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

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 proposedmethod 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.


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][9][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][9][10][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.Section 2 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.

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][19][20][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, Ifunction, 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.

Formulation of Multilevel Fuzzy Kapur's Entropy
MT needs to set a set of thresholds   and the image can be segmented into different regions based on different gray levels.Let image  is be segmented; in order to achieve the final segmentation, the proposed method assumes that every pixel in the image  will have a degree of membership in each of the  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.

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 distribution  =   , where   is the probability of the system in possible state  [7].The probability for each gray level  is its relative occurrence frequency.Let the number of gray levels for image  be ℎ(); the probability distribution can be described as follows: Kapur's entropy is used to measure the compactness and separability of classes.For MT, Kapur's entropy can be described in Thus, the function of () can be obtained by ( 3), used as a parameter of MQABC's fitness function in Section 4.
where  represents a vector quantity of thresholds and  represents the stop value of the number of thresholds.

Concept of Fuzzy Kapur's Entropy for Soft Thresholding.
Let the set  be a collection of elements that can either belong to or not belong to region ; according to the fuzzy theory, it can be defined as follows: where   () is called membership function, which is used to measure the closeness between pixel  and region .The higher the value is, the bigger the probability that the pixel will be distributed into region  is, and  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.
The fuzzy trapezoidal membership function with four parameters can be easily extended to the scenario of  ( = 2 * ( + 1)), which can be formulated as in where  1 ,  2 , . . .,   represent the fuzzy parameters of the trapezoidal membership function and  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 In order to get the specific  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.
After  parameters are obtained through the solution, the corresponding thresholds can be represented as in (8)

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.

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.
where  , is the th dimensional data of the th food source.  and   stand for the lower limit and upper limit of the parameter   .  is the th food source.
In the employed bee phase, a search is conducted in her memory at a specific speed  , .The speed (as shown in (10)) determines the change rate of the  , 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.
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.fit where (  ) 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 where SN represents the number of food sources.

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 where  best  , is defined as the optimal solution of all the neighborhood food sources around the present food source and   ,   represents all the neighborhood food sources including   ; 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.
where  , is the search scope of the food source   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   is shorter than  , , then it is regarded as the neighborhood of the present solution   , which is different from the standard ABC algorithm.When an onlooker bee reaches the food source   , firstly it will investigate all the neighborhood food sources, choosing the best food source  best   , and improving her search by (13).In   , the best food source is defined by 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  and maximum cycle number CN, initialization of the population of source foods: Then, 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  = 1 to SN do Produce new solutions (food source positions) V  in the neighborhood of   for the employed bees using the formula: Step 4. Apply the greedy selection process between the corresponding   and V  ; Step 5. Calculate the probability values prob  for the solutions   by means of their fitness values using the formula: Step Step 7. Produce the new solutions (new positions) V best   for the onlookers from the solutions  best   using the formula: And then, selected depending on prob  , and evaluate them; Step 8. Apply the greedy selection process for the onlookers between  best   and V best   ; Step 9. Determine the abandoned solution (source food), if it exists, and replace it with a new randomly produced solution   for the scout; Step 10.Memorize the best food source position (solution) achieved so far; Step 11.End for Step 12. End while

Membership Initialization and Local Information Aggregation
After the optimized processing in the above Section 4, 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.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.

The Membership Initialization
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.Similar to the formulation of trapezoidal membership function, in the case that there are  centroids, PTS membership function can be formulated as follows: where   0 (),   1 (), . . .,    () are initialization functions of pixel fuzzy membership;  1 ,  2 , . . .,   are the fuzzy parameters of the Pseudo trapezoidal membership;  is the gray scale value for each pixel in the test image.
Figure 2 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  in  class is represented by   (()); after the initialization of the image's fuzzy membership by way of PTS membership function, every pixel corresponds to a membership vector   (()), as described in

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 exists  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  surrounding regions, which can be formulated as follows: where  aggr () represents fuzzy membership degree after aggregation, median{} represents the median operations of vector quantity, and (()) is the number of  neighboring pixels for number  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 (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  aggr (()  ) represents the computational results of ( 24) when iterations occur once.
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 , and convert the input image to gray scale and calculate the histogram of grayscale image.
Step 2. Set the number of fuzzy parameters 2 *  which is used in fuzzy trapezoidal membership function (Figure 1).Provide the number of thresholds  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.
Step 5. 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.

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-theart 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.

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.
In order to highlight the convergence performance of MQABC visually, Figure 5 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.

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-tosignal ratio) is employed to compare the segmentation results between the original image and the segmented image.It is defined as in PSNR (, ) = 20 log 10 ( 255 RMSE (, ) ) , 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.Table 3 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.
Figures 7 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  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.

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.Table 4 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 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.
Tables 3 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.

Comparison of Fuzzy Multilevel Image Segmentation
Result and Quality Assessment by PSNR.Figures 7 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.
In order to further compare the advantages and disadvantages between EMO, ABC, MQABC, FDE, FMQABC, and FQMABCA, Table 5 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  = 3; the lowest also increases by 0.8% for the Aeroplane image when  = 4. On average, the FMQABCA method shows more than 12% increase on the PSNR value for the eight figures.

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 Table 1, 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  = 5; the average also increases by 7.7% with  = 5 and 4.8% with  = 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.
In 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 when  = 4; the lowest also increases by 28% for the Lena image when  = 5.On average, the FMQABC method shows more than 44% of the time improvement for the eight figures when  = 4 and more than 41% when  = 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  = 4.And when  = 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  = 4 and more than 26% when  = 5.Therefore, the FMQABC and FMQABCA methods ensure the optimal threshold so as to obtain the best time efficiency.
From Tables 4-6 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.

Figure 4 :
Figure 4: The comparison of convergence with different number of iterations.

Figure 6 :
Figure 6: The results of image segmentation with different thresholds and aggregation.

Table 2 :
Comparison of image segmentation quality with different swarm size.

Table 3 :
Comparison of segmentation quality with different thresholds and aggregation.

Table 4 :
Comparison of best objective function values and their corresponding thresholds between EMO, ABC, and MQABC.

Table 5 :
PSNR metrics of the test images segmented with different threshold.