗tNt,ibest-tr,iAnd then, selected depending on probi, and evaluate them;
Step  8. Apply the greedy selection process for the onlookers between tNjbest and vNjbest;
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. Membership Initialization and Local Information Aggregation

After the optimized processing in the above Section4, a set of thresholds is obtained through (8), and the subsequent processing will take the thresholds as the basis. Based on fuzzy theory, every pixel will be assigned with a fuzzy membership. The process is called membership initialization. On the basis of the initialization value, this paper proposes to conduct the local information aggregation based on the pixel spatial relationship so as to achieve the final segmentation results.


### 5.1. The Membership Initialization Using Pseudo Trapezoid-Shaped (PTS) Membership Function

Literature [1] summarizes the initialization methods of image fuzzy sets into two kinds: histogram based methods and centroid-based methods. Histogram based methods are the most important and the most commonly used for thresholding segmentation, but such methods need the prediction of probability distribution of histograms (like the most commonly used Gaussian distribution). Unfortunately, the probability distribution of the histograms for different kinds of images follows different probability patterns, like MR data is known to follow a Rician distribution [34], ultrasound data have been described more accurately by a mixture of Gamma distributions [1]. As a result, applicability of histogram based methods is relatively poor; in contradistinction, centroid-based methods render the shape and probability of histogram unnecessary; what only needs to be done is to select a proper centroid. As previously described, the thresholds which are obtained through (8) are set as the centroids, based on that, the fuzzy degree of each pixel is distributed through a proper function. Figure 1 provides fuzzy membership function adopted by the fuzzy entropy. In the process of initialization of image fuzzy sets, Pseudo Trapezoid-Shaped membership function (PTS) which is proposed in literature [1] will be adopted, as shown in Figure 2. Comparing Figures 2 and 1, PTS has less fuzzy degree values of 1 which is more conducive to the use of fuzzy spatial information aggregation in the following step.Figure 2
Fuzzy Pseudo Trapezoid-Shaped membership function.Similar to the formulation of trapezoidal membership function, in the case that there areN centroids, PTS membership function can be formulated as follows:(21)μ0′k=1,k≤a1,k-a2a1-a2,a1<k≤a2,0,k>a2,μ1′k=0,k≤a1,k-a1a2-a1,a1<k≤a2,k-a3a2-a3,a2<k≤a3,0,k>a3⋮μm′k=0k≤an-1k-an-1an-an-1an-1<k≤an1k>an,where μ0′k,μ1′k,…,μm′(k) are initialization functions of pixel fuzzy membership; a1,a2,…,an are the fuzzy parameters of the Pseudo trapezoidal membership; k is the gray scale value for each pixel in the test image.Figure2 presents the distribution methods of fuzzy degree of each pixel when there are three centroids being set. The membership of any random pixel in image I in r class is represented by μr(I(i)); after the initialization of the image’s fuzzy membership by way of PTS membership function, every pixel corresponds to a membership vector μr(I(i)), as described in(22)μIi=μ0Iiμ1Ii⋯μmIi.


### 5.2. Membership Aggregation Based on Local Spatial Information

After the initialization of fuzzy membership function, one pixel is likely to be distributed into two regions at most (in (22), there are two nonzero numbers at most). Simply by adopting the biggest value, it will be enough to distribute the pixel into the corresponding region, but such methods only consider the single pixel’s fuzzy degree, to the neglect of the pixel spatial relationship. Therefore, analyzing the local information aggregation of adjacent pixels will improve segmentation precision so that the isolated pixels can be removed.Image segmentation is in essence the aggregation of adjacent pixels of the same features. The biggest drawback of thresholding segmentation lies in the fact that it only takes the gray scale features of the test image to the neglect of spatial relationship. In the process of initialization of fuzzy membership function, each pixel will be assigned with one or two fuzzy values. These indeterminate values provide optimal prerequisites for local aggregation. By means of average and median aggregation of the fuzzy degrees of the pixels in neighborhood, the quality of image segmentation can be improved effectively.When there existsN surrounding regions for each pixel, the value of the fuzzy degree of each pixel can maximize the fuzzy relevance of pixels in surrounding regions by average and median filtering. As depicted in literature [1], there exist three methods for local fuzzy aggregation.(a) Median Aggregation. The fuzzy membership degree of each pixel will get through median filter in N surrounding regions, which can be formulated as follows:(23)μaggrIi=medianIi∈NIiμ0Iiμ1Ii⋯μmIi,where μaggr() represents fuzzy membership degree after aggregation, median{} represents the median operations of vector quantity, and N(I(i)) is the number of N neighboring pixels for number i pixel.(b) Average Aggregation. In the scope of the neighborhood for each pixel (like 3∗3), search for the solution which includes the average values of all pixels (including itself) and take them as the fuzzy membership of the present pixel, as described in(24)μaggrIi=1NIi∗∑Ii∈NIiμIi.(c) Iterative Averaging Aggregation. Compared to median aggregation, average aggregation can make better use of neighboring pixel information. Therefore the latter performs better in removing isolated pixels. But the aggregation capability with one-time iteration is limited; therefore, iterative averaging aggregation can strengthen aggregation effects. At the same time we should bear in mind that too large the number of iterations will make the segmentation boundary blurry; consequently, it is necessary to set different weights for adjacent pixels with different spatial distances as depicted in (25). Accordingly, the iterative formulation is shown in (26), where μaggr(I(i)t) represents the computational results of (24) when iterations occur once.(25)w=13∗00.500.510.500.50,(26)μaggrIit+1=w∗μaggrIit.After the fuzzy optimization and fuzzy aggregation, each pixel in the test image will get a vector by (22). In the set of vector quantities, selecting the biggest fuzzy value will determine which region the pixel belongs to. The whole procedures of computations are represented by Algorithm 2.Algorithm 2 (fuzzy multilevel image thresholding).
Step  1. Initialization of the number of region N, and convert the input image to gray scale and calculate the histogram of grayscale image.
Step  2. Set the number of fuzzy parameters 2∗N which is used in fuzzy trapezoidal membership function (Figure 1). Provide the number of thresholds m and maximum cycle number CN.
Step  3. Use QABC (as described in Algorithm 1) to compute the fuzzy parameters. The objective function follows (7) which merges the fuzzy theory and Kapur’s entropy, as described in Section 3.2.
Step  4. The soft threshold calculated by fuzzy parameters obtained in Step 3.(27)t0=a1+a22,t1=a3+a42⋮tm=an-1+an2.Step5. Regard the soft thresholds as the centroids, assigning membership to each pixel using fuzzy Pseudo Trapezoid-Shaped membership function (Figure 2).
Step  6. According to the local neighborhood information, perform aggregation operations on the membership degree of each pixel as described in Section 5.2
Step  7. According to the results of aggregation, determining the region for each pixel by way of maximum value.
Step  8. According to the regional labels, output the segmentation result.


## 5.1. The Membership Initialization Using Pseudo Trapezoid-Shaped (PTS) Membership Function

Literature [1] summarizes the initialization methods of image fuzzy sets into two kinds: histogram based methods and centroid-based methods. Histogram based methods are the most important and the most commonly used for thresholding segmentation, but such methods need the prediction of probability distribution of histograms (like the most commonly used Gaussian distribution). Unfortunately, the probability distribution of the histograms for different kinds of images follows different probability patterns, like MR data is known to follow a Rician distribution [34], ultrasound data have been described more accurately by a mixture of Gamma distributions [1]. As a result, applicability of histogram based methods is relatively poor; in contradistinction, centroid-based methods render the shape and probability of histogram unnecessary; what only needs to be done is to select a proper centroid. As previously described, the thresholds which are obtained through (8) are set as the centroids, based on that, the fuzzy degree of each pixel is distributed through a proper function. Figure 1 provides fuzzy membership function adopted by the fuzzy entropy. In the process of initialization of image fuzzy sets, Pseudo Trapezoid-Shaped membership function (PTS) which is proposed in literature [1] will be adopted, as shown in Figure 2. Comparing Figures 2 and 1, PTS has less fuzzy degree values of 1 which is more conducive to the use of fuzzy spatial information aggregation in the following step.Figure 2
Fuzzy Pseudo Trapezoid-Shaped membership function.Similar to the formulation of trapezoidal membership function, in the case that there areN centroids, PTS membership function can be formulated as follows:(21)μ0′k=1,k≤a1,k-a2a1-a2,a1<k≤a2,0,k>a2,μ1′k=0,k≤a1,k-a1a2-a1,a1<k≤a2,k-a3a2-a3,a2<k≤a3,0,k>a3⋮μm′k=0k≤an-1k-an-1an-an-1an-1<k≤an1k>an,where μ0′k,μ1′k,…,μm′(k) are initialization functions of pixel fuzzy membership; a1,a2,…,an are the fuzzy parameters of the Pseudo trapezoidal membership; k is the gray scale value for each pixel in the test image.Figure2 presents the distribution methods of fuzzy degree of each pixel when there are three centroids being set. The membership of any random pixel in image I in r class is represented by μr(I(i)); after the initialization of the image’s fuzzy membership by way of PTS membership function, every pixel corresponds to a membership vector μr(I(i)), as described in(22)μIi=μ0Iiμ1Ii⋯μmIi.


## 5.2. Membership Aggregation Based on Local Spatial Information

After the initialization of fuzzy membership function, one pixel is likely to be distributed into two regions at most (in (22), there are two nonzero numbers at most). Simply by adopting the biggest value, it will be enough to distribute the pixel into the corresponding region, but such methods only consider the single pixel’s fuzzy degree, to the neglect of the pixel spatial relationship. Therefore, analyzing the local information aggregation of adjacent pixels will improve segmentation precision so that the isolated pixels can be removed.Image segmentation is in essence the aggregation of adjacent pixels of the same features. The biggest drawback of thresholding segmentation lies in the fact that it only takes the gray scale features of the test image to the neglect of spatial relationship. In the process of initialization of fuzzy membership function, each pixel will be assigned with one or two fuzzy values. These indeterminate values provide optimal prerequisites for local aggregation. By means of average and median aggregation of the fuzzy degrees of the pixels in neighborhood, the quality of image segmentation can be improved effectively.When there existsN surrounding regions for each pixel, the value of the fuzzy degree of each pixel can maximize the fuzzy relevance of pixels in surrounding regions by average and median filtering. As depicted in literature [1], there exist three methods for local fuzzy aggregation.(a) Median Aggregation. The fuzzy membership degree of each pixel will get through median filter in N surrounding regions, which can be formulated as follows:(23)μaggrIi=medianIi∈NIiμ0Iiμ1Ii⋯μmIi,where μaggr() represents fuzzy membership degree after aggregation, median{} represents the median operations of vector quantity, and N(I(i)) is the number of N neighboring pixels for number i pixel.(b) Average Aggregation. In the scope of the neighborhood for each pixel (like 3∗3), search for the solution which includes the average values of all pixels (including itself) and take them as the fuzzy membership of the present pixel, as described in(24)μaggrIi=1NIi∗∑Ii∈NIiμIi.(c) Iterative Averaging Aggregation. Compared to median aggregation, average aggregation can make better use of neighboring pixel information. Therefore the latter performs better in removing isolated pixels. But the aggregation capability with one-time iteration is limited; therefore, iterative averaging aggregation can strengthen aggregation effects. At the same time we should bear in mind that too large the number of iterations will make the segmentation boundary blurry; consequently, it is necessary to set different weights for adjacent pixels with different spatial distances as depicted in (25). Accordingly, the iterative formulation is shown in (26), where μaggr(I(i)t) represents the computational results of (24) when iterations occur once.(25)w=13∗00.500.510.500.50,(26)μaggrIit+1=w∗μaggrIit.After the fuzzy optimization and fuzzy aggregation, each pixel in the test image will get a vector by (22). In the set of vector quantities, selecting the biggest fuzzy value will determine which region the pixel belongs to. The whole procedures of computations are represented by Algorithm 2.Algorithm 2 (fuzzy multilevel image thresholding).
Step  1. Initialization of the number of region N, and convert the input image to gray scale and calculate the histogram of grayscale image.
Step  2. Set the number of fuzzy parameters 2∗N which is used in fuzzy trapezoidal membership function (Figure 1). Provide the number of thresholds m and maximum cycle number CN.
Step  3. Use QABC (as described in Algorithm 1) to compute the fuzzy parameters. The objective function follows (7) which merges the fuzzy theory and Kapur’s entropy, as described in Section 3.2.
Step  4. The soft threshold calculated by fuzzy parameters obtained in Step 3.(27)t0=a1+a22,t1=a3+a42⋮tm=an-1+an2.Step5. Regard the soft thresholds as the centroids, assigning membership to each pixel using fuzzy Pseudo Trapezoid-Shaped membership function (Figure 2).
Step  6. According to the local neighborhood information, perform aggregation operations on the membership degree of each pixel as described in Section 5.2
Step  7. According to the results of aggregation, determining the region for each pixel by way of maximum value.
Step  8. According to the regional labels, output the segmentation result.


## 6. Experiments and Result Discussions

This paper, on the basis of numerical comparative experiments, through contrasts of segmented images, data, and graph analysis, testifies to the superiority of the proposed algorithm. From relevant literature, it can be seen that some methods are superior to others. The present paper will compare the proposed algorithm with the state-of-the-art methods; proven inferior ones will be sidelined. In the following sections, the proposed algorithm will be compared to Electromagnetism method, standard ABC, and the latest intelligent optimization that takes Kapur’s entropy as the objective function proposed in literature [7, 27, 35]. Electromagnetism optimization with Kapur’s entropy as the objective function is so far the newest intelligent optimization. PDE is one of the most effective fuzzy entropy thresholding segmentation methods. The comparison with literature [35] is mainly to testify that MQABC outperforms standard ABC.


### 6.1. The Discussion of Initialization Parameters and Convergence in MQABC

The proposed algorithm has been tested under a set of benchmark images. Some of these images are widely used in the multilevel image segmentation literature to test different methods (Lena, Baboon, and so forth) [7, 8, 35]. Others are chosen purposefully from the Berkeley Segmentation Data Set and Benchmarks 500 (BSDS500 for short, see [36]), as shown in Figure 3. The experiments were carried out on a LENOVO Laptop with Intel(R) Core i3 processor and 4 GB memory. And the algorithm was developed via the signal processing toolbox, image processing toolbox, and global optimization toolbox of MatlabR2011b. the parameters used for ABC algorithms [35] were presented in Table 1. In order to test the specific effects of these parameters in MQABC, many experiments have been conducted. Figure 4 presents the convergence curve of objective functions with iterations ranging from 50 to 500. When the iterations range from 50 to 150, the convergence is not so apparent. But when the iterations reach 200, the convergence is evident enough. However, when the number of iterations increases further, the convergence speed is hardly improved. Table 2 presents the assessment parameters for the segmentation quality of image Baboon with different swarm-sizes (detailed PSNR is provided at Section 6.4). From the table, it can be seen that when swarm-size reaches 30, the quality of image segmentation reaches the highest. At the same time, we also test the parameter changes of max trial limit. In the following tests, the biggest parameter value is 7, but most of them are below 5. Therefore, the parameters presented in Table 1 can be applied directly to MQABC.Table 1
Parameters used for MQABC.
Parameters Swam size Number of iterations Lower bound Upper bound Max trial limit Value 30 200 1 256 10Table 2
Comparison of image segmentation quality with different swarm size.
Swam size 10 20 30 40 50 PSNR 20.3650 20.4675 20.7193 20.4243 20.4071Figure 3
The original images and their histograms: (a) Baboon, (b) Lena, (c) Aeroplane, (d) Swan, (e) Ladyhand, (f) Starfish, (g) Mountain, and (h) Building.
(a) (b) (c) (d) (e) (f) (g) (h)Figure 4
The comparison of convergence with different number of iterations.In order to highlight the convergence performance of MQABC visually, Figure5 shows the convergence curves of MQABC and ABC for test images Baboon, Lena, and starfish when the thresholds are 5, respectively. From the curve lines, MQABC not only reaches the optimal objective earlier, but also is speedier toward the optima. More importantly, compared with ABC algorithm, the objective value obtained by MQABC is equal or even bigger, which means MQABC can get better thresholds. Therefore, MQABC shows better convergence performance and processing results.Figure 5
The comparison of convergence (a)–(c) with 5 thresholds, respectively.
(a)
Baboon with 5 th (b)
Lena with 5 th (c)
Starfish with 5 th


### 6.2. The Fuzzy Image Segmentation Result Based on MQABC with Different Thresholds

In order to make comparisons with other thresholding segmentation algorithms, one of the most popular performance indicators PSNR (the peak-to-signal ratio) is employed to compare the segmentation results between the original image and the segmented image. It is defined as in(28)PSNRi,j=20log10⁡255RMSEi,j,where RMSE is the root mean-squared error.In order to compare the influence of different thresholds and different aggregation methods, we conducted numerical experimental tests. Visual effects and qualitative data indicate that with the increase of the number of thresholds, image segmentation results will get finer. Note that the number of thresholds cannot increase without limit. The number of information units in each image is finite; too much thresholds cannot improve segmentation results; on the contrary it will add to the complexity. Based on the data sets adopted by this paper, we test the segmentation results when the number of thresholds ranges from 2 to 5 and adopt three different membership aggregations to conduct neighboring aggregations. The experimental results indicate that in the case of the same number of thresholds, the median aggregation gets the best segmentation results. When the average aggregation is compared to iterative average aggregation, in most cases, iterative average aggregation performs better except for some extremely rare occasions. Table3 presents different PSNR value of Baboon image with different thresholds and aggregation. Figure 6 shows the segmentation results for the images in Table 3. From Figure 6, we can see that with the increase of the number of thresholds, the segmentation results get finer gradually, but the visual effects are not obvious enough for different aggregation methods. To compensate for the insufficiency of visual comparisons, Table 3 clearly indicates that the median aggregation is far superior to the other aggregation methods.Table 3
Comparison of segmentation quality with different thresholds and aggregation.
M 2 3 4 5 Median aggregation 18.8793 19.9228 20.4195 21.0405 Average aggregation 18.2370 18.7917 19.3352 19.7773 Iterative averaging 18.6391 19.5356 20.0112 20.5206Figure 6
The results of image segmentation with different thresholds and aggregation.Figures7 and 8 show the segmentation results of the original images in Figure 3 after median aggregation is done with the thresholds being 2, 3, 4, and 5, respectively. Visually, it is adequate enough to discriminate the object from its background when the thresholds are 2. With the increase of the number of thresholds, the segmentation results get finer. In actual applications, we can vary the number of thresholds to obtain the desired output in response to different functions and scenes. Surely, if the size of the image is much larger or needs dividing into more regions, to that end, it is quite necessary to increase the number of thresholds. What needs to be done in the proposed method is just to set the number of thresholds M to get the desired results. Compared with other segmentation methods, it is hard to tell visually the difference of segmentation results. As for more detailed and comprehensive assessment concerning the segmentation results and performance, we will provide qualitative analysis in the subsequent sections.Figure 7
The segmentation results of (a)–(d) in Figure1.Figure 8
The segmentation results of (e)–(h) in Figure1.


### 6.3. Comparison of Best Objective Function Values and Their Corresponding Thresholds

In this section, the results of best objective function values and their corresponding thresholds acquired by various images are discussed. Table4 depicts the number of thresholds, objective values, and the corresponding optimal thresholds obtained by EMO, ABC, and MQABC methods. From the tables, it can be observed that all the images when processed by different methods have obtained relatively high objective function values. Notably, with the increase of the number of thresholds, the three methods all get high objective function values. Generally, MQABC and EMO obtain similar and even higher objective function values, while ABC appears a little inferior, but the difference range is less than 0.1.Table 4
Comparison of best objective function values and their corresponding thresholds between EMO, ABC, and MQABC.
Image M Optimum thresholds Best objective function values EMO ABC MQABC EMO ABC MQABC Baboon 2 78 143 80 144 80 144 17.6799 17.6802 17.6802 3 46 100 153 46 101 153 46 100 153 22.1331 22.1331 22.1331 4 46 100 153235 45 100 153 236 45 100 152 235 26.5254 26.5249 26.5265 5 34 75 115 160 235 34 75 116 161 236 33 74 115 162 235 30.6432 30.4394 30.6509  Lena 2 97 164 97 164 97 164 17.8234 17.8234 17.8234 3 81 126 177 83 127 178 82126 176 22.1102 22.1091 22.1201 4 16 82 127 177 16 87 123 174 16 83 125 176 26.1107 26.1064 26. 1101 5 16 64 97 137 179 16 65 96 133 178 16 64 97 137 178 30.0052 29.9870 30.0029  Aeroplane 2 54 161 54 161 54 161 17.8618 17.8618 17.8618 3 54 157 229 54 134 186 54 132 184 22.3106 22.2964 22.3112 4 54 127 167 206 47 109 165 227 48 110 165 228 26.3540 26.6389 26.6427 5 45 77 109 155 202 41 96 109 168 229 45 109 149 189 229 30.8109 30.8083 30.9162  Swan 2 66 150 66 150 66 150 18.1274 18.1274 18.1274 3 64 132 200 64 133 198 64 131 199 22.9607 22.9621 22.9887 4 64 117 161 210 64 117 164 212 64 117 162 210 27.3410 27.3493 27.3509 5 63 108 140 179 218 63 97 132 170 212 63 98 134 174 212 31.2520 31.2512 31.2654  Ladyhand 2 86 171 86 171 86 171 18.4477 18.4477 18.4477 3 63 116 182 64 118 180 61 115 180 22.9799 22.9883 22.9890 4 52 95 137 180 53 96 136 180 51 96 138 182 27.1901 27.1956 27.1958 5 52 97 136 168 202 56 98 137 168 202 55 98 137 174 204 31.1297 31.1302 31.1321  Starfish 2 90 170 90 170 90 170 18.7518 18.7518 18.7518 3 73 131 185 73 127 182 72 127 185 23.3105 23.3192 23.3241 4 68 117 166 208 63 112 159 207 67 118 167 206 27.5621 27.5509 27.5675 5 57 96 135 173 211 55 94 132 170 210 53 90 130 170 210 31.5536 31.5338 31.5519  Mountain 2 117 196 117 196 117 196 17.6737 17.6737 17.6737 3 62 119 196 62 119 196 62 119 196 22.1255 22.1255 22.1255 4 20 62 119 196 20 62 120 198 20 62 119 194 26.4474 26.4338 26.4427 5 20 61 96 137 197 20 60 96 139 203 20 61 100 159 204 30.5513 30.5448 30.5562  Building 2 90 156 90 156 90 156 17.8071 17.8071 17.8071 3 84 139 191 84 139 190 83 141 194 22.3895 22.3893 22.3876 4 20 84 141 193 20 84 141 192 20 83 139 192 26.7108 26.7105 26.7106 5 20 81 131 170 213 20 83 136 177 218 20 81 134 173 214 30.7119 30.8631 30.8983Table4 provides only the comparisons of the best objective function values and their corresponding thresholds between EMO, ABC, and MQABC. After the introduction of fuzzy entropy, we also tested the optimized results in terms of thresholds and objective function values based on DE (differential evolution) and FMQABC.Tables3 and 4 indicate that MQABC and FMQABC are able to obtain relatively good objective function values. Although, in some cases, these objective function values appear a little smaller in comparison with EMQ and FDE, the difference is insignificantly trivial. This paper lays more emphasis on time efficiency of MQABC and FMQABC; detailed comparisons are presented in Section 6.5.


### 6.4. Comparison of Fuzzy Multilevel Image Segmentation Result and Quality Assessment by PSNR

Figures7 and 8 demonstrate the segmentation results of FMQABCA. In order to further compare the proposed method with MQABC and other fuzzy optimized methods, Figure 9 presents the segmentation results of FDE, FMQABC, and FMQABCA with different thresholds. For the sake of convenience, the results in Figure 7 will have to be grayscaled; the gray level in the same region will be replaced by the average value. Analyzed visually, all the three methods get relatively fine segmentation results, but it is hard to tell visually which one is superior or inferior.Figure 9
The comparison of grayscale segmentation results about (a)–(d) in Figure1.In order to further compare the advantages and disadvantages between EMO, ABC, MQABC, FDE, FMQABC, and FQMABCA, Table5 demonstrates PSNR values these methods can achieve on fuzzy entropy, which indicate that MQABC gets the high assessment value, testifying to the superiority of MQABC in segmentation quality results. Furthermore, when the number of thresholds increases, MQABC gets better PSNR values. The comparison of fuzzy entropy (FDE, FMQABC, and FQMBCA) with nonfuzzy entropy (EMO, ABC, and MQABC) shows that in most of the cases, PSNR value is higher when fuzzy entropy is applied; noticeably FMABCA is far superior to the rest of the methods. The FMQABCA method gets the highest increase of 42% for the Swan image when M=3; the lowest also increases by 0.8% for the Aeroplane image when M=4. On average, the FMQABCA method shows more than 12% increase on the PSNR value for the eight figures.Table 5
PSNR metrics of the test images segmented with different threshold.
Image M PSNR EMO ABC MQABC FDE FMQABC FMQABCA Baboon 2 15.9947 16.0070 16.0070 15.9533 16.2043 18.8793 3 18.5921 18.5921 18.5921 17.8855 19.1648 19.9228 4 18.5921 18.5435 18.6718 17.0614 19.8360 20.4195 5 20.5234 20.5058 20.7193 19.5717 21.0266 21.0405  Lena 2 14.5901 14.5901 14.5901 16.7293 16.6456 18.6328 3 17.2122 17.1178 17.2328 17.8298 17.8884 19.0366 4 18.5488 18.3798 18.6011 18.3482 19.3349 20.9990 5 20.3250 20.2440 20.3360 20.8519 20.9714 22.2737  Aeroplane 2 12.7187 12.7187 12.7187 14.5518 15.3454 16.9651 3 15.9198 16.1541 16.2681 18.4402 18.3086 20.0782 4 18.4136 18.6846 18.7095 21.0749 20.6326 21.2494 5 19.1549 18.6876 21.0436 21.2442 21.1950 21.5475  Swan 2 14.8785 14.8785 14.8785 16.2123 16.3309 17.3657 3 16.2369 16.2373 16.2407 16.2334 17.3286 23.0785 4 16.6445 16.6571 16.6953 16.7511 16.9194 23.2771 5 16.9943 16.9482 17.0144 23.3818 23.7829 24.6911  Ladyhand 2 15.4928 15.4928 15.4928 15.7940 15.8844 16.9998 3 17.5422 17.5446 17.5744 17.9525 17.9778 20.4394 4 18.7518 18.7950 18.9135 19.0744 20.5095 21.3241 5 20.6768 20.7171 20.7392 20.7710 21.7443 22.0850  Starfish 2 14.3952 14.3952 14.3952 17.0598 16.5631 17.3641 3 16.9727 17.1199 17.1430 17.3605 17.9680 19.4480 4 18.1811 18.4836 18.5054 18.5546 18.5492 19.7407 5 20.0649 20.2304 20.3400 21.2142 21.4146 21.8224  Mountain 2 10.8782 10.8782 10.8782 13.2357 15.0018 17.6530 3 17.4652 17.4652 17.4652 21.6381 21.6748 21.8547 4 18.0535 18.0726 18.2056 20.1892 20.9634 22.4085 5 20.6893 20.4889 21.7589 20.3026 20.4539 23.3328  Building 2 14.7242 14.7242 14.7242 12.2905 13.9163 15.3027 3 16.2205 16.2261 16.2340 18.1586 17.8589 18.6818 4 17.9914 18.2581 18.3348 17.4129 18.9629 20.3899 5 18.8962 19.1442 20.8872 18.4646 19.0172 20.9096


### 6.5. Comparison of the Characteristics of Running Time

On the prior condition that segmentation quality is guaranteed, we examine the running time to test the performance. This is the biggest highlight of this paper. As shown in Table1, the experiments are set to the same number of iterations for testing. But in order to obtain the operating time for convergence as described in Table 6, when the fitness function remains unchanged for 20 consecutive times, the experimental iteration is forced to be terminated. Moreover, in order to better ensure the effectiveness of MQABC, its consecutive times are set to 50 specifically. Table 6 presents data concerning running time to convergence. The running time that EMO takes is much longer than the others. First, when comparing ABC with MQABC, we find that MQABC method gets much less running time than the ABC. And the highest increase of 12% is for the Lena image with M=5; the average also increases by 7.7% with M=5 and 4.8% with m=4. Then, the comparison of fuzzy methods to nonfuzzy methods shows that the running time for fuzzy methods takes a little longer, but if we take the segmentation quality assessment (PSNR) into account as shown in Table 5 the segmentation result is much better. All taken together, fuzzy methods are much more valuable.Table 6
The running time to convergence for EMO, ABC, MQABC, FDE, FMQABC, and FMQABCA.
Image M Running time (second) EMO ABC MQABC FDE FMQABC FMQABCA Baboon 4 6.700095 2.176596 2.131156 3.665105 2.234492 2.983749 5 8.389706 2.269232 2.021364 5.288985 3.032174 3.766464  Lena 4 6.598210 2.202426 2.000834 3.847913 2.179925 3.307005 5 8.202104 2.275810 2.001621 5.351902 3.809324 4.252020  Aeroplane 4 7.877475 2.439630 2.318803 4.966922 2.562252 2.957771 5 9.311095 2.590907 2.448975 6.188636 3.045369 3.810145  Swan 4 7.778865 2.520282 2.305300 4.741355 2.194570 2.958491 5 9.368673 3.253198 3.045478 6.227627 3.575132 4.198971  Ladyhand 4 7.637196 2.582458 2.580046 4.742558 3.066803 3.515309 5 9.254623 3.219210 2.962865 6.399271 4.435080 5.078334  Starfish 4 7.722705 3.271955 3.013681 4.995923 2.806032 3.178964 5 9.123489 3.358983 3.075762 6.647341 4.417617 4.978053  Mountain 4 7.732155 2.408338 2.340408 4.723301 2.517399 3.096332 5 9.190040 3.063349 2.940541 6.345679 3.073803 4.989546  Building 4 7.612425 2.532897 2.452882 4.825658 2.951447 3.996561 5 9.296582 3.184240 2.981036 6.318346 3.586839 4.960509In the three fuzzy methods, when FDE is compared with FMQABC and FBCA, it shows the latter two enjoy higher time efficiency significantly. The FMQABC method gets the highest increase of 53% for the Swan image whenM=4; the lowest also increases by 28% for the Lena image when M=5. On average, the FMQABC method shows more than 44% of the time improvement for the eight figures when M=4 and more than 41% when M=5. From the perspective of FMQABCA, the maximum and minimum running time increase by 40% and 14% for the Aeroplane and Lena image, respectively, when M=4. And when M=5, the maximum and minimum running time increase by 38% and 21% for the Lena and Aeroplane image, respectively. On average, the FMQABCA method shows more than 28% of the time improvement for the eight figures when M=4 and more than 26% when M=5. Therefore, the FMQABC and FMQABCA methods ensure the optimal threshold so as to obtain the best time efficiency.From Tables4–6 and Figures 4–9, together with visual effect analysis, it can be seen that FMQABCA delivers better segmentation results and enjoys obvious advantages in running time. More importantly, MQABC is far superior to ABC over convergence performance. Therefore it can be safely concluded that FMQABCA is a desirable multilevel segmentation method with high speed and of high quality.


## 6.1. The Discussion of Initialization Parameters and Convergence in MQABC

The proposed algorithm has been tested under a set of benchmark images. Some of these images are widely used in the multilevel image segmentation literature to test different methods (Lena, Baboon, and so forth) [7, 8, 35]. Others are chosen purposefully from the Berkeley Segmentation Data Set and Benchmarks 500 (BSDS500 for short, see [36]), as shown in Figure 3. The experiments were carried out on a LENOVO Laptop with Intel(R) Core i3 processor and 4 GB memory. And the algorithm was developed via the signal processing toolbox, image processing toolbox, and global optimization toolbox of MatlabR2011b. the parameters used for ABC algorithms [35] were presented in Table 1. In order to test the specific effects of these parameters in MQABC, many experiments have been conducted. Figure 4 presents the convergence curve of objective functions with iterations ranging from 50 to 500. When the iterations range from 50 to 150, the convergence is not so apparent. But when the iterations reach 200, the convergence is evident enough. However, when the number of iterations increases further, the convergence speed is hardly improved. Table 2 presents the assessment parameters for the segmentation quality of image Baboon with different swarm-sizes (detailed PSNR is provided at Section 6.4). From the table, it can be seen that when swarm-size reaches 30, the quality of image segmentation reaches the highest. At the same time, we also test the parameter changes of max trial limit. In the following tests, the biggest parameter value is 7, but most of them are below 5. Therefore, the parameters presented in Table 1 can be applied directly to MQABC.Table 1
Parameters used for MQABC.
Parameters Swam size Number of iterations Lower bound Upper bound Max trial limit Value 30 200 1 256 10Table 2
Comparison of image segmentation quality with different swarm size.
Swam size 10 20 30 40 50 PSNR 20.3650 20.4675 20.7193 20.4243 20.4071Figure 3
The original images and their histograms: (a) Baboon, (b) Lena, (c) Aeroplane, (d) Swan, (e) Ladyhand, (f) Starfish, (g) Mountain, and (h) Building.
(a) (b) (c) (d) (e) (f) (g) (h)Figure 4
The comparison of convergence with different number of iterations.In order to highlight the convergence performance of MQABC visually, Figure5 shows the convergence curves of MQABC and ABC for test images Baboon, Lena, and starfish when the thresholds are 5, respectively. From the curve lines, MQABC not only reaches the optimal objective earlier, but also is speedier toward the optima. More importantly, compared with ABC algorithm, the objective value obtained by MQABC is equal or even bigger, which means MQABC can get better thresholds. Therefore, MQABC shows better convergence performance and processing results.Figure 5
The comparison of convergence (a)–(c) with 5 thresholds, respectively.
(a)
Baboon with 5 th (b)
Lena with 5 th (c)
Starfish with 5 th


## 6.2. The Fuzzy Image Segmentation Result Based on MQABC with Different Thresholds

In order to make comparisons with other thresholding segmentation algorithms, one of the most popular performance indicators PSNR (the peak-to-signal ratio) is employed to compare the segmentation results between the original image and the segmented image. It is defined as in(28)PSNRi,j=20log10⁡255RMSEi,j,where RMSE is the root mean-squared error.In order to compare the influence of different thresholds and different aggregation methods, we conducted numerical experimental tests. Visual effects and qualitative data indicate that with the increase of the number of thresholds, image segmentation results will get finer. Note that the number of thresholds cannot increase without limit. The number of information units in each image is finite; too much thresholds cannot improve segmentation results; on the contrary it will add to the complexity. Based on the data sets adopted by this paper, we test the segmentation results when the number of thresholds ranges from 2 to 5 and adopt three different membership aggregations to conduct neighboring aggregations. The experimental results indicate that in the case of the same number of thresholds, the median aggregation gets the best segmentation results. When the average aggregation is compared to iterative average aggregation, in most cases, iterative average aggregation performs better except for some extremely rare occasions. Table3 presents different PSNR value of Baboon image with different thresholds and aggregation. Figure 6 shows the segmentation results for the images in Table 3. From Figure 6, we can see that with the increase of the number of thresholds, the segmentation results get finer gradually, but the visual effects are not obvious enough for different aggregation methods. To compensate for the insufficiency of visual comparisons, Table 3 clearly indicates that the median aggregation is far superior to the other aggregation methods.Table 3
Comparison of segmentation quality with different thresholds and aggregation.
M 2 3 4 5 Median aggregation 18.8793 19.9228 20.4195 21.0405 Average aggregation 18.2370 18.7917 19.3352 19.7773 Iterative averaging 18.6391 19.5356 20.0112 20.5206Figure 6
The results of image segmentation with different thresholds and aggregation.Figures7 and 8 show the segmentation results of the original images in Figure 3 after median aggregation is done with the thresholds being 2, 3, 4, and 5, respectively. Visually, it is adequate enough to discriminate the object from its background when the thresholds are 2. With the increase of the number of thresholds, the segmentation results get finer. In actual applications, we can vary the number of thresholds to obtain the desired output in response to different functions and scenes. Surely, if the size of the image is much larger or needs dividing into more regions, to that end, it is quite necessary to increase the number of thresholds. What needs to be done in the proposed method is just to set the number of thresholds M to get the desired results. Compared with other segmentation methods, it is hard to tell visually the difference of segmentation results. As for more detailed and comprehensive assessment concerning the segmentation results and performance, we will provide qualitative analysis in the subsequent sections.Figure 7
The segmentation results of (a)–(d) in Figure1.Figure 8
The segmentation results of (e)–(h) in Figure1.


## 6.3. Comparison of Best Objective Function Values and Their Corresponding Thresholds

In this section, the results of best objective function values and their corresponding thresholds acquired by various images are discussed. Table4 depicts the number of thresholds, objective values, and the corresponding optimal thresholds obtained by EMO, ABC, and MQABC methods. From the tables, it can be observed that all the images when processed by different methods have obtained relatively high objective function values. Notably, with the increase of the number of thresholds, the three methods all get high objective function values. Generally, MQABC and EMO obtain similar and even higher objective function values, while ABC appears a little inferior, but the difference range is less than 0.1.Table 4
Comparison of best objective function values and their corresponding thresholds between EMO, ABC, and MQABC.
Image M Optimum thresholds Best objective function values EMO ABC MQABC EMO ABC MQABC Baboon 2 78 143 80 144 80 144 17.6799 17.6802 17.6802 3 46 100 153 46 101 153 46 100 153 22.1331 22.1331 22.1331 4 46 100 153235 45 100 153 236 45 100 152 235 26.5254 26.5249 26.5265 5 34 75 115 160 235 34 75 116 161 236 33 74 115 162 235 30.6432 30.4394 30.6509  Lena 2 97 164 97 164 97 164 17.8234 17.8234 17.8234 3 81 126 177 83 127 178 82126 176 22.1102 22.1091 22.1201 4 16 82 127 177 16 87 123 174 16 83 125 176 26.1107 26.1064 26. 1101 5 16 64 97 137 179 16 65 96 133 178 16 64 97 137 178 30.0052 29.9870 30.0029  Aeroplane 2 54 161 54 161 54 161 17.8618 17.8618 17.8618 3 54 157 229 54 134 186 54 132 184 22.3106 22.2964 22.3112 4 54 127 167 206 47 109 165 227 48 110 165 228 26.3540 26.6389 26.6427 5 45 77 109 155 202 41 96 109 168 229 45 109 149 189 229 30.8109 30.8083 30.9162  Swan 2 66 150 66 150 66 150 18.1274 18.1274 18.1274 3 64 132 200 64 133 198 64 131 199 22.9607 22.9621 22.9887 4 64 117 161 210 64 117 164 212 64 117 162 210 27.3410 27.3493 27.3509 5 63 108 140 179 218 63 97 132 170 212 63 98 134 174 212 31.2520 31.2512 31.2654  Ladyhand 2 86 171 86 171 86 171 18.4477 18.4477 18.4477 3 63 116 182 64 118 180 61 115 180 22.9799 22.9883 22.9890 4 52 95 137 180 53 96 136 180 51 96 138 182 27.1901 27.1956 27.1958 5 52 97 136 168 202 56 98 137 168 202 55 98 137 174 204 31.1297 31.1302 31.1321  Starfish 2 90 170 90 170 90 170 18.7518 18.7518 18.7518 3 73 131 185 73 127 182 72 127 185 23.3105 23.3192 23.3241 4 68 117 166 208 63 112 159 207 67 118 167 206 27.5621 27.5509 27.5675 5 57 96 135 173 211 55 94 132 170 210 53 90 130 170 210 31.5536 31.5338 31.5519  Mountain 2 117 196 117 196 117 196 17.6737 17.6737 17.6737 3 62 119 196 62 119 196 62 119 196 22.1255 22.1255 22.1255 4 20 62 119 196 20 62 120 198 20 62 119 194 26.4474 26.4338 26.4427 5 20 61 96 137 197 20 60 96 139 203 20 61 100 159 204 30.5513 30.5448 30.5562  Building 2 90 156 90 156 90 156 17.8071 17.8071 17.8071 3 84 139 191 84 139 190 83 141 194 22.3895 22.3893 22.3876 4 20 84 141 193 20 84 141 192 20 83 139 192 26.7108 26.7105 26.7106 5 20 81 131 170 213 20 83 136 177 218 20 81 134 173 214 30.7119 30.8631 30.8983Table4 provides only the comparisons of the best objective function values and their corresponding thresholds between EMO, ABC, and MQABC. After the introduction of fuzzy entropy, we also tested the optimized results in terms of thresholds and objective function values based on DE (differential evolution) and FMQABC.Tables3 and 4 indicate that MQABC and FMQABC are able to obtain relatively good objective function values. Although, in some cases, these objective function values appear a little smaller in comparison with EMQ and FDE, the difference is insignificantly trivial. This paper lays more emphasis on time efficiency of MQABC and FMQABC; detailed comparisons are presented in Section 6.5.


## 6.4. Comparison of Fuzzy Multilevel Image Segmentation Result and Quality Assessment by PSNR

Figures7 and 8 demonstrate the segmentation results of FMQABCA. In order to further compare the proposed method with MQABC and other fuzzy optimized methods, Figure 9 presents the segmentation results of FDE, FMQABC, and FMQABCA with different thresholds. For the sake of convenience, the results in Figure 7 will have to be grayscaled; the gray level in the same region will be replaced by the average value. Analyzed visually, all the three methods get relatively fine segmentation results, but it is hard to tell visually which one is superior or inferior.Figure 9
The comparison of grayscale segmentation results about (a)–(d) in Figure1.In order to further compare the advantages and disadvantages between EMO, ABC, MQABC, FDE, FMQABC, and FQMABCA, Table5 demonstrates PSNR values these methods can achieve on fuzzy entropy, which indicate that MQABC gets the high assessment value, testifying to the superiority of MQABC in segmentation quality results. Furthermore, when the number of thresholds increases, MQABC gets better PSNR values. The comparison of fuzzy entropy (FDE, FMQABC, and FQMBCA) with nonfuzzy entropy (EMO, ABC, and MQABC) shows that in most of the cases, PSNR value is higher when fuzzy entropy is applied; noticeably FMABCA is far superior to the rest of the methods. The FMQABCA method gets the highest increase of 42% for the Swan image when M=3; the lowest also increases by 0.8% for the Aeroplane image when M=4. On average, the FMQABCA method shows more than 12% increase on the PSNR value for the eight figures.Table 5
PSNR metrics of the test images segmented with different threshold.
Image M PSNR EMO ABC MQABC FDE FMQABC FMQABCA Baboon 2 15.9947 16.0070 16.0070 15.9533 16.2043 18.8793 3 18.5921 18.5921 18.5921 17.8855 19.1648 19.9228 4 18.5921 18.5435 18.6718 17.0614 19.8360 20.4195 5 20.5234 20.5058 20.7193 19.5717 21.0266 21.0405  Lena 2 14.5901 14.5901 14.5901 16.7293 16.6456 18.6328 3 17.2122 17.1178 17.2328 17.8298 17.8884 19.0366 4 18.5488 18.3798 18.6011 18.3482 19.3349 20.9990 5 20.3250 20.2440 20.3360 20.8519 20.9714 22.2737  Aeroplane 2 12.7187 12.7187 12.7187 14.5518 15.3454 16.9651 3 15.9198 16.1541 16.2681 18.4402 18.3086 20.0782 4 18.4136 18.6846 18.7095 21.0749 20.6326 21.2494 5 19.1549 18.6876 21.0436 21.2442 21.1950 21.5475  Swan 2 14.8785 14.8785 14.8785 16.2123 16.3309 17.3657 3 16.2369 16.2373 16.2407 16.2334 17.3286 23.0785 4 16.6445 16.6571 16.6953 16.7511 16.9194 23.2771 5 16.9943 16.9482 17.0144 23.3818 23.7829 24.6911  Ladyhand 2 15.4928 15.4928 15.4928 15.7940 15.8844 16.9998 3 17.5422 17.5446 17.5744 17.9525 17.9778 20.4394 4 18.7518 18.7950 18.9135 19.0744 20.5095 21.3241 5 20.6768 20.7171 20.7392 20.7710 21.7443 22.0850  Starfish 2 14.3952 14.3952 14.3952 17.0598 16.5631 17.3641 3 16.9727 17.1199 17.1430 17.3605 17.9680 19.4480 4 18.1811 18.4836 18.5054 18.5546 18.5492 19.7407 5 20.0649 20.2304 20.3400 21.2142 21.4146 21.8224  Mountain 2 10.8782 10.8782 10.8782 13.2357 15.0018 17.6530 3 17.4652 17.4652 17.4652 21.6381 21.6748 21.8547 4 18.0535 18.0726 18.2056 20.1892 20.9634 22.4085 5 20.6893 20.4889 21.7589 20.3026 20.4539 23.3328  Building 2 14.7242 14.7242 14.7242 12.2905 13.9163 15.3027 3 16.2205 16.2261 16.2340 18.1586 17.8589 18.6818 4 17.9914 18.2581 18.3348 17.4129 18.9629 20.3899 5 18.8962 19.1442 20.8872 18.4646 19.0172 20.9096


## 6.5. Comparison of the Characteristics of Running Time

On the prior condition that segmentation quality is guaranteed, we examine the running time to test the performance. This is the biggest highlight of this paper. As shown in Table1, the experiments are set to the same number of iterations for testing. But in order to obtain the operating time for convergence as described in Table 6, when the fitness function remains unchanged for 20 consecutive times, the experimental iteration is forced to be terminated. Moreover, in order to better ensure the effectiveness of MQABC, its consecutive times are set to 50 specifically. Table 6 presents data concerning running time to convergence. The running time that EMO takes is much longer than the others. First, when comparing ABC with MQABC, we find that MQABC method gets much less running time than the ABC. And the highest increase of 12% is for the Lena image with M=5; the average also increases by 7.7% with M=5 and 4.8% with m=4. Then, the comparison of fuzzy methods to nonfuzzy methods shows that the running time for fuzzy methods takes a little longer, but if we take the segmentation quality assessment (PSNR) into account as shown in Table 5 the segmentation result is much better. All taken together, fuzzy methods are much more valuable.Table 6
The running time to convergence for EMO, ABC, MQABC, FDE, FMQABC, and FMQABCA.
Image M Running time (second) EMO ABC MQABC FDE FMQABC FMQABCA Baboon 4 6.700095 2.176596 2.131156 3.665105 2.234492 2.983749 5 8.389706 2.269232 2.021364 5.288985 3.032174 3.766464  Lena 4 6.598210 2.202426 2.000834 3.847913 2.179925 3.307005 5 8.202104 2.275810 2.001621 5.351902 3.809324 4.252020  Aeroplane 4 7.877475 2.439630 2.318803 4.966922 2.562252 2.957771 5 9.311095 2.590907 2.448975 6.188636 3.045369 3.810145  Swan 4 7.778865 2.520282 2.305300 4.741355 2.194570 2.958491 5 9.368673 3.253198 3.045478 6.227627 3.575132 4.198971  Ladyhand 4 7.637196 2.582458 2.580046 4.742558 3.066803 3.515309 5 9.254623 3.219210 2.962865 6.399271 4.435080 5.078334  Starfish 4 7.722705 3.271955 3.013681 4.995923 2.806032 3.178964 5 9.123489 3.358983 3.075762 6.647341 4.417617 4.978053  Mountain 4 7.732155 2.408338 2.340408 4.723301 2.517399 3.096332 5 9.190040 3.063349 2.940541 6.345679 3.073803 4.989546  Building 4 7.612425 2.532897 2.452882 4.825658 2.951447 3.996561 5 9.296582 3.184240 2.981036 6.318346 3.586839 4.960509In the three fuzzy methods, when FDE is compared with FMQABC and FBCA, it shows the latter two enjoy higher time efficiency significantly. The FMQABC method gets the highest increase of 53% for the Swan image whenM=4; the lowest also increases by 28% for the Lena image when M=5. On average, the FMQABC method shows more than 44% of the time improvement for the eight figures when M=4 and more than 41% when M=5. From the perspective of FMQABCA, the maximum and minimum running time increase by 40% and 14% for the Aeroplane and Lena image, respectively, when M=4. And when M=5, the maximum and minimum running time increase by 38% and 21% for the Lena and Aeroplane image, respectively. On average, the FMQABCA method shows more than 28% of the time improvement for the eight figures when M=4 and more than 26% when M=5. Therefore, the FMQABC and FMQABCA methods ensure the optimal threshold so as to obtain the best time efficiency.From Tables4–6 and Figures 4–9, together with visual effect analysis, it can be seen that FMQABCA delivers better segmentation results and enjoys obvious advantages in running time. More importantly, MQABC is far superior to ABC over convergence performance. Therefore it can be safely concluded that FMQABCA is a desirable multilevel segmentation method with high speed and of high quality.


## 7. Conclusion

In this paper, MQABC has been employed to optimize fuzzy Kapur’s entropy in order to obtain a set of thresholds. Based on the set values, a set of fuzzy degrees is assigned to each pixel through Pseudo Trapezoid-Shaped membership function, and then aggregation operation is conducted in the way that fuzzy degrees of neighborhood are aggregated to achieve the final segmentation. When compared with ABC, MQABC, by improving on the distance strategies in the onlooker bee phase, not only achieves the higher objective function and betters PSNR value, but also shows obvious improvement in convergence performance and running time. In the process of fuzzy aggregation, FMQABCA gets higher PSNR values obtained by median, average, and iterative methods than EMO, ABC, MQABC, FDE, and FMQABC and show more efficiency in running time than EMO and FDE. Although compared with ABC, MQABC, and FMQABC, FMQABCA takes a slightly longer time, but such a little price is completely acceptable if we take the high quality of image segmentation into account. In conclusion, FMQABC and FMQABCA are able to obtain better image segmentation quality and also show obvious superiority in terms of running time and convergence speed.



---

*Source: 5985616-2016-12-25.xml*