Multilevel thresholding by fuzzy type II sets using evolutionary algorithms

Journal Pre-proof Multilevel thresholding by fuzzy type II sets using evolutionary algorithms Diego Oliva, Sayan Nag, Mohamed Abd Elaziz, Uddalok Sarkar, Salvador Hinojosa PII:

S2210-6502(18)30335-3

DOI:

https://doi.org/10.1016/j.swevo.2019.100591

Reference:

SWEVO 100591

To appear in:

Swarm and Evolutionary Computation BASE DATA

Received Date: 25 April 2018 Revised Date:

3 May 2019

Accepted Date: 30 September 2019

Please cite this article as: D. Oliva, S. Nag, M.A. Elaziz, U. Sarkar, S. Hinojosa, Multilevel thresholding by fuzzy type II sets using evolutionary algorithms, Swarm and Evolutionary Computation BASE DATA (2019), doi: https://doi.org/10.1016/j.swevo.2019.100591. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.

Multilevel thresholding by Fuzzy Type II Sets using evolutionary algorithms 1

Diego Oliva a*, Sayan Nag b*, Mohamed Abd Elaziz c*, Uddalok Sarkar b*, Salvador Hinojosa d* b

Department of Electrical Engineering, UG-IV Jadavpur University, Kolkata, India {nagsayan112358, uddaloksarkar}@gmail.com

a

Departamento de Ciencias Computacionales, Universidad de Guadalajara, CUCEI, Av. Revolución 1500, Guadalajara, Jal, México 1 [email protected]

d

Dpto. Ingeniería del Software e Inteligencia Artificial, Facultad Informática, Universidad Complutense de Madrid, 28040 Madrid, Spain [email protected]

c

Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt [email protected]

*International Research Team Abstract The image segmentation based on Multilevel thresholding has attracted more attention in recent years, they have been used in different applications. Therefore, several evolutionary computation methods have been proposed to determine the optimal threshold values. However, these approaches suffer from some limitations such as the stagnation point which leads to degradation the quality of the segmented image. In addition, most of them used either Otsu or Kapur as a fitness function, and the complexity of these methods is increased with increasing the threshold levels. Moreover, they don’t provide accurate results. To overcome such situations, in this paper is proposed the use of evolutionary computation algorithms combined with the Type II Fuzzy Entropy as the objective function. Such methods are the Backtracking Search Optimization Algorithm (BSA) and the Salp Swarm Algorithm (SSA). The BSA and SSA are able to avoid the limitation of similar techniques for image threshold because the objective function removes the ambiguities helping to find more accurate results. The BSA and SSA are used to find the best parameters of the Type II Fuzzy Entropy that extracts the optimal thresholds that properly segment the histogram of a digital image. To evaluate the performance of the proposed two methods, a set of experiments are performed using a set of benchmark images which have different characteristics. Moreover, the experiments are also performed over medical images from blood cells. The experimental results indicate that the proposed two methods have a good performance. However, they provide better performance than other algorithms in terms of quality and accuracy. Keywords: Type II Fuzzy Entropy; Backtracking Search Algoritmm; Salp Swarm Algorithm, Multilevel Thresholding.

1. Introduction Segmentation is one of the most important tasks in image analysis; it is used to separate the objects in the image. The impact of segmentation techniques has been growing since digital cameras are becoming more accessible for implementations. Some examples of the use of segmentation are: their implementation for topographic maps [1], biometric systems [2], medical image [3,4], precision agriculture [5], and satellite images [6]. Thresholding (TH) is a segmentation method and consists in the separate the pixels into different groups depending on their intensity level according to one or more threshold values [7,8]. The main advantages of thresholding are that is easy to implement [9]. There exist two kinds of thresholding methods, when is used only one threshold it is called bi-level. Meanwhile, in the case of using two or more thresholds, it is called multilevel [10]. In general, in thresholding the pixels that have an intensity value higher than the

1

Corresponding author, Tel +52 33 1378 5900, ext. 27714, E-mail: [email protected]

1

threshold are labeled as an object, and the remaining pixels are part of the background. Those mentioned above can be applied to bi-level or multilevel segmentation. Due to its simplicity, for the segmentation of real-life images, the best alternative is the use of multilevel thresholding. The thresholding problem can be summarized as the search for the best threshold values on an image. It should be noticed that the threshold points depend on the histogram of the image; thus, each image has its own set of best threshold values. There are two classical methods for bi-level thresholding: the first one was proposed by Otsu [11] and maximizes the variance between classes. The second one was proposed by Kapur in [12], it proposes the maximization of entropy as a measure of the homogeneity among classes. The two methods are extensively used in image processing, and both of them have proved to be efficient and accurate alternatives to segment pixels into two classes [13]. Otsu and Kapur can be extended for multilevel thresholding; however, their computational complexity increases, while their accuracy decreases with each new threshold added into the searching process [13,14]. There exist other approaches based on entropies as the Shannon entropy [15], Tsallis entropy [16], Renyi’s entropy [17,18] and Fuzzy entropy [19]. However, their complexity and computational effort increase in multilevel thresholding. To overcome the problems presented by the traditional thresholding methods, there is a trend to use the Fuzzy techniques. The Fuzzy based approaches are able to remove greyness ambiguities providing more accurate results [20]. The Type II Fuzzy sets (TII-FS) are a generalization of Type I Fuzzy sets. In TII-FS is used the concept of ultrafuzziness that determine and eliminate the uncertainties that the classical Fuzzy sets possess [21]. The efficiency of TII-FS in image processing has been verified in different problems in this field. An interesting review of TII-FS in image processing is presented in [22]. The Type II Fuzzy entropy (TII-FE) is used in this article to find the best thresholds in digital images. It is an information-theoretical approach where is optimized the measure of fuzziness and the fuzzy entropy. According to the related literature, the TII-FE is one of the easiest and faster techniques based on TII-FS [20,23]. However, the main drawback of TII-FE is that it requires estimating the fuzzy parameters of the membership functions. This problem becomes worst in determining the number of thresholds [24], this fact is reflected in the computational effort required to compute the entropy. To face the problems previously described, some interesting solutions had been developed. Such as the Evolutionary Computation Algorithms are interesting search strategies designed to solve complex optimization problems. ECAs has been extensively used to find the best thresholds using standard metrics as Otsu or Kapur [25,26]. However, only a few amounts of papers use ECA with TII-FE as the objective function. For example, in [23] the Differential Evolution (DE) is applied to find the best parameters of TII-FE that provide the best thresholds. Here is important to mention, that the field of ECA is growing every year and different algorithms are proposed to solver more complex problems. This situation occurs due to the No-FreeLunch (NFL) theorem that states that not all the optimization algorithms can be applied to the same problem [27]. In this article, is proposed the use of the Salp Swarm Algorithm (SSA) [28] and the Backtracking Search Optimization Algorithm (BSA) [29] for finding the best thresholds in a digital image using the TII-FE. From the best of our knowledge, those methods have not been used in combination with TII-FS. Moreover, since both SSA and BSA were published between the past five years, their performance in real problems has not been completely tested, especially in image processing. The SSA was introduced in 2017 for solving numerical problems [28], it imitates the salp chain behavior that occurs in the deep ocean. SSA has been tested for numerical and benchmark engineering problems [28]. Moreover, it has also been used for Feature Selection in data mining [30], where the SSA obtains a better result for biomedical datasets. Another interesting implementation of SSA is for extracting the best parameters of PEM fuel cells providing good performance in comparison with similar approaches [31]. In the same context, the BSA was introduced in 2013 for numerical optimization [29]. Different to other ECAs the BSA does not imitate any natural behavior but it has different operators used in an iterative process to find the optimal values. For example, the random mutation scheme that only considers a single direction, this mechanism is different than in DE. The BSA has been used in engineering for the synthesis of concentric circular antenna arrays [32], to find the best allocation of electrical generators [33] and for the classification

2

of electroencephalogram signals [34]. Moreover, the BSA has been recently used for image segmentation using a modified version of the Type I Fuzzy Entropy for color segmentation in digital images [35]. In this sense, the aims of this paper can be summarized as follows: 1. The introduction of two alternative and accurate methods for multilevel thresholding based on SSA and BSA considering the TII-FE. 2. Perform a comparative study of the results of these new optimization approaches in the selected problem. 3. Test the SSA and BSA in a multidimensional image processing problem and verify their accuracy. 4. Verify that the SSA and BSA are able to segment images from blood cells using TII-FE. The methodology for MTH using SSA and BSA with TII-FE is similar. Both approaches consider as an input (search space) the histogram previously computed. The candidate solutions are generated based on the number of thresholds to find for each image; the double of this amount is the number of dimensions of the problem. SSA and BSA employ different operators (separately) to find the best configuration of the fuzzy parameter. The quality of the candidate solution is evaluated using the TII-FE criteria. The best solutions from SSA and BSA are compared with both classical approaches such as Particle Swarm Optimization (PSO) [36], Genetic Algorithms (GA) [37], Differential Evolution (DE) [38], Plant Propagation Algorithm (PPA) [39], Crow Search Algorithm (CSA) [40], Stochastic Fractal Search [41], and Cuckoo Search [42]. The comparisons include an analysis of the segmented images from each ECA. Also, two non-evolutionary segmentation approaches are considered for comparison; soft thresholding [43], and k-means clustering [44]. The segmented images generated by both evolutionary and non-evolutionary techniques are evaluated in terms for image quality. For this task, the Peak Signal-to-Noise Ratio (PSNR), the Root Mean Squared Error (RMSE), the Feature Similarity (FSIM) and the Structural Similarity Index (SSIM) [25,45–47] are evaluated. The reminder paper is organized as follows: Section 2 presents the problem of multilevel thresholding and the Type II Fuzzy Entropy. Section 3 explains the proposed algorithms based on SSA and BSA. Meanwhile, in Section 4 are presented the experimental results. Finally, Section 5 discusses the conclusions. 2. Multilevel thresholding using Fuzzy Type II Sets Multilevel Thresholding is a process used to classify the pixels into various groups thereby creating multiple regions finally yielding finer segmentation results useful for better image analysis. Thresholding methods essentially include two categories, namely, parametric and nonparametric. To get rid of the drawbacks that the parametric methods suffer from which include excessive time consumption and computational burden, the nonparametric approaches are widely appreciated and used. Non-parametric methods being more robust enjoy advantages over the parametric methods. Optimal thresholds for these techniques can be decided by optimizing few existing standards like between-class variance, the entropy, and the error rate. Several entropy-based methodologies have been proposed in the recent years for image thresholding, including the Shannon entropy [15], Tsalli’s entropy [16], Renyi’s entropy [18] and a fuzzy entropy-based algorithm [19], yet not devoid of the shortcomings, like longer computational time and complexity. Tao et al. [48] proposed a fuzzy entropy-based technique as a modification to that proposed by Zhao et al. [49], in this approach the image is thresholded using histogram partitions with definite fuzzy membership values, such partitions are used to extract the objects contained in the image. 2.1 Type II Fuzzy Sets Type-I fuzzy set A, in a finite set, X = {x1, x2,…, xn} can be defined as in Eq. (1):

A = {x, µ A ( x) | x ∈ X, 0 ≤ µ A ( x) ≤ 1}

3

(1)

Where µ A ( x ) is the membership function of the set A. In order to handle more uncertainty, in Type-II fuzzy a range of membership values are used instead of a single value. It may be defined as: high low A = { x, µAhigh ( x) , µlow A ( x) x ∈ X, 0 ≤ µA ( x) , µA ( x) ≤ 1}

(2)

from Eq. (2) µ Ahigh ( x ) and µ Alo w ( x ) are the upper and lower membership functions respectively. 2.2 Image segmentation using Type II Fuzzy Sets Thresholding is the easiest way to segment an image. The ease of thresholding consists in the use of threshold values (th) and apply them over the histogram until an optimal criterion is met. The process of thresholding using the histogram is summarized in the following rule:

 IGr ( r, c ) if IGr ( r , c ) ≤ th1  I s ( r , c ) =  thk −1 if thk −1 < IGr ( r , c ) ≤ thk ,  I ( r, c ) if IGr ( r , c ) > thnt  Gr

k = 2,3,K nt

(3)

where I s ( r , c ) is the gray value of the segmented image, I Gr ( r , c ) is the gray value of the original image both in the pixel position r, c. Since most applications require the segmentation of more than two classes, the normalized histogram is divided into nt+1 classes using nt thresholds, where thn is the n-th threshold value used for the segmentation process. The problem is to find the best configuration of thresholds that properly segment the image in different levels. A digital image I Gr contains M × N . pixels, where each pixel has a position defined by (m,n). The image has L intensity values that are stored into the pixels. In this context, the distribution of the intensity levels on the image can be represented by a histogram that can also be normalized as H = {h0, h1,…, hL-1}. Each value hi is computed using Eq. (4). hi =

npi , ( NP )

NP

∑h i =1

i

=1

(4)

Where i is a level of intensity ( 0 ≤ i ≤ L − 1 ) , NP = M × N is the total number of pixels contained in I Gr . npi is the number of pixels that correspond to the i intensity level in the image. The ultra-fuzziness then can be used as a metric associated to a fuzzy set which gives a 0 value when the membership values can be represented without any uncertainty. Whereas the value rises to 1 when membership values can be specified within an interval. For a digital image, the ultra-fuzziness for the i-th level of intensity is defined as follows: L −1

(

)

Pk = ∑ hi * ( µ khigh ( i ) − µ klow ( i ) ) , k = {1, 2,K , nl} , i =0

(5)

where µ k is the trapezoidal fuzzy membership function that generates the sets where the pixels belong depending on their intensity value. The Eq. (6) provides a better explanation of the membership function used in this approach.

4

0   i−a k −1   ck −1 − ak −1  1 µk ( i ) =   i −c k   ak − ck  0 

if

i ≤ ak −1

if

ak −1 < i ≤ ck −1

if

ck −1 < i ≤ ak

if

ak < i ≤ ck

if

i > ck

(6)

In Eq. (6) a k and c k are depicted as the fuzzy parameters where k ={1,2,…,nl}. The values for the bounds are given as: a0 = c0 = 0 and anl +1 = cnl +1 = nl + 1 , where nl denotes the number of thresholds used for segmentation. The fuzzy type-II entropy for a k-th threshold thk is therefore given as:

(

 hi ∗ ( µkhigh ( i ) − µklow ( i ) ) Fek = −∑  Pk i =1   L −1

)  *ln  ( h ∗ ( µ i

 

high k

 

( i ) − µklow ( i ) ) )  Pk

 

, k = {1, 2,..., nl + 1}

(7)

The sum of all the entropies for the nl+1 levels is the total entropy defined as: nt +1

TFe ( a1 , c1 ,K, an , cn ) = ∑ Fei

(8)

k =1

The problem of Eq. (8) is to define the best values for the fuzzy parameters, this process can be performed if the total entropy is maximized as in Eq. (9).

( a1 , c1 ,K , an , cn ) = max (TFe )

(9)

In this approach a threshold to segment the image I Gr is a point in which the membership value of k falls to 0.5 an the membership functions are linear. Since for multilevel segmentation, it is necessary to find a set of nt thresholds defined as TH = [th1 , th2 ,..., thnt ] , each element can be computed as in Eq. (10).

thn =

1 ( an + cn ) , n = {1, 2,K, nt} 2

(10)

In Eq. (10) each threshold is computed using the parameter of the TII-FE, it is an easy task that is performed once the best configuration of a and c parameters is found.

3. Multilevel thresholding using evolutionary approaches This section introduces the basic information for the Salp Swarm Algorithm (SSA) and the Backtracking Search Optimization Algorithm (BSA). Moreover, it is explained the adaptation of such methods for multilevel thresholding. The SSA and the BSA work exploring the search space defined by the image

5

histogram, as an objective function it is used the Type II Fuzzy entropy (TII-FE) that was introduced in section 2. In this context, the 8 bits digital images have 255 intensity levels (L), it means that the search space is defined between the bounds [0, 255]. In the SSA and BSA, it is necessary to initialize a set of candidate multidimensional solutions. This set is called population, and each solution contains the parameters ( a1 , c1 , K , a n , c n ) used by the TII-FE. It is important to mention that due to the definition of the TII-FE it is necessary to define two dummy thresholds 0 and L-1 such values are also included in the candidate solutions. In general terms, the population is defined in Eq. (11).

G = [Gf1 , Gf2 ,..., Gf N ], Gfi = [ 0,0, a1 , c1 , a2 , c2 ,..., an , cn , L − 1, L − 1]

(11)

T

In Eq. 9, Gfi ⊆ G and it is a vector that contains the fuzzy parameters. The dimensions of the optimization problem are defined by K = 2 × nt that means the double number of thresholds. In the initialization procedure of both SSA and BSA the solutions are randomly taken for a feasible space bounded by the lower ( lb ) and higher ( ub ) intensity level of the histogram. In this sense, for an 8-bit image the limits are defined as [lb=0, ub=255]. To randomly generate the values [ a j , c j ] of the solution Gfi it is used the Eq. (12):

aj , cj  = lbj + rand ×( ubj − ubj ) , ∀j = [1,2,..., K ]

(12)

where rand is a random number uniformly distributed between 0 and 1. j corresponds to a dimension of the search space. Once the initial solutions are generated, the SSA and the BSA are then used to modify the thresholds using their own search operators. 3.1 Salp swarm algorithm The Salp Swarm Algorithm (SSA) was introduced as a novel alternative for solve complex optimization problems [28]. The SSA imitates the swarming behavior of salps in the deep ocean. Such behaviour is called salp chain and was mathematically modeled by Mirjalili in [28]. In the SSA, one element of the population is considering as a leader that is located at the front of the chain. Meanwhile, the rest of the population of salps are followers. In the salp chain, a follower goes behind other follower creating a chain that is finally guided by the leader. Once the population G of salps is generated it is evaluated on the objective function defined in section 2. The salp that possesses the value of the objective function is selected and assigned as a good source (F). In the SSA, the set of candidate solutions represents the salp chain, and the leader is defined by the position of F. The position of the leader is updated using the following equation:

( (

 F j + q1 ( ub j − lb j ) q2 + lb j  Gf 1j =   F j − q1 ( ub j − lb j ) q2 + lb j

) )

q3 ≥ 0

(13)

q3 < 0

From Eq. (13) Gf 1j is the first element of the population G (the leader), Fj is the position of the food source and j is the j-th dimension of the problem. q2 and q3 are randomly generated between [0, 1]. Finally, q1 is used to control the exploration and exploitation of the SSA and it is modified using Eq. (14):

6

q1 = 2e−( 4 it Maxit)

2

(14)

Where it is the current iteration and Maxit corresponds to the maximum number of iterations. On the other hand, to modify the values of the rest of the salps (followers) in the original SSA it is proposed a modified version of Newton’s motion law [28] that is described in the next equation:

Gf ji =

1 ( Gf ij + Gf ij −1 ) 2

In Eq. (15) Gf ij is the position of the i-th follower salp at the dimension j, notice that

(15)

i ≥ 2 to consider all the

followers. Figure 1 presents a flowchart of the SSA for image segmentation.

Figure 1. Flowchart of SSA.

3.2 Backtracking Search Algorithm Backtracking Search Algorithm (BSA) is a simple evolutionary algorithm which is effective for solving multi-modal optimization problems successfully and swiftly. It was introduced in 2013 by Civicioglu [29]. The basic difference between BSA and Differential Evolution (DE) lies in the mutation strategies adopted by them. BSA uses one directional candidate solution for each target candidate solution for its random mutation strategy which is in contrast with DE. BSA randomly chooses the direction individual from individuals of a randomly chosen previous generation. BSA uses a non-uniform complex crossover strategy for generating trial solutions and it has a single control parameter and is not overly susceptible to the initial value of this parameter. BSA is an adaptive algorithm having better control of search direction and better exploitation and exploration of parameter space. Backtracking Search Algorithm (BSA) abstains from using candidates in the population with better fitness values more often than others. BSA has a unique boundary control mechanism and is also a dual-population algorithm which uses present population at the same time remembering its past population. BSA consists of five steps: Trial Initialization, Trial Selection (I), Mutation, Crossover and Trial Selection (II). The steps are mentioned in detail as follows:

7

Initialization BSA starts by initialization step similar to other EA, here N candidate solutions are randomly generate using the bound of the K-dimensional search space. To construct the population G the BSA uses the Eq. (12). Then the first selection step (also, called Selection-I) decides the historical population oldG which is to be used for calculating the search direction. The initial historical population is determined beforehand for i = 1,…., N and j = 1,…., K using a uniform distribution U as in Eq. (16).

oldG ij

U ( lb j , ub j )

(16)

At the beginning of each iteration the oldG can be redefined using the following rule:

oldG = G , if

r1 < r2

(17)

where r1, r2 are random numbers lying between 0 and 1. BSA uses a memory-based approach by identifying a population belonging to a randomly chosen previous generation as the historical population. Thereby recognizing this historical population until and unless it is altered. The order of candidates is randomly changed in oldG by permutation, that is, by random shuffling of the order. Mutation After that the Mutation step generates a Mutant solution given as:

Mut = G + MF ( oldG − G )

(18)

where MF is the mutation factor which controls the amplitude of the search-direction vector (oldG – G). It is clear that BSA generates a trial population incorporating the advantages it gains from the experiences in the earlier generations. The value of MF commonly used is MF = 3. After that, the crossover step generates the final form of the population of trial solutions or off-springs (T). The initial value of the offspring population is Mut which was set in the previous step of mutation. Fittest candidate solutions are used to evolve the target population candidates. The crossover process includes a step where a binary integer-valued matrix (map) of size N × K is calculated that indicates the individuals of the trial population, T which is to be manipulated by using the germane candidates of G. If mapi,j = 1, where i ϵ {1, 2, 3, ….., N} and j ϵ {1, 2, 3, ….., K}, T is updated to G as Ti,j := Gf ij , where := represents the update operator. The crossover strategy adopted by BSA is more complex and is different from that of DE-in BSA’s crossover step there exists a DIM_rate parameter which controls the number of elements of individuals which will mutate in a trial using ceil(DIM_rate . rand . K) where rand is any random number between 0 and 1 and ceil function is the ceiling function, and K is the dimension of the optimization problem. In BSA’s map there exist two cases (given by an if-else structure)-the first case is just defined before; the second step allows only one randomly selected candidate to mutate in each trial.

8

BSA is modified to be an adaptive one in view of the bounds of the search space. In the event that the upper and lower bounds are disregarded the point is regenerated in order to lie within the search region where, lbj and ubj are the respective lower and upper boundaries of the j-th coordinate of the search domain. This boundary control mechanism assigns the value according to the Eq. (12) if either of the limits is crossed, thus, preventing any overflow of search space limits. In the second selection step (which called Selection-II), those Ti s which are superior to the corresponding Gfi in terms of fitness scores are used to update the Gfi based on a greedy selection procedure. The best individual of G, Gfbest is found out based on the fitness values and if this Gfbest has a better fitness score than the global optimum value achieved so far, the global optimizer is updated to Gfbest, and the global optimum value is updated to the fitness value of Gfbest. A description of the BSA for image segmentation in the form of a flowchart (Figure 2) is presented below.

Figure 2. Flowchart of BSA.

Here is important to mention that once the optimization algorithm (BSA or SSA) obtains the global best, it is used to compute each threshold value using the Eq. (10). When all the thresholds are generated they are applied to the histogram to generate the classes and the segmented image according to Eq. (3). 4. Experiments and results The SSA and BSA for multilevel thresholding are tested over ten benchmark images which contain different complexity levels. The images have different resolutions but their sizes are not higher than 512 × 512 pixels, and they are in JPEG format [50]. To graphically show the results of the proposed algorithm they are selected only five images. Meanwhile, for an analytical analysis, there are used the entire set of ten images. The selected images and their respective histogram are shown in Figure 3. In the supplementary material file (SMF) there is presented the full set of images used for the experiments. Moreover, all the figures and tables in this section are extended in the SMF using the same numbers.

9

Image

Histogram

41004

147091

385028

225017

176035

Im

Figure 3. Selected images for graphical analysis.

To verify the efficacy of the proposed approaches they have been compared against different metaheuristics from the state-of-the-art. Such methods are the Genetic Algorithms (GA) [37], the Differential Evolution (DE) [38], the Particle Swarm Optimization (PSO) [36], the Plant Propagation Algorithm (PPA) [39], the Crow Search Algorithm (CSA) [40], Stochastic Fractal Search [41], and Cuckoo Search [42]. All the selected ECA have different control parameters that are set based on their respective authors. According to the NoFree-Lunch (NFL) theorem, not all the optimization algorithms can be applied to the same problem [27]. In this context, the experiments and comparisons of the ECA for image thresholding using TII-FE help to prove that the BSA and SSA are competitive for real problems. Despite the focus of this article is metaheuristic algorithms coupled with TII-FE two non-evolutionary approaches for image segmentation are considered to broaden this study; the soft thresholding method [43], and k-means clustering [44]. Considering that the ECA involves the use of random variables it is necessary to statistically analyze their results. A set of 35 independent experiments were performed for each selected algorithm over all the images contained in the dataset. For the segmentation in this paper, there are used 2, 3, 4, and 5 thresholds are used over the energy curve instead of the histogram. Such threshold values are commonly used in the related literature [51–53]. The stop criteria and the number of elements of the population are set to 100 and 50, respectively for each algorithm. Such values are selected after exhaustive experiments that provide evidence

10

that after 100 iterations the algorithms converge using a population of 50 elements. All experiments were performed using MATLAB 9.3 on an AMD Ryzen 7 1700 CPU @ 3Ghz with 16GB of RAM. Performance Metrics The results of the MT algorithms based on BSA and SSA are measured not only considering the objective function values but also considering the quality of the segmented image. The metrics used are: 1- The Standard Deviation (STD) that represents the stability of the results obtained by the algorithms [51]. The STD is computed as follows: Iterm ax

(σ i − µ )

i =1

Ru

∑

ST D =

(19)

2-The Peak-Signal-to-Noise Ratio (PSNR) it is also used to compare the similarity of the segmented image with the original image. The PSNR is based on the mean square error (MSE) of each pixel [14,53,54]. The PSNR and RMSE are defined as in Eq. (20).  255  PSNR = 20 log10   , ( dB )  RMSE  ro

RMSE =

(20)

co

∑ ∑ ( I ( i, j ) − I ( i, j ) ) Gr

i =1 j =1

s

ro × co

From Eq. 20 I Gr is the original image, I s is the segmented image. Meanwhile, ro and co are the maximum number of rows and columns of the image. 3-The Structure Similarity Index (SSIM) is used to compare the structures of the original segmented image [55], and it is defined in Eq. (21). Where the higher SSIM value represents a better segmentation of the original image. SSIM ( I or , I th ) =

σI

th I or

=

( µI

(2µ 2 or

I or

µ I + C1)( 2σ I th

)(

or I th

+ C2

)

+ µ I th + C1 σ I or + σ Ith 2 + C 2

N

2

(

1 ∑ I ori + µ Ior N − 1 i =1

)( I

2

thi

+ µ I th

)

(21)

)

In Eq. (21) the mean of the original image is µ I and the mean of the thresholded image is represented by µ Ith or

.In the same way, for each image, the values of σ IGr and σ Ith correspond to the standard deviation. C1 and C2 are constants used to avoid the instability when µ IGr 2 + µ Ith 2 ≈ 0 . The values of C1 and C2 are set to 0.065 considering the experiments of [14]. 4-The Feature Similarity Index (FSIM) [56] is another interesting metric that calculates the similarity between two images. In this paper they are used the original grayscale image, and the segmented image. As PSNR and SSIM the higher value is interpreted as better performance of the thresholding method. The FSIM is then defined in Eq. (22).

11

FSIM =

∑ S ( w ) PC ( w ) ∑ PC ( w )

w∈Ω

L

w∈Ω

m

(22)

m

On the FSIM the entire domain of the image is defined by Ω, and their values are computed by Eq. (23). S L ( w ) = S PC ( w ) SG ( w ) S PC ( w ) = SG ( w ) =

2 PC1 ( w ) PC2 ( w ) + T1

PC12 ( w ) + PC2 2 ( w ) + T1

(23)

2G1 ( w ) G2 ( w ) + T2

G12 ( w ) + G2 2 ( w ) + T2

G is the gradient magnitude (GM) of a digital image and is defined, and the value of PC that is the phase congruence is defined as follows: G = Gx 2 + G y 2 PC ( w ) =

E ( w)

(24)

   ε + ∑ An ( w )  n  

From Eq. (24) An ( w ) is the local amplitude on scale n and E ( w) is the magnitude of the response vector in w

(

on n. ε is a small positive number and PCm ( w) = max PC1 ( w) , PC2 ( w)

)

4.1 Results and Discussions This section presents and analyzes the results obtained by the SSA and the BSA to optimize the TII-FE for multilevel image thresholding. As was previously mentioned, the SSA and BSA based approaches are applied over the entire set of benchmark images. Table 1 presents the best fuzzy parameters and the best thresholds generated by the SSA and the BSA for different segmentation levels (2, 3, 4, 5) on five selected images. Moreover, for comparative purposes in Table 1 are included the best results obtained by Genetic Algorithms (GA) [37], Particle Swarm Optimization (PSO) [36], Differential Evolution (DE) [38], Plant Propagation Algorithm (PPA) [39] and Crow Search Algorithm (CSA) [40], Stochastic Fractal Search [41], and Cuckoo Search [42]. Besides the soft thresholding method [43], and k-means clustering [44] are evaluated as nonevolutionary methods. Even though all the algorithms perform image segmentation, the non-evolutionary approaches cannot be evaluated as optimization methods. Thus, the non-evolutionary are only listed in comparisons where the quality of the image is assessed.

12

Table 1. Results after applying the optimization algorithms to segment set of selected benchmark images using TII-FE.

225017

51,179

3

12,106,107, 142,147, 255

59,125,201

15, 89, 90, 194,194,255

52, 142, 225

9, 97, 97, 192,192,253

53, 145, 223

8,94, 95, 139, 140,255

51,117, 198

9,86, 86, 180, 181,254

48,133, 218

15,103,160, 171,184,246

59,165,215

4,79,79, 171,171,256

41,125,213

52, 111, 162, 224

10,87, 87,141, 142,194,194, 253

49,114, 168,224

12,91, 92,98, 98,195, 195,253

52,95, 147,224

6,89,96,195, 200,218,238, 252

47,145,209,2 45

3,85,88,128, 128,192,192, 255

44,108,160,2 23

52, 93, 139, 84, 221

8,85, 86,87, 88,166,167, 197,197,254

47,87,127, 182,226

4

4

5

2 3

18,100,101, 106,112,186, 195, 252 14, 83, 85, 103,104,185, 187, 92,194, 247 0, 126, 129, 254 0, 120,121, 135, 139,252 1,113,114, 123,123,158, 160,253 0,33,35, 38,40, 115, 118,126,140, 254 23,122, 122, 255 22,139,142, 143,144,254

4

30,126,130, 167,169,171, 171,253

5

22,77, 81, 87, 91,146, 152,167,171, 253

2 3 4

5

2 3 4

5

1,153, 153, 254 4,155,157, 162,163,253 5, 76, 83, 143,145,145, 154,253 4, 74, 81, 82, 84,133, 147,172,175, 249 1, 61, 62, 27 2, 70, 71, 117, 120,237 4, 88, 89, 98,101,167, 169,236 3, 70, 77, 89,93,159, 161,206,214, 252

59,104,149, 224

49, 94,145, 190,221 63,192 60,128, 196 57,119,141, 207 17,37, 78, 122, 197 73,189 81,143, 199

8, 95, 95, 95,95, 186, 188, 255 8, 100,100, 100,100,181, 181,181,181, 255 0, 123, 126, 255 0, 139,140, 156, 158,255 0,138,139, 143,144,204, 205,255 3,120, 120, 122,122,172, 172,183,185, 255 18,127, 127, 255 18,134, 144, 150,153,255

78,149,170 ,212

23,93,96, 154,155,160, 163,255

50,84, 119, 160,212

28,87, 89, 103,108,164, 164,168,171, 255

77,204 80,160, 208 41,113, 145, 204 39,82, 109, 160,212 31,145 36, 94,179 46,94,134, 203 37,83, 126, 184,233

0,141, 141, 255 0,156, 159, 160,162,255 0,122, 122, 140,140,172, 172,255 0,119, 121, 123,123,184, 190,192,192, 255 0, 58, 59, 55 0, 74, 75, 120, 123,255 0, 82, 82, 149,149,170, 170,255 12,61, 62, 126,126,139, 139,215,215, 255

52, 95, 141, 222

54, 100,141, 181, 218 62, 191 70, 148, 207 69,141, 174, 230 62,121, 147, 178,220 73,191 76,147, 204

11, 93, 93, 128, 128, 95, 197, 251 14, 90, 91, 95, 95, 182, 183, 85, 187, 254 1, 114, 115, 255 0, 124,124, 129,129, 255 0,107, 108, 125,127,174, 175,252 1,38,38,107, 108,147,147, 176,178,255 18,125, 125, 255 21,142, 144, 161,161,255

58,125,158, 209

18,90, 91, 150,152,169, 170,255

58,96,136, 166,213

19,86, 86, 97, 98,147, 150,175,176, 254

71,198 78,160, 209 61,131, 156, 214 60,122, 154, 191,224 29,157 37,98, 189 41,116, 160, 213 37,94, 133, 177,235

11,140, 140, 255 8,142, 143, 152,152,255 10,95,95, 136,139,164, 166,255 8, 85, 88, 91,92,143, 143,173,173, 255 0,62, 62,212 0,69,69, 125,125,229 0, 73, 73, 116,116,210, 210,255 1, 69, 69, 135,135,135, 135,210,211, 255

58, 185 62, 127, 192 54,117, 151, 214 20,73, 128, 162,217 72,190 82,153, 208

1,115,115, 255 0,129,129, 157,157,255 12,108, 109, 147,147,195, 195,254 0,56, 59,91, 97,163, 165, 178,179,255 19,129,129, 254 19,140, 140, 162,162,255

54,121, 161, 213

21,92,93, 140,140,170, 170,254

53,92, 123, 163,215

24,81, 81, 115,116,138, 139,174,176, 255

76,198 75,148, 204 53,116, 152, 211 47,90, 118, 158,214 31,137 35,97, 177 37,95, 163, 233 35,102, 135, 173,233

11,139,139, 255 10,147, 148, 156,157,254 12,94, 95, 142,142,169, 169,252 4,53, 53, 81,81,140, 140,172,172, 248 1,62, 63,209 0,69, 69, 130, 130,236 0,73, 73, 74,75,163, 163,239 4,72, 81, 132,133,152, 152,209,211, 252

58,185 65,143, 206 60,128, 171, 225 28,75, 130, 172,217 74,192 80,151, 209

28,85, 85, 86, 86,175, 177,195,197, 254 1,111,111, 255 2,119,119, 143,144,255 17,102, 102, 155,156,180, 181,255 2,44, 44,111, 112,121,122, 189,189,253 19,123,123, 254 25,114, 116, 116,118,255

57,117,155, 212

20,85,86, 142,142,173, 174,255

53,98, 127, 157,216

24,75, 76, 91, 91,166, 169,171,173, 254

75,197 79,152, 206 53,119,156, 211 29,67, 111, 156,210 32,136 35,100, 183 37,74,119, 201 38,107, 143, 181,232

9,139,140, 254 8,145, 146, 153,153,255 4,70,70, 146,146,153, 153,252 8,78, 78, 79,79,142, 142,166,167, 251 0,61, 61,212 0,69, 70, 125, 125,238 1,69, 75, 99,104,203, 208,254 5,65, 65, 115,117,117, 117,209,210, 254

13

57,86,131, 186,226 56,183 61,131, 200 60,129, 168, 218 23,78, 117, 156,221 71,189 70,116, 187

9,40,50, 104,121,136, 139,186,212, 250 14,104,106, 210 24,100,101, 104,105,242 34,76,83, 163,187,195, 205,256 11,37,57, 124,132,176, 185,222,239, 254 44,120,137, 216 26,50,50, 75,94,247

53,114,158, 215

5,84,91, 107,140,200, 212,232

50,84, 129, 170,214

58,66,75, 108,110,132, 155,170,183, 239

74,197 77,150, 204 37,108,150, 203 43,79,111, 154,209 31,137 35,98, 182 35,87,154, 231 35,90, 117, 163,232

5,189,196, 223 20,52,62, 113,115,232 6,102,106, 119,140,192, 203,256 22,70,79, 83,93,114, 116,126,136, 244 45,98,128, 227 13,74,82, 153,162,219 9,73,91, 124,155,213, 216,241 17,50,50, 106,119,131, 137,153,172, 216

24,77,128, 162,231 59,158 62,102,173 55,123,191, 230 24,90,154, 203,246 82,176 38,62,170

12,91,97, 117,118,154, 155,185,187, 256 1,114,114, 256 5,113,113, 141,142,256 1,42,51,124, 125,167,167, 256 3,59,60,115, 116,174,175, 196,197,256 2,125,125, 256 2,142,142, 171,171,256

44,99,170, 222

12,87,88, 133,138,182, 184,256

62,91,121, 162,211

5,100,107, 127,127,168, 169,178,178, 255

97,209 36,87,173 54,112,166, 229 46,81,103, 121,190 71,177 43,117,190 41,107,184, 228 33,78,125, 145,194

7,142,142, 256 1,148,149, 165,167,254 1,52,53, 153,159,162, 163,256 11,91,93, 137,139,171, 178,206,208, 256 1,61,61,204 2,76,77, 205,211,256 6,72,78, 119,122,209, 211,256 1,71,84, 138,139,148, 150,204,209, 256

51,107, 136,170,221 57,185 59,127,199 21,87,146, 211 31,87,145,18 5,226 63,190 72,156,213

8, 101, 101, 255 8, 84, 84, 166, 166, 255 8, 94, 94, 130, 130,192, 192, 255 11, 98, 98, 111,112,182, 182,194,195, 255 0, 117, 117, 255 3, 121,121, 156, 156,255 4,108, 108, 140,141,173, 173,253 0, 43,43, 104,105,142, 143,185,185, 252 18,125, 125, 255 20,134,135, 136,136,253

49,110,160, 220

18,128, 128, 128,132,197, 197,255

52,117,147, 173,216

22,82, 82, 88,88,150, 151,164,165, 253

74,199 74,157,210 26,103,160, 209 51,115,155, 192,232 31,132 39,141,233 39,98,165, 233 36,111,143, 177,232

9,140, 140, 255 4,150, 150, 153,155,254 9, 91, 91, 144,144,165, 165,254 10,70, 71, 96,96,138, 138,171,172, 250 0,62, 62,209 0, 71,71, 206, 210,255 1, 81, 82, 109,109,209, 209,255 2, 75, 76, 77,78,142, 143,205,209, 250

55, 178 46, 125, 211

51, 112, 161, 224

55, 105,147, 188, 225 59, 186 62, 139, 206 56,124, 157, 213 22, 74,124, 164,219 72,190 77,136, 195

8, 96, 96, 255 8, 108, 108,130, 130, 255 8, 100, 100,109, 109,188, 188, 255 8, 96, 96, 96,96, 176, 176,179,179, 255 0, 121, 121, 255 2, 121,121, 123, 123,255 0,115, 115, 151,151,187, 187,255 0,57, 57,98, 98,132, 132, 171,171,255 18,125, 125, 255 18,143, 143, 149,149,255

Thresholds

1,102,102, 256

Fuzzy Parameter s

134,243

SSA TII-FE Thresholds

36,232,234, 252

Fuzzy Parameter s

55,178

Thresholds

Fuzzy Parameter s

8,101,101, 255

Fuzzy Parameter s

53,176

Thresholds

8,97, 97,255

Fuzzy Parameter s

54, 178

Thresholds

8, 99, 100, 255

Fuzzy Parameter s

50, 174

Fuzzy Parameter s

Thresholds

BSA TII-FE

Thresholds

CS TII- FE

Fuzzy Parameter s

SFS TII-FE

Thresholds

Thresholds

CSA TII-FE

8, 92, 92, 255

3

385028

PPA TII-FE

62,186

2

147091

DE TII-FE

10,114,116, 255

5

41004

PSO TII-FE

2

Fuzzy Parameter s

Im 176035

Levels

GA TII-FE

52, 176 58,119, 193

54, 105, 149, 222

52,96,136, 178,217 61, 188 62, 122, 189 58,133, 169, 221 29,78, 115, 152,213 72,190 81,146, 202

73,128, 165, 226

18,99,99, 145,145,170, 170,255

59,122, 158, 213

52,85, 119, 158,209

18,100, 100, 101,101,183, 183,183,183, 255

59,101, 142, 183,219

75,198 77,152, 205 50,118, 155, 210 40,84,117, 155,211 31,136 36,139, 233 41,96, 159, 232 39,77, 110, 174,230

8,139, 139, 255 5,148, 148, 153,153,255 8, 79, 79, 145,145,158, 158,255 8, 71, 71, 71,71,144, 144,155,155, 255 0,61, 61,212 0, 68, 68, 210, 210,255 0, 73, 73, 111,111,210, 210,255 0, 69, 70, 130,130,130, 130,209,210, 255

74,197 77,151, 204 44,112, 152, 207 40,71,108, 150,205 31,137 34,139, 233 37,92, 161, 233 35,100, 130, 170,233

To validate the results presented in Table 1, Table 2 presents the Mean and the Standard deviation (STD) of the objective function values. From Table 2, the values of the mean and STD are better for BSA and SSA; this fact indicates that the stability of such algorithms is higher than GA, PSO, DE, PPA, CSA, SFS and CS. A good solution for this problem is the higher in terms of objective function values. In this context, the BSA and SSA are more accurate than the other algorithms used for comparisons, it is verified with the mean of the objective function. In general terms, the BSA and SSA are good alternatives to find the best thresholds for image segmentation using the TII-FE. The results obtained by them are very similar, and in some cases, SSA is better. This situation occurs due to random values used in the iterative process. Moreover, considering that each image is an optimization problem and according to the NFL theorem is completely normal that BSA produces good solutions in some cases and SSA in other cases. From Table 2, the PPA, CSA and DE are also competitive approaches for image segmentation, however the accuracy is lower than BSA and SSA. Finally, the less accurate optimization algorithms are the SFS, CS, GA and PSO. On the other hand, the STD represents the stability of the algorithms in the experiments, from Table 2 the lower values of STD are obtained by the BSA and SSA. The DE, PPA and SCA are also stable alternatives for this problem, meanwhile, the SFS, CS, GA and PSO are not the best in terms of the STD. Table 2. Mean and standard deviation of the objective function values obtained by GA, PSO, DE, PPA, CSA, SFS, CS, BSA, and SSA for the segmentation of digital images using TII-FE.

STD

Mean

STD

13.6194 16.9518 20.0061 22.7892 13.7184 17.0767 19.8855 22.8354 13.9184 17.1719 19.9791 22.6475 14.1017 17.4499 20.2992 23.1055 13.2917 16.5164 19.4277 22.0986

1.69E-01 2.98E-01 3.80E-01 5.22E-01 2.01E-01 3.01E-01 3.43E-01 3.73E-01 2.27E-01 3.26E-01 3.71E-01 4.80E-01 2.04E-01 3.45E-01 3.30E-01 4.27E-01 1.54E-01 1.86E-01 3.22E-01 4.58E-01

13.6884 17.6687 21.0254 23.8722 14.2044 17.8961 20.9344 23.8640 13.5104 18.1433 21.1018 23.8147 14.6264 18.3059 21.2782 24.0291 13.7440 17.1683 20.3673 22.9802

7.90E-03 4.15E-02 1.82E-01 3.02E-01 1.13E-02 1.32E-01 1.52E-01 2.44E-01 1.95E-02 1.25E-01 1.55E-01 2.26E-01 1.11E-02 1.07E-01 1.76E-01 1.96E-01 2.16E-02 7.65E-02 1.87E-01 3.31E-01

13.9755 17.7446 21.5080 25.0463 14.213 18.0610 21.4147 24.7731 14.4485 18.3218 21.5470 24.8549 14.6501 18.5825 21.9178 24.9697 13.8053 17.4041 20.9338 24.1169

1.98E-09 6.63E-09 5.47E-09 2.99E-10 4.66E-09 5.43E-10 9.02E-10 3.38E-09 3.39E-10 4.92E-11 8.34E-10 9.92E-09 9.26E-09 7.21E-09 5.11E-11 2.21E-10 6.55E-10 1.13E-10 1.02E-10 3.41E-11

13.9753 17.7393 21.5307 25.1351 14.212 18.0699 21.4644 24.7688 14.4486 18.3229 21.5755 25.0504 14.6491 18.6154 21.9293 25.2030 13.8060 17.4463 21.0051 24.2365

1.12E-09 9.92E-10 9.45E-11 3.57E-10 6.11E-10 7.68E-10 9.23E-09 8.68E-11 8.01E-10 7.41E-12 5.55E-11 3.23E-10 1.03E-12 2.43E-11 4.40E-13 5.09E-10 6.41E-10 4.59E-11 3.53E-11 8.20E-12

STD

CSA TII-FE Mean

STD

Mean

13.9738 1.77E-08 17.7232 1.78E-07 21.4673 3.81E-08 25.0274 3.41E-09 14.2111 4.27E-08 18.0607 5.58E-08 21.3454 3.87E-09 24.6500 3.42E-08 14.4214 4.92E-08 18.2929 5.52E-08 21.5425 1.77E-08 24.5634 4.27E-09 14.6473 1.66E-08 18.5795 8.51E-08 21.9110 3.44E-08 24.8906 6.61E-08 13.7927 5.21E-09 17.2805 1.74E-07 20.6862 3.36E-08 23.9246 2.02E-08 SSA TII-FE

STD

13.9612 9.82E-08 17.706 9.98E-09 21.429 4.11E-09 25.0281 1.13E-08 14.208 5.28E-07 18.0627 1.19E-09 21.3442 2.90E-08 24.7229 8.08E-08 14.4415 6.66E-09 18.2667 2.90E-08 21.4935 1.92E-11 24.7933 8.82E-10 14.6413 3.79E-09 18.5746 7.09E-07 21.8037 2.61E-08 24.9926 8.11E-09 13.8052 9.00E-09 17.2855 5.19E-09 21.004 1.60E-08 24.2317 2.01E-08 BSA TII-FE

Mean

STD

PPA TII-FE

Mean

13.9693 4.06E-01 17.6663 3.64E-02 21.5076 5.72E-03 25.1044 7.87E-02 14.1621 1.16E-03 18.0306 7.63E-02 21.2360 5.22E-01 24.6756 2.98E-03 14.4464 9.23E-03 18.1588 1.69E-03 21.4033 4.02E-04 24.5714 4.47E-01 14.6201 3.07E-01 18.5204 5.44E-01 21.7900 6.62E-03 24.9156 5.80E-03 13.6950 9.97E-02 17.2041 4.94E-01 20.5436 6.12E-03 23.9075 8.90E-02 CS TII-FE

Mean

STD

Mean

STD

Mean

13.9346 4.57E-03 17.6466 7.13E-01 21.1822 2.23E-01 24.5134 3.03E-04 14.155 2.70E-02 17.954 1.82E-02 21.227 2.79E-03 24.2328 4.29E-01 14.403 1.34E-03 18.2320 6.91E-04 21.1261 2.54E-01 24.3452 8.85E-04 14.607 6.06E-04 18.5165 4.36E-03 21.5427 8.49E-03 24.4194 2.28E-02 13.7526 1.96E-01 17.2079 2.15E-04 20.4073 7.09E-02 23.5115 5.98E-03 SFS TII-FE

DE TII-FE

STD

2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5

PSO TII-FE

Mean

Im 176035 225017 385028 147091 41004

2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 Levels

Im 176035 225017 385028 147091 41004

Levels

GA TII-FE

13.9667 17.7180 21.4686 24.806 14.2012 18.0596 21.2997 24.7072 14.4336 18.2030 21.5475 24.6160 14.6419 18.6027 21.8295 25.1500 13.8034 17.2790 20.7303 23.9434

4.36E-08 1.34E-08 9.82E-08 5.02E-08 3.10E-08 8.24E-09 2.00E-08 2.03E-08 1.94E-08 6.15E-09 1.58E-08 1.16E-07 3.44E-08 5.96E-08 1.06E-07 9.82E-09 3.04E-08 3.60E-08 2.59E-08 2.15E-09

To graphically show the efficacy of BSA and SSA, Figure 4 shows the resultant images after applying the thresholds generated by the BSA using the TII-FE. This figure also shows the corresponding histogram of each image with the thresholds obtained by the BSA. The resultant images provide evidence that increasing the number of thresholds the quality of the segmented image also increases. This fact is desirable for some implementation due that is possible to segment more details of the objects contained in the image.

14

147091

385028

225017

176035

Im 2 3 Level 4

15 5

41004

Figure 4. Segmented images and thresholded histograms obtained using BSA with TII-FE.

Similar to the results of BSA, the images of Figure 5 are obtained by the MTH algorithm based on SSA combined with the TII-FE. This table shows the histograms with their respective thresholds. The graphical quality of the segmented images increases with the number of th used for segmentation. From both Figure 4 and Figure 5 is possible to analyze that visually the classes are well defined because the thresholds values are properly selected. This effect can be observed for example in image 225017 where the classes and objects are segmented homogeneously using 3 thresholds. This situation also occurs in images with more complex histograms as 176035 in which with 4 or 5 thresholds are segmented most of the objects preserving the features of the scene. Level 2

3

4

176035

Im

16

5

17

41004

147091

385028

225017

Figure 5. Segmented images and thresholded histograms obtained using SSA with TII-FE. The images and plots from Figure 4 and Figure 5 provide evidence about the performance of BSA and SSA for multilevel thresholding. To support the accuracy of the solutions, in Figure 6 they are also included the plots of the objective function of the GA, PSO, DE, PPA, CSA, BSA, and SSA. The results in Figure 4 are for the five selected images using 2, 3, 4, and 5 segmentation levels. Level 2

3

4

41004

147091

385028

225017

176035

Im

18

5

Figure 6. Comparison of the objective function values for PSO (red line), DE, (cyan line), GA (magenta line), PPA (olive line), CSA (black line), SFS (pink line), CS (purple line), BSA (blue line) and SSA (green line) applied for multilevel thresholding using TII-FE.

In Figure 6, the fitness values obtained for the five selected images are presented. For this experiment, each algorithm runs 100 times, and the best values are stored at the end of each iteration. This number is selected because experimentally we noticed that for all the algorithms the TII-FE value decrease in the first 100 iterations. From Figure 6 it is possible to deduce that the proposed MTH algorithms based on BSA and SSA require fewer iterations than other similar approaches to obtain the best thresholds. In this context, the stop criteria could also be modified. The results presented in Figure 6 show that for few dimensions (level 2) all the algorithms have the same performance with the exception of SFS which fails on every test. However, when the number of thresholds increases, all the selected ECAs start having problems to find the best solutions. For example, in the image 385028 the GA, PSO, DE, PPA, SFS, CS, and CSA cannot reach the maximum value for levels 3,4, and 5. Meanwhile, the BSA and SSA can obtain more accurate solutions due to their optimization operators. Moreover, from Figure 6 it can also be analyzed that the DE, PPA and CSA are also able to find the best solution in a reduced number of iterations. Meanwhile, the PSO and GA in some cases get stuck in suboptimal and cannot reach the global best. An example for PSO can be observed in for image 41004 using 2 thresholds and for GA in the same image using 4 thresholds, in such examples both PSO and GA converges in a local optimal value and are not able to continue searching for the global best.

4.2 Evaluation of the results The results from Tables 1-2 and Figure 4-6 provide evidence of the capabilities of BSA and SSA to find the best parameters of the TII-FE and generate the accurate threshold. However, such information doesn’t measure the quality of the segmented images. Table 3 presents the values of the quality metrics obtained after applying the thresholds over the test images. Such values provide evidence that the segmented images obtained using the thresholds computed by the TII-FE have better quality, in specific the computed by BSA and SSA. To further analyse the quality of the segmented images the two non-evolutionary segmentation methods are included in the following sections.

19

Table 2. Comparison of the PSNR, SSIM and SSIM values of the GA, PSO, DE, PPA, CSA, SFS, CS, Soft Threholding, k-means segmentation, BSA and SSA applied over the selected images using the TII-FE.

FSIM

PSNR

SSIM

FSIM 0.6670 0.7065 0.7367 0.7606 0.6600 0.7398 0.7911 0.8199 0.6163 0.6875 0.7401 0.7852 0.7228 0.7780 0.7848 0.8077 0.7389 0.7603 0.7785 0.7946

FSIM

SSIM

0.5877 0.6781 0.7464 0.7758 0.5587 0.6720 0.7625 0.8064 0.5780 0.6609 0.7153 0.7745 0.6027 0.7187 0.7356 0.7806 0.5834 0.7134 0.7688 0.7911

SSIM

PSNR

13.3542 15.5577 17.0884 18.4710 13.1418 15.0303 17.1912 18.4903 12.9814 14.9623 16.6729 18.5646 12.9758 15.8468 17.5558 19.2857 13.9545 16.4418 17.6737 18.9807

PSNR

FSIM

0.6416 0.7084 0.7741 0.8069 0.6687 0.7859 0.8328 0.8458 0.6410 0.7463 0.7729 0.8305 0.7320 0.7427 0.7989 0.8186 0.7542 0.7471 0.7875 0.8234

FSIM

0.7629 0.8207 0.8457 0.8657 0.7577 0.8290 0.8524 0.9001 0.7336 0.7908 0.8171 0.8573 0.8354 0.8406 0.8792 0.9088 0.8532 0.8701 0.8837 0.8840 SSA TII FE

SSIM

17.9874 21.3558 24.4420 26.1183 17.9141 22.4243 23.0914 25.4758 19.6508 22.6367 24.7582 26.6768 20.3821 22.6270 24.7837 26.7568 19.4944 22.3010 24.2122 25.4706

PSNR

0.6434 0.7281 0.7593 0.7970 0.6706 0.7830 0.8260 0.8518 0.6437 0.7388 0.7782 0.8305 0.7312 0.7423 0.7955 0.8136 0.7565 0.7471 0.7644 0.8117

FSIM

0.7632 0.8306 0.8518 0.8658 0.7560 0.8388 0.8582 0.8835 0.7323 0.7900 0.8124 0.8288 0.8342 0.8372 0.8787 0.9114 0.8516 0.8703 0.8876 0.8891 BSA TII FE

SSIM

17.9878 21.5974 24.7064 26.8769 18.1382 22.5610 23.1801 25.9302 19.4853 23.2970 24.7839 26.0405 20.3866 22.5581 25.1892 26.6842 20.4476 22.2802 23.2834 24.4526

SSIM

PSNR

FSIM

SSIM

PSNR

18.0356 0.7524 0.6352 21.1274 0.8081 0.7019 24.7989 0.8532 0.7671 26.9951 0.8703 0.8039 18.1382 0.7560 0.6706 22.5192 0.8235 0.7832 23.0772 0.8580 0.8165 25.7913 0.8913 0.8520 19.5921 0.7331 0.6413 23.3451 0.7900 0.7417 24.2296 0.8000 0.7694 26.6758 0.8600 0.8499 20.3821 0.8354 0.7300 22.5496 0.8403 0.7426 25.1018 0.8837 0.7980 26.5525 0.9001 0.8166 20.4476 0.8616 0.7543 22.5321 0.8743 0.7580 24.8516 0.8989 0.7928 25.5574 0.9071 0.8223 k-means segmentation PNSR

FSIM 0.6453 0.7003 0.7762 0.8055 0.6689 0.7877 0.8166 0.8623 0.6431 0.7407 0.7685 0.8454 0.7303 0.7466 0.7739 0.7978 0.7435 0.7499 0.7927 0.8115

FSIM

SSIM 0.7713 0.8086 0.8590 0.8725 0.7561 0.8302 0.8574 0.8804 0.7332 0.7920 0.7979 0.8485 0.8357 0.8362 0.8622 0.8787 0.8357 0.8731 0.8912 0.9023 Soft thresholding

SFS TII FE

SSIM

PSNR 17.0091 21.2180 24.5086 26.8053 18.1911 22.3081 22.6964 24.1597 19.540 23.140 24.2377 26.6690 20.2020 22.1811 24.3278 26.1018 18.5531 22.3455 24.4831 25.6153

CSA TII-FE

PNSR

0.6373 0.7224 0.7753 0.7967 0.6701 0.7829 0.8195 0.8031 0.6413 0.7424 0.7653 0.8433 0.7289 0.7447 0.7998 0.8185 0.7602 0.7525 0.8025 0.7771

PPA TII FE

FSIM

FSIM

PSNR

Im

SSIM 0.7637 0.8148 0.8534 0.8686 0.7578 0.8255 0.8514 0.8566 0.7331 0.7885 0.8094 0.8591 0.8303 0.8337 0.8762 0.9068 0.8530 0.8714 0.9037 0.8872 CS TII FE

DE TII FE

SSIM

2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5

17.9549 22.5176 24.8165 26.7716 18.5465 22.3515 23.0478 24.6285 19.5921 23.3907 23.6901 26.5423 20.2726 22.2429 24.6913 26.4308 19.7685 22.3076 24.7260 25.1391

PSO TII FE

PNSR

2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 Levels

176035 225017 385028 147091 41004

Im 176035 225017 385028 147091 41004

Levels

GA TII FE

14.5538 16.2893 17.5711 19.1858 12.8672 15.6078 17.1968 18.5796 12.3915 15.2786 16.9347 19.0106 12.6177 16.5128 18.4792 19.8841 14.9073 15.7177 17.7221 18.6913

0.6956 0.7216 0.7662 0.8036 0.6059 0.6891 0.7847 0.8286 0.5792 0.6691 0.7136 0.7841 0.6059 0.6826 0.7664 0.8065 0.7233 0.7461 0.7869 0.8080

0.6667 0.7143 0.7409 0.7698 0.6418 0.7665 0.7903 0.8262 0.5984 0.7074 0.7396 0.7916 0.7244 0.7580 0.8033 0.8223 0.7704 0.7711 0.7953 0.7949

11.3206 13.0592 15.3468 17.1629 9.4899 13.5636 15.4757 15.5711 8.0739 13.3993 15.5290 17.3043 9.4915 14.5971 16.3831 18.9929 10.4256 12.0596 15.3829 15.3111

0.3506 0.4090 0.4968 0.5904 0.1909 0.4618 0.5096 0.5196 0.1589 0.4786 0.5536 0.5835 0.4806 0.5581 0.5921 0.6153 0.3084 0.3427 0.7172 0.7777

0.5798 0.6290 0.6519 0.6979 0.4808 0.5507 0.6486 0.6998 0.4465 0.5199 0.5897 0.6384 0.6291 0.6679 0.7028 0.7305 0.7178 0.7384 0.7350 0.7502

9.4140 11.3909 12.9628 13.7084 9.0959 11.7723 12.8643 13.5878 7.8720 11.4086 12.2610 13.7792 10.3537 12.2539 13.7102 14.4225 10.5821 12.1249 13.9827 14.6943

0.3260 0.4008 0.5432 0.5977 0.2334 0.4927 0.5671 0.6093 0.1832 0.5091 0.5686 0.6914 0.4504 0.5080 0.5713 0.5986 0.2971 0.3591 0.6181 0.6397

0.6181 0.6819 0.7378 0.7917 0.6065 0.6764 0.7889 0.8367 0.5455 0.6963 0.7503 0.8299 0.6757 0.7220 0.7598 0.7841 0.7486 0.7867 0.7932 0.8190

17.9874 22.4894 24.8073 26.8933 18.2946 22.5791 23.2756 26.0158 19.6997 23.1799 25.0070 26.7771 20.4061 22.6372 25.1233 26.7859 20.4343 22.3271 24.8768 25.5737

0.7667 0.8248 0.8528 0.8690 0.7568 0.8284 0.8601 0.8900 0.7351 0.7665 0.8091 0.8533 0.8335 0.8402 0.8805 0.9060 0.8619 0.8733 0.8991 0.8913

0.6516 0.7325 0.7846 0.8196 0.6800 0.7950 0.8296 0.8542 0.6431 0.7458 0.7813 0.8534 0.7320 0.7527 0.8012 0.8297 0.7647 0.7756 0.8127 0.8318

17.9723 22.8508 24.7575 26.8942 18.5791 22.6931 23.3818 26.5461 19.6997 23.4291 24.8123 26.9780 20.4105 22.6274 25.1925 26.7953 20.4476 22.6880 24.8936 25.6666

0.7677 0.8264 0.8578 0.8744 0.7583 0.8449 0.8616 0.8919 0.7351 0.7933 0.8000 0.8631 0.8334 0.8423 0.8813 0.9144 0.8616 0.8796 0.9055 0.9042

0.6544 0.7313 0.7856 0.8154 0.6790 0.7977 0.8285 0.8565 0.6466 0.7498 0.7805 0.8500 0.7339 0.7524 0.8074 0.8370 0.7647 0.7728 0.8222 0.8338

20

Table 3 presents the result of the image quality metrics, being PSNR, SSIM and FSIM. On all metrics a higher value is expected for a good segmentation. Moreover, the value of the three metrics increases according with the number of thresholds. The values of PSNR, FSIM and SSIM obtained by the BSA and SSA are the higher followed by the results obtained by PPA, CSA and DE. Then, the lower values scored by an ECA were generated by GA, PSO, and SFS. As depicted by Figure 6, SFS is ranked as the worst ECA in this study. Both non-evolutionary approaches score lower results than the ECAs on most cases. With this test is validated the quality of the images segmented using the thresholds computed by the BSA and SSA using the TII-FE. 4.3 Wilcoxon’s rank test The fitness (also called objective function) values of seven methods are statistically compared using a nonparametric significance proof known as the Wilcoxon’s rank test [57] that is conducted with 35 independent samples. Such proof allows assessing result differences among two related methods. The analysis is performed considering a 5% significance level over the best fitness value data corresponding to the five threshold points. Table 4 reports the p-values produced by Wilcoxon’s test for pair-wise comparison of the fitness function between two groups; the first group is defined as GA vs. BSA, PSO vs. BSA, DE vs. BSA, PPA vs. BSA, CSA vs. BSA, SFS vs. BSA, and CS vs. BSA. Meanwhile, the second group is established as GA vs. SSA, PSO vs. SSA, DE vs. SSA, PPA vs. SSA, CSA vs. SSA, BSA vs. SSA, SFS vs. SSA, and CS vs. CS. As a null hypothesis, it is assumed that there is no difference between the values of the two algorithms tested. The alternative hypothesis considers an existent difference between the values of both approaches. All p-values reported in Table 4 are less than 0.05 (5% significance level) which is strong evidence against the null hypothesis, indicating that the BSA and SSA fitness values for the performance are statistically better, and it has not occurred by chance. Table 4. p-values from the Wilcoxon rank test of the compared algorithms GA vs. BSA, PSO vs. BSA, and DE vs. BSA, PPA vs. BSA, CSA vs. BSA, SFS vs. BSA, CS vs. BSA, GA vs. SSA, PSO vs. SSA, and DE vs. SSA, PPA vs. SSA, CSA vs. SSA, SFS vs. SSA, CS vs. SSA and BSA vs. SSA.

41004

147091

385028

225017

176035

Im

Level

2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5

GA vs. BSA 6.29E-13 3.45E-12 6.07E-12 5.81E-13 6.16E-12 5.55E-13 7.79E-13 8.23E-12 5.77E-13 5.50E-13 7.45E-12 7.10E-13 6.83E-13 6.77E-13 6.16E-13 6.88E-13 1.34E-12 5.67E-13 6.46E-13 6.91E-13

PSO vs. BSA 4.62E-12 2.65E-12 7.91E-13 9.28E-12 6.61E-13 6.90E-12 8.94E-12 7.08E-13 4.99E-12 5.51E-13 6.01E-13 9.48E-12 9.80E-12 2.18E-12 6.15E-13 6.88E-13 4.19E-12 8.74E-12 6.46E-13 6.91E-13

DE vs. BSA 6.29E-13 5.70E-13 7.91E-13 5.81E-13 6.61E-13 5.55E-13 7.79E-13 7.08E-13 5.77E-13 5.51E-13 6.01E-13 7.10E-13 6.83E-13 6.77E-13 6.16E-13 6.88E-13 6.77E-13 5.67E-13 6.46E-13 6.91E-13

PPA vs. BSA 4.62E-12 2.61E-12 7.91E-13 9.26E-12 6.61E-13 6.99E-12 8.85E-12 7.08E-13 4.93E-12 5.51E-13 6.01E-13 9.54E-12 9.81E-12 2.23E-12 6.52E-13 6.88E-13 4.27E-12 8.76E-12 6.46E-13 6.91E-13

CSA vs. BSA 6.85E-13 6.51E-13 7.91E-13 6.48E-13 6.61E-13 5.36E-13 7.30E-13 7.08E-13 5.23E-13 5.51E-13 6.01E-13 6.76E-13 6.60E-13 7.03E-13 6.48E-13 6.88E-13 5.98E-13 5.34E-13 6.46E-13 6.91E-13

SFS vs. BSA 6.55E-13 6.55E-13 6.55E-13 6.55E-13 6.55E-13 6.55E-13 6.55E-13 6.55E-13 6.55E-13 6.55E-13 6.55E-13 6.55E-13 6.55E-13 6.55E-13 6.55E-13 6.55E-13 6.55E-13 6.55E-13 6.55E-13 6.55E-13

CS vs. BSA 1.45E-09 1.09E-12 6.55E-13 6.55E-13 7.48E-04 1.00E-12 6.55E-13 6.55E-13 1.22E-11 3.21E-11 6.55E-13 6.55E-13 1.33E-11 7.77E-13 6.55E-13 6.55E-13 6.55E-13 5.39E-12 7.13E-13 6.55E-13

GA vs. SSA 6.29E-13 3.48E-12 5.97E-12 5.97E-13 6.11E-12 5.55E-13 7.17E-13 8.25E-12 6.02E-13 6.22E-13 7.45E-12 6.18E-13 5.99E-13 6.77E-13 5.66E-13 5.91E-13 1.26E-12 5.67E-13 6.00E-13 6.09E-13

PSO vs. SSA 4.62E-12 2.69E-12 6.93E-13 9.29E-12 6.03E-13 6.90E-12 8.87E-12 7.32E-13 5.02E-12 6.23E-13 6.01E-13 9.39E-12 9.72E-12 2.18E-12 5.65E-13 5.91E-13 4.11E-12 8.74E-12 6.00E-13 6.09E-13

DE vs. SSA 6.29E-13 6.06E-13 6.93E-13 5.97E-13 6.03E-13 5.55E-13 7.17E-13 7.32E-13 6.02E-13 6.23E-13 6.01E-13 6.18E-13 5.99E-13 6.77E-13 5.66E-13 5.91E-13 5.97E-13 5.67E-13 6.00E-13 6.09E-13

PPA vs. SSA 4.68E-12 2.78E-12 6.93E-13 9.24E-12 6.03E-13 6.82E-12 8.86E-12 7.32E-13 4.96E-12 6.23E-13 6.01E-13 9.31E-12 9.67E-12 2.28E-12 6.30E-13 5.91E-13 4.13E-12 8.68E-12 6.00E-13 6.09E-13

CSA vs. SSA 5.49E-13 6.01E-13 6.93E-13 6.36E-13 6.03E-13 5.06E-13 7.64E-13 7.32E-13 5.76E-13 6.23E-13 6.01E-13 6.68E-13 6.29E-13 6.29E-13 6.15E-13 5.91E-13 6.30E-13 6.19E-13 6.00E-13 6.09E-13

SFS vs. SSA 6.55E-13 6.55E-13 6.55E-13 6.55E-13 6.55E-13 9.22E-13 6.55E-13 6.55E-13 6.55E-13 6.55E-13 6.55E-13 6.55E-13 6.55E-13 6.55E-13 6.55E-13 6.55E-13 6.55E-13 6.55E-13 6.55E-13 6.55E-13

CS vs. SSA 1.66E-10 3.16E-08 6.55E-13 6.55E-13 1.41E-02 2.79E-09 6.55E-13 6.55E-13 1.22E-11 5.59E-11 7.77E-13 6.55E-13 1.93E-10 7.77E-13 3.87E-12 6.55E-13 2.58E-08 4.13E-10 7.49E-12 8.47E-13

4.4 Computational time For experimental purposes the stop criteria for all algorithms is defined as a predefined number of iterations. Such value is selected after an exhaustive experimental procedure in which is detected that all the algorithms are able to reach its optimal in the first 100 iterations. However, due to the internal operators each algorithm requires a different amount of computational effort that is reflected in the time needed for find the optimal. In Table 5 is presented the computational time for GA, PSO, DE, PPA, CSA, SFS, CS, BSA, and SSA. The computational effort of the Soft Thresholding and k-means segmentation is also presented.

21

BSA vs. SSA 6.29E-13 6.06E-13 6.93E-13 5.97E-13 6.03E-13 5.55E-13 7.17E-13 7.32E-13 6.02E-13 6.23E-13 6.01E-13 6.18E-13 5.99E-13 6.77E-13 5.66E-13 5.91E-13 5.97E-13 5.67E-13 6.00E-13 6.09E-13

Table 5. Comparison of the computational time in seconds.

41004

147091

385028

225017

176035

Im

Level

GA TII FE

PSO TII FE

DE TII FE

PPA TII FE

CSA TII FE

SFS TII FE

CS TII FE

Soft Thresholding

k-means segmentation

BSA TII FE

SSA TII FE

2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5

6.8490 12.0134 17.8217 23.4835 6.4540 10.7077 17.3218 22.6779 6.6160 11.0108 17.6483 22.8566 6.6232 11.1041 17.6582 23.0714 6.7756 11.6450 17.7020 23.1366

2.2559 4.0499 6.1221 7.3950 1.8334 3.7969 5.7436 7.0746 1.8500 3.9016 5.7527 7.0907 1.8524 3.9016 5.7915 7.0921 2.1520 3.9622 5.9751 7.2522

1.9612 3.5515 5.7783 6.9704 1.6986 3.2913 4.9498 6.7191 1.7024 3.3480 5.2090 6.7222 1.7046 3.4100 5.2415 6.7473 1.8931 3.4658 5.2889 6.8172

2.1592 3.8653 6.0526 7.2350 1.7921 3.5972 5.5474 6.9255 1.8080 3.6652 5.5878 6.9332 1.8520 3.7023 5.6114 6.9775 2.0548 3.7741 5.6613 7.0957

2.0540 3.6383 5.9532 7.2018 1.7753 3.4528 5.3598 6.9042 1.7964 3.4807 5.4608 6.9153 1.8359 3.5598 5.4623 6.9415 1.9883 3.5884 5.5523 7.0331

6.6838 7.2931 7.5688 7.9598 6.5141 7.0883 7.3656 7.8625 6.5115 7.0774 7.3493 7.8583 6.6872 7.2697 7.5326 8.0514 6.5020 7.0985 7.3705 7.8698

4.9689 8.2049 10.5237 4.8136 3.4583 6.7121 10.0686 7.4410 7.6519 9.6061 10.3264 11.1209 8.4475 9.6791 5.2846 7.3339 8.7881 5.7153 4.6322 8.5422

5.3666 19.7116 24.8160 31.1126 6.1449 10.9652 17.1447 24.4380 6.9810 13.1060 20.5617 23.9101 5.4892 11.6416 32.8652 40.7489 5.4070 18.6089 22.6163 29.2993

6.2485 4.3081 4.6610 3.3599 2.8266 4.3213 5.1665 6.2933 4.6222 3.7778 5.4174 4.5821 2.4880 3.4120 7.0761 3.8486 2.6301 3.7718 5.6023 6.7196

1.9069 3.3772 5.5719 6.8019 1.6815 3.1853 4.7820 6.4863 1.6863 3.2162 4.8843 6.5036 1.6928 3.2311 4.9380 6.5116 1.8693 3.2481 5.0762 6.7145

1.8096 3.1804 5.2288 6.5690 1.6125 2.9374 4.7149 6.2189 1.6324 2.9997 4.7361 6.3765 1.6462 3.0713 4.7705 6.3868 1.6888 3.1239 4.8745 6.4760

From Table 5 it is possible to conclude that the BSA, SSA and DE requires less time to find the more accurate solution. Meanwhile, the GA is the slower ECA method in this test. Moreover, as is expected, the time increases with the number of thresholds to find. The time required by Soft Thresholding grows much more than the ECAs as the number of thresholds increases, while the time required by k-means presents smaller increments. 4.5 Segmentation of leukocytes in medical images The aim of this subsection is to verify that the use of the TII-FE in combination with BSA and SSA can be applied in images from other sources. In this sense, they are analyzed images that contain blood cells from a public dataset called Acute Lymphoblastic Leukemia Image database (ALL-IDB). All the images were collected by M. Tettamanti Research Centre for childhood leukemia and hematological diseases located in Monza, Italy [58]. The images that contains this data set are separated into two groups, for experimental purposes in this article it is considered the ALL-IDB1 in which the main challenge is the segmentation of the leukocytes (darker cells). Since, segmentation is one of the main tasks in computer vision and image processing, it is desired to obtain the best output at this step. The problem of the blood cells images is that the cells have different abnormalities that can affect the segmentation procedure. In the experiments, the nine ECAs (GA, PSO, DE, PPA, CSA, SFS, CS, BSA and SSA) are used to find the best thresholds of ten images from the dataset using the TII FE. Besides, the Soft thresholding and k-means segmentation are used. In order to graphically compare the algorithms, they are considered only five images (see Figure 7) but in the supplementary material all the resultant values are available. Im

Image

Histogram 4

x 10 6

1_ALLIDB1

5

Frequency

4

3

2

1

0 0

50

100

150

200

250

150

200

250

Gray level 4

x 10

2_ALLIDB1

6

Frequency

5

4

3

2

1

0 0

50

100 Gray level

22

4

x 10 6

3_ALLIDB1

5

Frequency

4

3

2

1

0 0

50

100

150

200

250

150

200

250

150

200

250

Gray level 4

x 10 4.5

4_ALLIDB1

4 3.5

Frequency

3 2.5 2 1.5 1 0.5 0 0

50

100 Gray level

4

x 10 5

5_ALLIDB1

4.5 4

Frequency

3.5 3 2.5 2 1.5 1 0.5 0 0

50

100 Gray level

Figure 7. Selected images from ALLIDB1 used for graphical analysis.

Similar to the experiments with benchmark images, Table 6 shows the best thresholds and fuzzy parameters obtained by each method using 2, 3, 4, and 5 thresholds. The experiments were performed following the methodology previously explained. In Table S6, the results are presented for all the images used in this section.

23

Table 6. Results after applying the optimization algorithms to segment the images from the ALLIDB1 using TII-FE.

1_ALLIDB1

2 3 4

5

2_ALLIDB1

2 3 4

5

3_ALLIDB1

2 3 4

5

4_ALLIDB1

2 3 4

5

5_ALLIDB1,

2 3 4

5

32,122,125, 160 32,101,102, 105, 110,171 19,63, 65,105, 112,118,121, 164 22,54, 61,76, 79,114,117, 119, 121,159 23,104,110, 160 26,96,101, 106, 108,166 27,68, 72, 98,99,100, 103,170 21,62, 64, 70,70,108, 119,120,129, 176 17,107,111, 156 17,101,101, 107,118,157 14,93,94, 103,103,118, 119,160 61,70,75, 108,108,111, 114,156 25,118,123, 179 36,118,122, 123, 124,189 19,62,65, 96,102,127, 129,184 20,60,62, 102,106,123, 124,140,141, 188 23,109,110, 167 28,111,113, 116, 116,180 25,50, 52, 89,90,100, 104,168 17,61,63, 84,85,111, 114,132,136, 164

77, 143 67,104, 141 41,85,115, 143 38,69, 97,118, 140 64,135 61,104, 137 48,85,100, 137 42,67,89, 120, 153 62,134 59,104,138 54,99,111, 140 38,66,92, 110, 135 72,151 77,123,157 41,81,115, 157 40,82,115, 132, 165 66,139 70,115,148 38,71,95, 136 39,74,98, 123, 150

14,124,126, 161 15,117,117, 119, 119,180 14,61, 64,119, 120,120,121, 163 14,62, 62,79, 80,122,122, 122, 123,180 17,111,112, 160 17,115,116, 116, 116,162 17,74, 74, 100,102,106, 107,184 17,44, 44, 81,81,112, 113,113,113, 184 14,99,99,160 14,112,113, 114, 117,171 14,66, 67, 106,106,106, 106,171 14,68, 68, 68,68,105, 105,107,108, 171 16,127,128, 184 16,103,104, 106, 106,202 16,55, 55, 117,117,120, 121,202 20,68, 68, 78,79,123, 124,124,124, 186 14,110,111, 168 20,106,107, 110, 117,186 14,62, 62, 94,96,104, 108,172 19,86, 86, 95,99,131, 135,160,163, 186

69, 144 66,118, 150 38,92,120, 142 38,71,101, 122, 152 64,136 66,116, 139 46,87,104, 146 31,63,97, 113, 149 57,130 63,114, 144 40,87, 106,139 41,68,87, 106, 140 72,156 60,105, 154 36,86,119, 162 44,73,101, 124,155 62,140 63,109, 152 38,78,100, 140 53,91,115, 148, 175

14,125, 126, 161 14,112,113, 121, 122,163 14,63, 63, 110,110,117, 121,164 15,66, 66, 86,86,117, 117,119, 120, 163 20,117,117, 160 20,115,116, 118,118,160 17,58, 59, 111,111,112, 113,167 17,53,55, 70,70,112, 113,113,125, 161 14,115,115, 157 14,109,109, 111, 111,159 14,63, 65, 110,111,111, 112,159 18,59, 59, 66,66,115, 116,117,118, 161 16,111,112, 182 21,110,110, 127, 127,179 16,66, 67, 115,115,115, 116,183 17,53,56, 84,84,120, 120,122,123, 184 16,115,116, 168 15,109,111, 112,113,167 14,48,48, 109,110,112, 116,167 14,54, 54, 69,70,112, 115,115,116, 170

70, 144 67,117, 143 39,87,114, 143 41,76,102, 118 142 69,139 68,117, 139 38,85,112, 140 35,63,91, 113, 143 65,136 62,110, 135 39,88,111, 136 39,63,91, 117, 140 64,147 66,119, 153 41,91,115, 150 35, 70,102, 121,154 66,142 62,112,140 31,79,111, 142 34,62,91, 115, 143

14,58,58, 124 14,117,117, 117,117,164 14,64, 64, 122, 122,122, 122,164 14,61, 61, 75,75,108, 108,115,115, 164 24, 115,115, 160 24,115,115, 115, 115,161 17,58, 58, 114,114,114, 114,161 18,61, 61, 61,61,110, 110,110,110, 161 14, 114,114, 156 20,116,116, 116,116,158 16,70, 70, 109,109,110, 110,157 15,62, 62, 68,68,114, 114,114,115, 159 23,129,129, 178 19,119,119, 119, 119,182 17,128,128, 129, 129,176, 176,202 17,77, 77, 107,107,133, 133,174,174, 200 20,115,115, 166 15,112,112, 112, 112,168 15,92, 92, 120,120,121, 121,165 15,53, 53, 73,73,113, 113,113,113, 167

39,91 66,117,141 39,93,122, 143 38,68,92, 112, 140 70,138 70,115, 138 38,86,114, 138 40,61,86, 110, 136 64,135 68,116, 137 43,90,110, 134 39,65,91, 114, 137 76,154 69,119, 151 73,129,153, 189 47,92,120, 154, 187 68,141 64,112, 140 54,106,121, 143 34,63,93, 113, 140

14,121,121, 162 19,119,119, 118,119, 164 14,54, 55, 115,116,122, 122,165 18,60, 61, 75,75,119, 120,120,121, 164 23,116,116, 160 23,116,116, 116, 116,161 17, 58, 58, 114, 114,114,114, 161 18,61, 61, 61,61,110, 110,110,110, 161 15,58, 59, 116 23,113,113, 113, 113,160 15,55, 55, 114,114,115, 115,164 19,60, 61, 62,63,112, 112,112,113, 162 17,129,129, 178 24,124,124, 124, 124,183 19,56, 57, 119,119,120, 120,183 21,70, 71, 96,96,122, 123,123,124, 179 20,116,116, 166 22,115,115, 115,115,167 17,57,57, 113,113,113, 114,168 20,60, 60, 65,66,111, 112,112,113, 172

24

68,142 69,119,142 34,85119, 144 39,68,97, 120, 143 70,138 70,116, 139 44,90,112, 137 44,65,81, 101, 135 37,88 68,113, 137 35,85,115, 140 40,62,88, 112, 138 73,54 74,124, 154 38,88,120, 152 46,84,109, 123, 152 68,141 69,115, 141 37,85,113, 141 40,63,89, 112, 143

5,85,101, 226 2,42,55, 66,100,174 37,64,65, 67,78,103, 114,202 20,32,34, 79,94,102, 129,169,174, 196 20,104,121,157 9,55,87, 104,106,221 27,51,68, 73,76,135, 149,224 12,40,63, 88,93,116, 123,151,191, 236 10,54,104,226 12,31,79, 106,109,146 4,19,23, 46,70,81, 120,171 15,68,74, 106,111,118, 128,213,213, 254 36,93,103, 204 7,65,77, 98,139,186 2,81,88, 123,130,149, 172, 205 38,81,82, 83,99,159, 166,180,187, 245 27,152,162,230 24,85,103, 110,127,249 7,61,114, 143,146,161, 162,174 15,62,72, 86,88,95, 127,143,161, 181

45,163 32,60,137 50,66,90, 158 26,56,98, 149,185 62,139 32,95,163 39,70,105, 186 26,75,104, 137,213 32,165 21,92,127 11,34,75, 145 41,90,114, 170,233 64,153 36,87,162 41,105,139, 188 59,82,129, 173,216 89,196 54,106,188 34,128,153, 168 38,79,91, 135,171

1,54,55, 124 1,112,112, 114,114,186| 7,82,82, 100,105,116, 117, 170 1,55,56, 98,98,127, 133,159,160, 251 15,111,112,161 1,91,97, 114,117,162 10,63,69, 113,119,119, 120,252 9,68,68, 84,89,110, 110,168,172, 231 18,113,114, 157 8,102,104, 113,114,169 6,54,56, 101,105,105, 111,207 14,61,62, 76,77,83, 84,107,115, 217 3,127,127, 179 2,103,106, 120,122,180 1,67,77, 96,98,115, 118,186 5,66,68, 112,112,119, 119,137,138, 235 23,112,112, 169 2,94,95,121, 122,167 3,44,45, 108,109,114, 116,211 2,48,52, 99,99,117, 117,130,131, 253

27,89 56,113,150 44,91,110, 143 28,77,112, 146,205 63,136 46,105,139 36,91,119,1 86 38,76,99,13 9,201 65,135 55,108,141 30,78,105, 159 37,69,80,95 ,166 65,153 52,113,151 34,86,106,1 52 35,90,115,1 28,186 67,140 48,108,144 23,76,111,1 63 25,75,108,1 23,192

14,120,120, 163 14,117,118, 121,121,164 14,62, 63, 116, 117,117,118, 164 14,58,59, 87,89,115, 115,116,116, 164 23,116,116, 161 23,114,115, 115, 115,161 17,64, 65, 113,113,113, 113,160 17,55,57, 65,66,111, 113,113,113, 162 14,114,114, 156 14,110,111, 112,112,161 14,58,59, 105,106,109, 111,161 14,60, 60, 78,80,116, 116,117,119, 160 17,127,127,178 16,124,125, 125, 126,180 16,64, 65, 123,123,124, 125,180 21,61, 62, 90,92,126, 128,12,129, 190 17,115,115, 167 19,115,116, 116, 116,170 14,63, 64, 111,112,112, 113,169 14,54, 55, 68,69,110, 119,119,120,1 69

67, 142 66,120,143 38,90, 117, 141 36,73,102, 116, 140 70,139 69,115, 138 41,89,113, 137 36,61,89, 113, 138 64,135 62,112, 137 36,82,108, 136 37,69,98, 117, 140 72,153 70,125, 153 40,94,124, 153 41,76,109, 128,160 66,141 67,116, 143 39,88,112, 141 34,62,90, 119, 145

14,123,23, 162 14,124,124, 124,124,163 14,64,64, 122,122,122, 122,164 14,67, 67, 67,67,121, 121,121,121, 163 21,117,117, 160 24,119,119, 119, 119,160 17, 61, 62, 114,114,114, 114,161 18,62, 62, 62,62,110, 110,110,110, 161 14,115,115,156 17,116,116, 116,116,157 14,62, 62, 116,116,116, 116,158 15,64, 64, 64,64,115, 115,116,116, 159 16,129,129,178 21,128,128, 128, 128,179 17,65, 66, 125,125,125, 125,180 17,68, 68, 69,69,123, 123,123,123, 180 16,116,116, 166 20,116,116, 116, 116,168 15,57, 57, 113,113,113, 113,169 15,59, 59, 59,59,111, 112,112,112, 170

Thresholds

Fuzzy Parameters

SSA TII-FE Thresholds

Fuzzy Parameters

BSA TII-FE Thresholds

Fuzzy Parameters

CS TII FE Thresholds

Fuzzy Parameters

SFS TII FE Thresholds

Fuzzy Parameters

CSA TII-FE Thresholds

Fuzzy Parameters

PPA TII-FE Thresholds

Fuzzy Parameters

DE TII-FE Thresholds

Fuzzy Parameters

PSO TII-FE Thresholds

Fuzzy Parameters

Levels

Im

GA TII-FE

69, 143 69,124,144 39,93,122, 143 41,67,94, 121, 142 69,139 72, 119,140 39,88,114, 138 40,62,86, 110, 136 65,136 67,116, 137 38,89,116, 137 40,64,90, 116, 138 73,154 75,128, 154 41,96,125, 153 43,69,96, 123, 152 66,141 68,116, 142 36,85,113, 141 37,59,85, 112, 141

The performance of the thresholds values obtained by each algorithm is evaluated according to the objective function value. In this context, Table 7 presents the Mean and the Standard deviation (STD) of the best solutions found by each algorithm in the 35 experiments over each image. Similar to the experiments with benchmark images, the best (higher) values of mean and STD are obtained by the BSA an SSA. It can be interpreted as both BSA and SSA are more accurate than GA, PSO, DE, PPA, CSA, SFS, and CS. However, it is possible to say that after BSA and SSA the following algorithms in terms of accuracy are the PPA, CSA and DE. Finally, the lest accurate algorithms are the GA and PSO. Table 7. Mean and standard deviation of the objective function values obtained by GA, PSO, DE, PPA, CSA,SFS, CS BSA, and SSA for the segmentation of images from blood celld using TII-FE.

STD

Mean

STD

11.5275 14.5362 16.7059 18.6665 11.2308 13.9932 16.0181 17.9974 11.3325 14.2369 16.2815 18.0741 12.1925 15.0609 17.5082 19.6716 11.8561 14.7369 16.8964 18.6923

1.82E-01 4.30E-01 5.11E-01 5.95E-01 1.87E-01 3.11E-01 3.61E-01 6.16E-01 1.60E-01 4.02E-01 6.44E-01 5.30E-01 2.15E-01 3.73E-01 3.60E-01 4.45E-01 1.74E-01 2.84E-01 5.03E-01 5.68E-01

11.9410 15.3292 17.9658 19.7266 11.6803 14.7757 17.1898 19.3829 11.7148 15.1525 17.7869 19.5820 12.6143 15.8048 18.3845 20.6568 12.2898 15.4504 18.0230 20.0461

4.83E-02 1.46E-01 2.83E-01 9.59E-01 5.25E-02 1.38E-01 2.31E-01 5.07E-01 4.78E-02 1.32E-01 2.44E-01 7.38E-01 5.62E-02 1.30E-01 2.45E-01 4.34E-01 5.06E-02 1.36E-01 1.96E-01 3.88E-01

12.0605 15.8887 18.9997 22.0303 11.8848 15.5056 18.5108 21.4156 11.8143 15.6510 18.8158 21.7328 12.4599 16.0993 19.1516 21.9013 23.3307 29.7622 29.7247 31.3009

8.17E-08 9.40E-10 1.26E-08 4.21E-08 3.39E-08 2.41E-08 1.00E-08 4.61E-08 3.37E-08 7.22E-08 1.58E-08 4.07E-08 4.84E-08 8.73E-08 6.86E-08 8.85E-08 0.9097 0.9317 0.9290 0.9424

12.0645 15.8840 19.1143 22.1994 11.8858 15.5188 18.6239 21.7045 11.8220 15.7368 18.8316 21.8853 12.4514 16.1098 19.1805 22.1115 23.3427 29.8026 29.8180 31.3089

3.40E-09 3.01E-09 7.70E-10 5.06E-09 2.53E-10 1.62E-09 2.19E-10 1.02E-08 2.34E-09 3.27E-09 3.21E-09 4.01E-11 3.30E-09 6.93E-09 4.75E-09 2.10E-09 0.9124 0.9387 0.9372 0.9430

STD

CSA TII-FE Mean

STD

Mean

12.0329 3.89E-08 15.8719 3.13E-08 18.9951 1.11E-07 21.8997 5.56E-08 11.8590 1.34E-08 15.5008 1.11E-08 18.1379 2.58E-09 20.6980 1.59E-08 11.7810 5.46E-08 15.5892 4.27E-08 18.6124 1.53E-08 21.5043 1.07E-07 12.4473 2.07E-08 16.0829 1.49E-08 19.1049 4.90E-08 21.2842 1.18E-07 23.2665 0.9097 29.3773 0.9337 29.4172 0.9313 30.3774 0.9406 SSA TII-FE

STD

12.0219 7.53E-08 15.7324 2.89E-08 18.8291 1.35E-07 21.7955 4.66E-08 11.8771 8.27E-09 15.3657 2.18E-08 18.3512 3.84E-08 20.9149 1.77E-08 11.8072 1.34E-08 15.6532 1.95E-08 18.6512 9.28E-08 21.6573 1.31E-08 12.4311 1.33E-08 15.8956 9.06E-08 18.8512 5.13E-08 21.6838 5.26E-09 23.0495 0.9095 29.1277 0.9310 27.2384 0.9215 31.0013 0.9426 BSA TII-FE

Mean

STD

PPA TII-FE

Mean

12.0294 6.22E-04 15.5756 1.31E-04 18.9133 3.02E-04 21.7399 4.81E-04 11.8249 4.51E-05 15.4169 6.09E-05 17.6675 3.52E-04 20.6304 1.04E-04 11.7217 1.89E-04 15.3847 8.34E-05 18.4437 7.82E-04 21.4419 2.25E-03 12.4120 2.22E-05 15.5546 1.50E-03 18.7068 7.24E-04 20.4295 2.25E-04 22.9229 0.9029 28.9263 0.9262 27.0916 0.9196 26.4572 0.8888 CS TII-FE

Mean

STD

Mean

STD

11.8551 2.61E-03 15.3211 1.27E-03 18.4635 5.91E-04 20.8189 6.32E-04 11.7025 6.11E-03 14.9986 1.41E-02 17.6118 3.28E-03 20.1314 6.91E-04 11.7056 4.47E-03 15.2643 6.66E-03 17.8684 1.34E-02 21.1859 6.49E-04 12.4089 8.66E-03 15.7627 2.05E-03 18.3306 1.80E-03 20.7844 2.80E-03 22.6394 0.9013 28.9424 0.9304 26.4073 0.9171 26.6486 0.9388 SFS TII-FE

DE TII-FE

STD

2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5

Mean

Levels Levels

2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5

PSO TII-FE

Mean

5_ALLI DB1

4_ALLI DB1

3_ALLI DB1

2_ALLI DB1

1_ALLI DB1

Im

5_ALLI DB1

4_ALLI DB1

3_ALLI DB1

2_ALLI DB1

1_ALLI DB1

Im

GA TII-FE

12.0523 15.8043 18.8893 21.6892 11.8652 15.4727 18.1314 21.3548 11.7908 15.6103 18.7133 21.5578 12.4404 16.7943 19.0273 21.5204 23.2665 29.4550 29.4809 30.7563

1.77E-08 7.61E-08 4.36E-08 9.91E-08 4.74E-08 5.88E-08 2.79E-09 2.26E-08 1.41E-08 6.14E-08 1.37E-08 7.80E-08 7.02E-08 9.61E-08 6.31E-08 5.90E-08 0.9121 0.9371 0.9321 0.9420

The best thresholds obtained by the BSA are presented in Figure 8 in which the thresholded histograms are plotted. Moreover, this figure also shows the images obtained by the segmentation process using the thresholds values generated using BSA and TII-FE. The aim of using the images from blood cells of the ALLIDB1 dataset is to detect leukocytes that are the darker objects contained in the scene. Considering this situation, the use of 2 thresholds probably is enough to divide the objects. However, for the detection of abnormalities in the cells they are necessary more details about the objects. In such case, there use of more than 2 thresholds is the best option. In Figure 8 it can be observed that the images segmented using more than 2 thresholds present more details of the blood cell.

25

4_ALLIDB1

3_ALLIDB1

2_ALLIDB1

1_ALLIDB1

Im 2 3 Level 4

26 5

5_ALLIDB1

Figure 8. Segmented images and thresholded histograms obtained using BSA with TII-FE over the imagres from ALLIDB1.

Figure 9 presents the thresholded histograms and the segmented images generated by the SSA and TII-FE. Similar to Figure 7 the results confirm that the SSA provide good results for this kind of images. As is expected using more thresholds represent more information segmented in the scene. Level 2

3

4

2_ALLIDB1

1_ALLIDB1

Im

27

5

3_ALLIDB1 4_ALLIDB1 5_ALLIDB1

Figure 9. Segmented images and thresholded histograms obtained using SSA with TII-FE over the imagres from ALLIDB1.

The results from Tables 6-7 and Figure 8-9 provide evidence of the capabilities of BSA and SSA to find the best parameters of the TII-FE and generate the accurate threshold. However, such information doesn’t measure the quality of the segmented images. Table 8 presents the values of the quality metrics obtained after

28

applying the thresholds over the test images. Such values provide evidence that the segmented images obtained using the thresholds computed by the TII-FE have better quality, in specific the computed by BSA and SSA. As is expected for the experiments in blood cells images, the second rank of algorithms is for the PPA, CSA and DE. The algorithm in third rank is GA followed by CS and SFS and that presents the lower value for PSNR, SSIM and FSIM for ECAs. The non-evolutionary algorithms behave as in the previous experimental setup obtaining the worst results.

29

Table 8. Comparison of the PSNR, SSIM and SSIM values of the GA, PSO, DE, PPA, CSA, SFS, CS, Soft Thresholding, k-means segmentation, BSA and SSA applied over the selected images from ALLIDB1 using the TII-FE.

PSNR

SSIM

FSIM

0.7017 0.8159 0.8520 0.8920 0.7331 0.8173 0.8739 0.8887 0.6897 0.8204 0.8708 0.8916 0.6976 0.7981 0.8695 0.8862 0.6763 0.7948 0.8548 0.8824

0.7203 0.7824 0.8063 0.8378 0.7597 0.7977 0.8369 0.8473 0.7297 0.7977 0.8276 0.8578 0.7741 0.8165 0.8543 0.8611 0.7334 0.7962 0.8278 0.8623

FSIM

FSIM

SSIM

13.5690 16.5492 17.8097 19.7905 14.6202 16.9695 19.1377 19.8238 13.5820 17.1723 19.3887 20.0388 13.6999 16.4406 18.7446 20.0017 13.6339 16.5399 18.6311 19.9212

SSIM

PSNR

0.8744 0.9193 0.9177 0.9495 0.8833 0.9272 0.8928 0.9423 0.8764 0.9379 0.9541 0.9646 0.8785 0.9234 0.9271 0.9405 0.8810 0.9296 0.9282 0.9394

PSNR

FSIM

0.9246 0.9464 0.9451 0.9542 0.9213 0.9423 0.9327 0.9581 0.9158 0.9502 0.9572 0.9617 0.9158 0.9502 0.9572 0.9617 0.9044 0.9207 0.9268 0.9352 SSA TII FE

FSIM

SSIM

24.7069 30.5662 30.5304 32.0894 24.0025 30.4042 28.7076 30.9396 23.3605 30.7537 31.8316 32.1602 23.3605 30.7537 31.8316 32.1602 22.7508 28.1308 28.8783 30.0121

SSIM

0.8645 0.9236 0.9097 0.9302 0.8774 0.9312 0.9027 0.9067 0.7047 0.9422 0.9542 0.9604 0.8782 0.9250 0.9126 0.9272 0.8818 0.9329 0.9166 0.9027

PSNR

0.9208 0.9483 0.9499 0.9535 0.9105 0.9426 0.9368 0.9444 0.9129 0.9565 0.9592 0.9630 0.9129 0.9565 0.9592 0.9630 0.9056 0.9231 0.9265 0.9333 BSA TII FE

FSIM

23.5930 30.6171 31.0612 31.6118 23.7036 30.4098 29.7748 29.5000 23.0760 31.9709 32.4850 33.6902 23.0760 31.9709 32.4850 33.6902 22.7654 28.3307 28.7476 25.7457

SSIM

PSNR

FSIM

SSIM

PSNR

0.8747 23.5225 0.9147 0.9316 28.9458 0.9388 30.1793 0.9419 0.9355 31.4680 0.9521 0.9414 24.0025 0.9213 0.8887 28.5401 0.9335 0.9427 29.8568 0.9403 0.9452 29.7899 0.9455 0.9571 23.3636 0.9225 0.8842 31.7512 0.9555 0.9484 31.6025 0.9569 0.9513 33.4840 0.9624 0.9718 23.3636 0.9225 0.8539 31.7512 0.9555 0.9288 31.6025 0.9569 0.9177 33.4840 0.9624 0.9433 22.4674 0.9024 0.8819 28.0986 0.9202 0.9325 28.9584 0.9281 0.9335 29.6920 0.9336 0.9554 k.means segmentation PNSR

0.8738 0.9457 0.9321 0.9599 0.8737 0.9431 0.9430 0.9504 0.8453 0.9553 0.9533 0.9596 0.8762 0.9250 0.9176 0.9516 0.8747 0.9265 0.9287 0.9506

FSIM

0.9198 0.9372 0.9491 0.9479 0.9073 0.9359 0.9279 0.9392 0.9064 0.9435 0.9544 0.9551 0.9064 0.9435 0.9544 0.9551 0.8990 0.9165 0.9259 0.9363 Soft thresholding

FSIM

SSIM

23.5847 28.1928 30.9477 30.8889 23.4853 29.1391 27.6229 26.2334 22.8470 30.2657 31.2983 31.9703 22.8470 30.2657 31.2983 31.9703 22.4582 27.5585 28.3832 30.2636

SFS TII FE

SSIM

PSNR

0.8830 0.9164 0.9374 0.9310 0.8698 0.9271 0.9263 0.9512 0.8715 0.9503 0.9595 0.9559 0.8776 0.9238 0.9322 0.9412 0.8793 0.9351 0.9150 0.9604

CSA TII FE

PNSR

FSIM

SSIM 0.9132 0.9365 0.9408 0.9464 0.9060 0.9332 0.9275 0.9289 0.8932 0.9412 0.9528 0.9541 0.8932 0.9412 0.9528 0.9541 0.8801 0.9176 0.9231 0.9325 CS TII FE

PPA TII FE

FSIM

23.4757 28.1355 29.6914 30.5978 23.0449 28.3112 27.3348 21.5738 22.5651 30.1724 31.2794 31.9257 22.5651 30.1724 31.2794 31.9257 21.7985 28.0794 27.4754 24.9578

DE TII FE

SSIM

2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5

PSNR

Levels Levels

2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5

PSO TII FE

PNSR

5_ALLI DB1

4_ALLI DB1

3_ALLI DB1

2_ALLI DB1

1_ALLI DB1

Im

5_ALLI DB1

4_ALLI DB1

3_ALLI DB1

2_ALLI DB1

1_ALLI DB1

Im

GA TII FE

13.2829 21.3756 21.8292 21.3708 16.3983 21.0546 21.3856 21.3960 13.7737 22.1312 21.7451 20.6775 14.4474 19.5100 20.7638 21.0415 15.3076 20.3690 20.4492 21.2277

0.7629 0.9367 0.9466 0.9246 0.7971 0.9270 0.9417 0.9325 0.7565 0.9470 0.9365 0.8970 0.7363 0.8955 0.9352 0.9250 0.7498 0.9076 0.9043 0.9121

0.7457 0.9155 0.9170 0.8755 0.8509 0.9258 0.9139 0.8921 0.7332 0.9415 0.9058 0.8315 0.8400 0.9021 0.9104 0.8987 0.8437 0.9190 0.8718 0.8824

8.0017 10.8763 10.7226 10.7834 8.5721 12.9383 12.7532 13.2085 7.6994 10.8867 10.6565 10.5660 7.3046 10.5951 10.2978 10.3294 7.7110 11.8269 11.6605 11.6450

0.7912 0.8137 0.7847 0.8067 0.7314 0.7658 0.7274 0.7737 0.7887 0.8240 0.7825 0.8063 0.4860 0.8224 0.7656 0.7926 0.7374 0.7854 0.7562 0.7557

0.7096 0.7311 0.7400 0.7605 0.6193 0.7519 0.7491 0.8058 0.6842 0.7451 0.7275 0.7783 0.5768 0.7165 0.6831 0.7306 0.6331 0.7337 0.7458 0.7671

10.5858 13.3704 13.3704 15.0958 10.3312 10.2434 12.5465 14.8158 10.3328 13.1690 13.1690 14.9453 5.2467 9.8017 13.3689 13.3873 10.6453 12.8895 14.7244 15.3785

0.7323 0.7937 0.7937 0.8523 0.6769 0.6731 0.7176 0.7934 0.7144 0.7886 0.7886 0.8245 0.0168 0.5344 0.7946 0.7949 0.6898 0.7375 0.7858 0.8229

0.7297 0.8790 0.8790 0.9458 0.6587 0.6589 0.8756 0.9398 0.6959 0.8904 0.8904 0.9118 0.6198 0.7168 0.8926 0.8886 0.6687 0.8804 0.9099 0.9469

24.6201 30.4130 31.0380 30.8503 23.9239 30.3776 30.2558 30.5211 23.4330 31.2976 31.5035 32.0101 23.4330 31.2976 31.5035 32.0101 22.6324 28.3181 28.9659 30.0034

0.9242 0.9451 0.9498 0.9474 0.9177 0.9419 0.9466 0.9481 0.9225 0.9545 0.9563 0.9595 0.9225 0.9545 0.9563 0.9595 0.9039 0.9209 0.9345 0.9345

0.8635 0.9337 0.9249 0.9272 0.8913 0.9388 0.9319 0.9483 0.8815 0.9558 0.9498 0.9703 0.8782 0.9242 0.9234 0.9482 0.8811 0.9354 0.9376 0.9601

25.0170 30.8877 31.4586 32.4356 24.2803 30.8353 30.6781 31.5073 23.4330 32.0465 32.5036 33.7286 23.4330 32.0465 32.5036 33.7286 22.9143 28.7768 29.1890 30.4525

0.9270 0.9524 0.9517 0.9596 0.9228 0.9454 0.9551 0.9598 0.9548 0.9583 0.9619 0.9641 0.9548 0.9583 0.9619 0.9641 0.9062 0.9277 0.9365 0.9370

0.8689 0.9376 0.9378 0.9472 0.8887 0.9461 0.9383 0.9342 0.8842 0.9559 0.9568 0.9692 0.8799 0.9195 0.9239 0.9508 0.8811 0.9335 0.9369 0.9539

30

5. Conclusions This paper introduces the implementation of two optimization algorithms for the problem of multilevel image thresholding. These two approaches have been recently proposed in the related literature; one of them is the BSA that uses three basic genetic operators. Meanwhile, SSA is inspired in the natural behavior of salps. In this paper, the BSA and the SSA are used to estimate the best parameters of the TII-FE that generate the best thresholds for image segmentation. The proposed approaches use the histogram of the image as a search space and by applying the optimization operators, they are able to find the best solutions. The proposed thresholding methods based on BSA and SSA have been tested over a set of benchmark images and over a set of images from blood cells extracted from the ALLIDB1 dataset. Moreover, considering that BSA and SSA are relatively novel approaches, they have been compared with GA, PSO, DE, PPA, CSA, SFS, CS, Soft Thresholding, and k-means segmentation; the last two being non-evolutionary approaches. From the experimental results, we can conclude that BSA and SSA outperform all the other algorithms in terms of accuracy and computational time. However, when the BSA and SSA results are compared, it is possible to conclude that they are very similar. Moreover, the BSA and SSA can be trapped in local optimal values if they are not properly tuned. This situation occurs in most of the optimization algorithms. However, according with the No-Free Lunch theorem both the BSA and SSA are good alternatives for image segmentation using TII-FE but this fact doesn’t represent that such methods are the best for all the optimization problems. On the other hand, the segmentation process in this paper is a multidimensional problem since 8-bit images are used; the number of thresholds is set to 2, 3, 4, and 5. For that is also proved that the BSA and SSA are able to find the best solutions in multidimensional optimization problems. In the supplementary material file, the reader can also see that the results over different images using BSA and SSA are better than other similar approaches based on TII-FE. Considering the results obtained, the future work might include the implementation of BSA or SSA for medical images (MRI, blood cells, or thermal).

References [1]

Q. Miao, P. Xu, T. Liu, J. Song, X. Chen, A novel fast image segmentation algorithm for large topographic maps, Neurocomputing. 168 (2015) 808–822. doi:10.1016/j.neucom.2015.05.043.

[2]

M. Elhoseny, E. Essa, A. Elkhateb, A.E. Hassanien, A. Hamad, Cascade Multimodal Biometric System Using Fingerprint and Iris Patterns, in: A. Hassanien, K. Shaalan, T. Gaber, M. Tolba (Eds.), Proc. Int. Conf. Adv. Intell. Syst. Informatics 2017. AISI 2017., Springer Cham, 2017. doi:10.1007/978-3-319-64861-3.

[3]

A. Mostafa, A.E. Hassanien, M. Houseni, H. Hefny, Liver segmentation in MRI images based on whale optimization algorithm, Multimed. Tools Appl. (2017) 1–24. doi:10.1007/s11042-017-4638-5.

[4]

R. Gharbia, A.E. Hassanien, A.H. El-Baz, M. Elhoseny, M. Gunasekaran, Multi-spectral and panchromatic image fusion approach using stationary wavelet transform and swarm flower pollination optimization for remote sensing applications, Futur. Gener. Comput. Syst. 88 (2018) 501–511. doi:https://doi.org/10.1016/j.future.2018.06.022.

[5]

I. Riomoros, G. Pajares, P.J. Herrera, M. Guijarro, X.P. Burgos-Artizzu, A. Ribeiro, Automatic image segmentation of greenness in crop fields, Proc. 2010 Int. Conf. Soft Comput. Pattern Recognition, SoCPaR 2010. (2010) 462–467. doi:10.1109/SOCPAR.2010.5685936.

[6]

A.K. Bhandari, A. Kumar, G.K. Singh, Tsallis entropy based multilevel thresholding for colored satellite image segmentation using evolutionary algorithms, Expert Syst. Appl. 42 (2015) 8707–8730.

31

doi:10.1016/j.eswa.2015.07.025. [7]

P.. Sahoo, S. Soltani, A.K.. Wong, A survey of thresholding techniques, Comput. Vision, Graph. Image Process. 41 (1988) 233–260. doi:10.1016/0734-189X(88)90022-9.

[8]

A. Shehab, M. Elhoseny, K. Muhammad, A.K. Sangaiah, P. Yang, H. Huang, G. Hou, Secure and robust fragile watermarking scheme for medical images, in: IEEE Access, 2018: pp. 10269–10278. doi:10.1109/ACCESS.2018.2799240.

[9]

R.C. Gonzalez, R.E. Woods, Digital Image Processing, Pearson, Prentice-Hall, New Jersey, 1992.

[10]

S. Kumar, P. Kumar, T.K. Sharma, M. Pant, Bi-level thresholding using PSO, Artificial Bee Colony and MRLDE embedded with Otsu method, Memetic Comput. 5 (2013) 323–334. doi:10.1007/s12293013-0123-5.

[11]

N. Otsu, THRESHOLD SELECTION METHOD FROM GRAY-LEVEL HISTOGRAMS., IEEE Trans Syst Man Cybern. SMC-9 (1979) 62–66.

[12]

A.K.C. J. N. Kapur, P. K. Sahoo, A. K. C. Wong, A new method for gray-level picture thresholding using the entropy of the histogram, (1985) 273–285.

[13]

P.D. Sathya, R. Kayalvizhi, Optimal multilevel thresholding using bacterial foraging algorithm, Expert Syst. Appl. 38 (2011) 15549–15564. doi:10.1016/j.eswa.2011.06.004.

[14]

S. Agrawal, R. Panda, S. Bhuyan, B.K. Panigrahi, Tsallis entropy based optimal multilevel thresholding using cuckoo search algorithm, Swarm Evol. Comput. 11 (2013) 16–30. doi:10.1016/j.swevo.2013.02.001.

[15]

R. Benzid, D. Arar, M. Bentoumi, A fast technique for gray level image thresholding and quantization based on the entropy maximization, in: 5th Int. Multi-Conference Syst. Signals Devices, IEEE, Amman, 2008: pp. 1–4. doi:10.1109/SSD.2008.4632831.

[16]

S. Sarkar, S. Das, S.S. Chaudhuri, Multilevel Image Thresholding Based on Tsallis Entropy and Differential Evolution., in: Swarm, Evol. Memetic Comput. SEMCCO 2012., 2012. doi:10.1007/9783-642-35380-2_3.

[17]

P.K. Sahoo, G. Arora, A thresholding method based on two-dimensional Renyi’s entropy, Pattern Recognit. 37 (2004) 1149–1161. doi:10.1016/j.patcog.2003.10.008.

[18]

S. Lan, L.I.U. Li, Z. Kong, J.G. Wang, Segmentation Approach Based on Fuzzy Renyi Entropy, (2010).

[19]

W. Tian, Y. Geng, J. Liu, L. Ai, Maximum Fuzzy Entropy and Immune Clone Selection Algorithm for Image Segmentation, in: 2009 Asia-Pacific Conf. Inf. Process., IEEE, Shenzhen, 2009: pp. 38–41. doi:10.1109/APCIP.2009.18.

[20]

H.R. Tizhoosh, Fuzzy Image Processing (In German), Springer Berlin Heidelberg, Heidelberg, 1998. doi:10.1007/978-3-642-58742-9.

[21]

H.R. Tizhoosh, Type II fuzzy image segmentation, Stud. Fuzziness Soft Comput. 220 (2008) 607– 619. doi:10.1007/978-3-540-73723-0_31.

32

[22]

O. Castillo, M. Sanchez, C. Gonzalez, G. Martinez, Review of Recent Type-2 Fuzzy Image Processing Applications, Information. 8 (2017) 97. doi:10.3390/info8030097.

[23]

H.R. Tizhoosh, Image thresholding using type II fuzzy sets, Pattern Recognit. 38 (2005) 2363–2372. doi:10.1016/j.patcog.2005.02.014.

[24]

R. Burman, S. Paul, S. Das, A differential evolution approach to multi-level image thresholding using type II fuzzy sets, Lect. Notes Comput. Sci. (Including Subser. Lect. Notes Artif. Intell. Lect. Notes Bioinformatics). 8297 LNCS (2013) 274–285. doi:10.1007/978-3-319-03753-0_25.

[25]

D. Oliva, E. Cuevas, G. Pajares, D. Zaldivar, V. Osuna, A Multilevel thresholding algorithm using electromagnetism optimization, Neurocomputing. 139 (2014) 357–381.

[26]

D. Oliva, E. Cuevas, G. Pajares, D. Zaldivar, M. Perez-Cisneros, Multilevel thresholding segmentation based on harmony search optimization, J. Appl. Math. 2013 (2013). doi:10.1155/2013/575414.

[27]

D.H. Wolpert, W.G. Macready, No free lunch theorems for optimization, IEEE Trans. Evol. Comput. 1 (1997) 67–82. doi:10.1109/4235.585893.

[28]

S. Mirjalili, A.H. Gandomi, S.Z. Mirjalili, S. Saremi, H. Faris, S.M. Mirjalili, Salp Swarm Algorithm: A bio-inspired optimizer for engineering design problems, Adv. Eng. Softw. 114 (2017) 163–191. doi:10.1016/j.advengsoft.2017.07.002.

[29]

P. Civicioglu, Backtracking Search Optimization Algorithm for numerical optimization problems, Appl. Math. Comput. 219 (2013) 8121–8144. doi:10.1016/j.amc.2013.02.017.

[30]

H.T. Ibrahim, W.J. Mazher, O.N. Ucan, O. Bayat, Feature Selection using Salp Swarm Algorithm for Real Biomedical Datasets, IJCSNS Int. J. Comput. Sci. Netw. Secur. 17 (2017) 13–20.

[31]

A.A. El-Fergany, Extracting optimal parameters of PEM fuel cells using Salp Swarm Optimizer, Renew. Energy. 119 (2018) 641–648. doi:10.1016/j.renene.2017.12.051.

[32]

K. Guney, A. Durmus, S. Basbug, Backtracking Search Optimization Algorithm for Synthesis of Concentric Circular Antenna Arrays, Int. J. Antennas Propag. 2014 (2014). doi:10.1155/2014/250841.

[33]

A. El-Fergany, Optimal allocation of multi-type distributed generators using backtracking search optimization algorithm, Int. J. Electr. Power Energy Syst. 64 (2015) 1197–1205. doi:10.1016/j.ijepes.2014.09.020.

[34]

S.K. Agarwal, S. Shah, R. Kumar, Classification of mental tasks from EEG data using backtracking search optimization based neural classifier, Neurocomputing. 166 (2015) 397–403. doi:10.1016/j.neucom.2015.03.041.

[35]

S. Pare, A.K. Bhandari, A. Kumar, V. Bajaj, Backtracking search algorithm for color image multilevel thresholding, Signal, Image Video Process. 12 (2018) 385–392. doi:10.1007/s11760-0171170-z.

[36]

J. Kennedy, R.C. Eberhart, Particle swarm optimization, Neural Networks, 1995. Proceedings., IEEE Int. Conf. 4 (1995) 1942–1948 vol.4. doi:10.1109/ICNN.1995.488968.

33

[37]

G. David, Genetic Algorithms in Search, Optimization, and Machine Learning, 1st ed., AddisonWesley, Boston, MA, USA, 1989.

[38]

R. Storn, K. Price, Differential evolution - a simple and efficient heuristic for global optimization over continuous spaces, J. Glob. Optim. (1997).

[39]

A. Salhi, E.S. Fraga, Nature-inspired optimisation approaches and the new plant propagation algorithm, (2011) 1–8.

[40]

A. Askarzadeh, A novel metaheuristic method for solving constrained engineering optimization problems: Crow search algorithm, Comput. Struct. 169 (2016) 1–12. doi:10.1016/j.compstruc.2016.03.001.

[41]

H. Salimi, Stochastic Fractal Search: A powerful metaheuristic algorithm, Knowledge-Based Syst. 75 (2015) 1–18. doi:10.1016/j.knosys.2014.07.025.

[42]

A.H. Gandomi, X.S. Yang, A.H. Alavi, Cuckoo search algorithm: A metaheuristic approach to solve structural optimization problems, Eng. Comput. 29 (2013) 17–35. doi:10.1007/s00366-011-0241-y.

[43]

S. Aja-Fernández, A.H. Curiale, G. Vegas-Sánchez-Ferrero, A local fuzzy thresholding methodology for multiregion image segmentation, Knowledge-Based Syst. 83 (2015) 1–12. doi:10.1016/j.knosys.2015.02.029.

[44]

M. Mignotte, Segmentation by fusion of histogram-based K-means clusters in different color spaces, IEEE Trans. Image Process. 17 (2008) 780–787. doi:10.1109/TIP.2008.920761.

[45]

S. Ouadfel, A. Taleb-Ahmed, Social spiders optimization and flower pollination algorithm for multilevel image thresholding: A performance study, Expert Syst. Appl. 55 (2016) 566–584. doi:10.1016/j.eswa.2016.02.024.

[46]

D. Oliva, V. Osuna-Enciso, E. Cuevas, G. Pajares, M. Pérez-Cisneros, D. Zaldívar, Improving segmentation velocity using an evolutionary method, Expert Syst. Appl. (2015). doi:10.1016/j.eswa.2015.03.028.

[47]

Sonali, S. Sahu, A.K. Singh, S.P. Ghrera, M. Elhoseny, An approach for de-noising and contrast enhancement of retinal fundus image using CLAHE, Opt. Laser Technol. (2018). doi:10.1016/j.optlastec.2018.06.061.

[48]

W.B. Tao, J.W. Tian, J. Liu, Image segmentation by three-level thresholding based on maximum fuzzy entropy and genetic algorithm, Pattern Recognit. Lett. 24 (2003) 3069–3078. doi:10.1016/S0167-8655(03)00166-1.

[49]

M. Zhao, A.M.N. Fu, H. Yan, A technique of three-level thresholding based on probability partition and fuzzy 3-partition, IEEE Trans. Fuzzy Syst. 9 (2001) 469–479. doi:10.1109/91.928743.

[50]

D. Martin, C. Fowlkes, D. Tal, J. Malik, A database of human segmented natural images and its application to\nevaluating segmentation algorithms and measuring ecological statistics, Proc. Eighth IEEE Int. Conf. Comput. Vision. ICCV 2001. 2 (2001) 416–423. doi:10.1109/ICCV.2001.937655.

[51]

P. Ghamisi, M.S. Couceiro, J.A. Benediktsson, N.M.F. Ferreira, An efficient method for segmentation of images based on fractional calculus and natural selection, Expert Syst. Appl. 39 (2012) 12407–

34

12417. doi:10.1016/j.eswa.2012.04.078. [52]

K. Hammouche, M. Diaf, P. Siarry, A comparative study of various meta-heuristic techniques applied to the multilevel thresholding problem, Eng. Appl. Artif. Intell. 23 (2010) 676–688. doi:10.1016/j.engappai.2009.09.011.

[53]

B. Akay, A study on particle swarm optimization and artificial bee colony algorithms for multilevel thresholding, Appl. Soft Comput. J. 13 (2013) 3066–3091. doi:10.1016/j.asoc.2012.03.072.

[54]

M.-H. Horng, R.-J. Liou, Multilevel minimum cross entropy threshold selection based on the firefly algorithm, Expert Syst. Appl. 38 (2011) 14805–14811. doi:10.1016/j.eswa.2011.05.069.

[55]

Z. Wang, A.C. Bovik, H.R. Sheikh, E.P. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Trans. Image Process. 13 (2004) 600–612. doi:10.1109/TIP.2003.819861.

[56]

D.Z. Lin Zhang, Lei Zhang, XuanqinMou, FSIM : A Feature Similarity Index for Image, IEEE Trans. Image Process. 20 (2011) 2378–2386.

[57]

S. García, D. Molina, M. Lozano, F. Herrera, A Study on the Use of Non-Parametric Tests for Analyzing the Evolutionary Algorithms’ Behaviour: A Case Study on the CEC 2005 Special Session on Real Parameter Optimization, J. Heuristics. 15 (2009) 617–644. doi:10.1007/s10732-008-9080-4.

[58]

R.D. Labati, V. Piuri, F. Scotti, ALL-IDB : THE ACUTE LYMPHOBLASTIC LEUKEMIA IMAGE DATABASE FOR IMAGE PROCESSING Ruggero Donida Labati IEEE Member , Vincenzo Piuri IEEE Fellow , Fabio Scotti IEEE Member Universit ` a degli Studi di Milano , Department of Information Technologies , IEEE Int. Conf. Image Process. (2011) 2045–2048.

35

Highlights • They are used two metaheuristics for multilevel thresholding. • The Type II Fuzzy entropy is used as objective function. • The selected algorithms find the optimal configuration of thresholds. • The quality of the segmentation results is higher than other algorithms. • The performance of the algorithms is tested in image processing problems.

A DISCLOSURE / CONFLICT OF INTEREST STATEMENT: We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome. We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. We further confirm that the order of authors listed in the manuscript has been approved by all of us. We confirm that we have given due consideration to the protection of intellectual property associated with this work and that there are no impediments to publication, including the timing of publication, with respect to intellectual property. In so doing we confirm that we have followed the regulations of our institutions concerning intellectual property. We further confirm that any aspect of the work covered in this manuscript that has involved either experimental animals or human patients has been conducted with the ethical approval of all relevant bodies and that such approvals are acknowledged within the manuscript. We understand that the Corresponding Author is the sole contact for the Editorial process (including Editorial Manager and direct communications with the office). He/she is responsible for communicating with the other authors about progress, submissions of revisions and final approval of proofs. We confirm that we have provided a current, correct email address which is accessible by the Corresponding Author and which has been configured to accept email from [email protected] Signed by all authors as follows: Name

Date

Dr. Diego Oliva

03 May 2019

Mr. Sayan Nag

01 May 2019

Dr. Mohamed Abd Elaziz

02 May 2019

Mr. Uddalok Sarkar

01 May 2019

Dr. Salvador Hinojosa

03 May 2019

Signature