Accepted Manuscript

Image thresholding segmentation based on a novel beta differential evolution

approach

Helon Vicente Hultmann Ayala, Fernando Marins dos Santos, Viviana Cocco

Mariani, Leandro dos Santos Coelho

PII: S0957-4174(14)00596-X

DOI: http://dx.doi.org/10.1016/j.eswa.2014.09.043

Reference: ESWA 9582

To appear in: Expert Systems with Applications

Please cite this article as: Ayala, H.V.H., Santos, F.M.d., Mariani, V.C., Coelho, L.d.S., Image thresholding

segmentation based on a novel beta differential evolution approach, Expert Systems with Applications (2014), doi:

http://dx.doi.org/10.1016/j.eswa.2014.09.043

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Image thresholding segmentation based on a novel beta differential evolution approach

Helon Vicente Hultmann Ayala1, Fernando Marins dos Santos2, Viviana Cocco Mariani2,3,* and Leandro dos Santos Coelho1,4

1 Industrial and Systems Engineering Graduate Program (PPGEPS), Pontifical Catholic University of Parana (PUCPR)

Imaculada Conceicao, 1155, Zip code 80215-901, Curitiba, Parana, Brazil 2 Department of Electrical Engineering, Electrical Engineering Undergraduate Program,

Federal University of Parana (UFPR), Polytechnic Center, C.P. 19011, Zip code 81531-970, Curitiba, Parana, Brazil 3 Department of Mechanical Engineering (PPGEM), Pontifical Catholic University of Parana (PUCPR)

Imaculada Conceicao, 1155, Zip code 80215-901, Curitiba, Parana, Brazil 4 Department of Electrical Engineering, Electrical Engineering Graduate Program (PPGEE),

Federal University of Parana (UFPR), Polytechnic Center, C.P. 19011, Zip code 81531-970, Curitiba, Parana, Brazil

email: helonayala@gmail.com, fm0504@gmail.com, viviana.mariani@pucpr.br, leandro.coelho@pucpr.br * corresponding author

Abstract Image segmentation is the process of partitioning a digital image into multiple regions that have some relevant semantic content. In this context, histogram thresholding is one of the most important techniques for performing image segmentation. This paper proposes a Beta Differential Evolution (BDE) algorithm for determining the n-1 optimal n-level threshold on a given image using Otsu criterion. The efficacy of BDE approach is illustrated by some results when applied to two case studies of image segmentation. Compared with a fractional-order Darwinian particle swarm optimization (PSO), the proposed BDE approach performs better, or at least comparably, in terms of the quality of the final solutions and mean convergence in the evaluated case studies. Keywords Image segmentation, Otsu’s method, optimization, evolutionary algorithms, differential evolution.

1. INTRODUCTION Image processing covers various techniques that are applicable to a wide range of applications. Among

the range of image processing tasks, image segmentation is considered as an important basic operation

for meaningful analysis and interpretation of acquired image. The image segmentation process the

partition of the image into a set of disjoint regions or sections. These regions usually have a strong

correlation with the objects in the image.

Threshold or multithreshold selection based segmentation routines constitute an important field of

research with many practical applications. A variety of thresholding approaches have been adopted for

image segmentation, including conventional methods (Pal & Pal, 2003) and intelligent techniques

(Osuna-Enciso, Cuevas, & Sossa, 2013). Among them, the histogram thresholding technique stands out

as a simple but effective tool. This procedure performs the image segmentation by choosing a threshold

value from its histogram. As a matter of fact, there are several thresholding techniques for its resolution.

One of them, the Otsu criterion, selects an optimum threshold by maximizing the variance intra-clusters

in a gray level image. In this context, the Otsu criterion reduces the thresholding problem to an

optimization problem.

Metaheuristics such as genetic algorithms (Manikandan, Ramar, Iruthayarajan, & Srinivasagan,

2014), particle swarm optimization (Ghamisi, Couceiro, Benediktsson, & Ferreira, 2012), ant colony

(Huang, Cao, & Luo , 2008), bacterial foraging algorithm (Sathya & Kayalvizhi, 2011), honey bee

mating optimization (Horng , 2010), firefly algorithm (Horng and Liou, 2011), wind driven

optimization (Bhandari & Singh, , 2014), cuckoo search (Bhandari, Singh, Kumar, & Singh, 2014;

Panda, Agrawal, & Bhuyan, 2013), artificial bee colony (Horng, 2011) and differential evolution

(Cuevas, Zaldivar, & Pérez-Cisneros, 2010) have already been utilized widely and successfully in image

segmentation. In this context, differential evolution (DE), proposed by Storn & Price (1995), is a fertile

research paradigm and its simple structure has encouraged the exploration of algorithmic variations. In

order to enhance the performance of DE-based methods a number of different techniques has been

proposed in literature (for a survey see e.g. Das and Suganthan (2011)).

DE algorithm performs the mutation, crossover and selection operations to achieve an optimization

search by using the weighted difference vector among current individuals. The DE performance depends

mainly on two components: (i) the trial vector generation strategy (i.e., mutation and crossover

operators) and (ii) its control parameters, that is, the population size (NP), scaling factor (F), and the

crossover rate (CR). The proposed beta differential evolution (BDE) approach adopts the beta

distribution to tune the F and CR in each generation. Results for two case studies in image thresholding

segmentation field are obtained to evaluate the proposed BDE.

The remainder of this paper is organized as follows. Section 2 presents the DE and BDE approaches.

Next, Section 3 presents a description of the image segmentation fundamentals. Section 4 presents a

brief description of two case studies and summarizes the segmentation results. In Section 5, the

conclusions are presented.

2. DIFFERENTIAL EVOLUTION

Currently, there exist several variants of DE. In the present work, we follow the DE/rand/1/exp

scheme which involves the following steps (Coelho, Mariani, & Leite, 2012):

Step 1 (Parameters’ definition): The user must choose the key parameters that control the DE, i.e., NP,

boundary constraints of optimized variables, F, CR, and the stopping criterion given by the maximum

number of generations (tmax).

Step 2 (Population of individuals’ initialization): To initialize a population of individuals (solution

vectors) in the n-dimensional problem space. All individuals in the first generation are generated with

uniformly distributed random numbers constrained by lower and upper bounds of the problem at hand.

Furthermore, set the current generation number to t = 1.

Step 3 (Objective function evaluation): For each individual in the population, the objective function

value is evaluated. The objective function is also referred to as the fitness function.

Step 4 (Mutation operation): The simplest way to produce a mutant vector is to multiply the scaling

factor F by the difference of two random vectors, and the result is added to another third random vector.

In other words, mutate individuals according to the following equation:

])(3)(2[)(1)1( trxtrxFtrxtiz −⋅+=+ (1)

In the above equation, the indices },...,1{3,2,1, NPrrri ∈ represent the individuals’ index of population,

randomly selected such that they are distinct; t is used to indicate the generation (time or iteration);

[ ]T)(...,),(2),(1)( tnixtixtixtix = stands for the position of the i-th individual of population of NP real-

valued n-dimensional vectors. The vector [ ]T)(...,),(2),(1)( tniztiztiztiz = stands for the position of the

i-th individual of a mutant vector.

Step 5 (Crossover operation): The aim of the crossover operation is to construct an offspring by

mixing the current components. In exponential crossover, we first choose an integer n randomly among

the numbers [1,D]. This integer acts as a starting point in the target vector, from where the crossover or

exchange of components with the donor vector starts. We also choose another integer L from the interval

[1,D]. L denotes the number of components the donor vector

actually contributes to the target vector. After choosing n and L the trial vector )1( +tiu is obtained

based on a given crossover rate, CR. Details about the crossover operation with exponential distribution

are presented in Das & Suganthan (2011).

Step 6 (Selection operation): In the present step it is decided whether the trial vector substitutes its

corresponding solution in the next iteration t+1. To do so, the fitness value obtained by ui(t+1) is

compared to the one which corresponds to xi(t). Thus, if f denotes the objective function in the context of

a maximization problem, it amounts to say that

⎩⎨⎧ >++

=+otherwise,)(

)),(())1((if)1()1(

tix

tixftiuftiutix (3)

Step 7 (Stopping criterion verification): Check the termination condition. If it is satisfied, terminate;

otherwise, increment the generation number as t = t + 1 and go to Step 3.

2.1. The proposed BDE approach The F is a scaling factor, which controls the length of the exploration vector )(3)(2 trxtrx − ,

determines how far from point )(1 trx the offspring will be generated. In the original DE (Storn & Price,

1997), F was chosen to lie in (0, 2]. In a relevant study (Kaelo & Ali, 2006) using 50 test problems the

value 0.5 was found to be a good choice.

The CR practically controls the diversity of the population. CR depends on the nature of the

problem, so CR with a value between 0.9 and 1 is suitable for non-separable and multimodal objective

functions, while a value of CR between 0 and 0.2 when the objective function is separable.

The original DE uses a constant F and CR values in Steps 4 and 5, respectively. However, instead of

tuning the control parameters for a specific problem, in order to improve the DE algorithm, we apply a

beta probability distribution (Johnson, Kotz, & Balakrishnan, 1995) in the tuning of the F and CR

parameters in BDE. The beta distribution is flexible for modeling data that are measured in a continuous

scale on a truncated interval in range [0,1] since its density is a versatile way to represent different

shapes depending on the values of the two parameters that index the distribution. In this context, other

stochastic global optimizers approaches have been proposed in the recent literature, see Ali (2007) and

Mendes and Kennedy (2007). The beta distribution on [0,1] has probability density given by

( ) 010,111)()(

)()( >≤≤−−−

ΓΓ+Γ= a,b,vbvav

ba

bavf (4)

with mean equal to )/( baa + and variance given by )1(2)/( +++ babaa . The gamma function is

represented by Γ(.). Clearly, the values of a and b determine the shape of the density function. For

symmetric distributions, a and b are the same. In this paper, it was adopted the command betarnd(a,b,1)

in MATLAB® to generate the beta distribution. In the proposed BDE, the tuning of the F and CR values

are given by the following MATLAB® script in each generation:

a = rand; b = (0.8*(t / tmax) + 0.2) * rand; CR = betarnd (a, b, [1 1]);

a = rand; b = (0.8*(t / tmax) + 0.1) * rand; F = betarnd (a, b, [1 1]);

where rand generates uniformly distributed pseudorandom numbers in range [0,1]. The choice of a and

b values was realized after several trial and error tests. Figure 1 shows an example of the F and CR

values generated using beta distribution in a run in BDE. We can see from this figure that the values for

both control parameters have higher frequencies in the two extremes of the range. In this way, the search

procedure is enhanced as the creation of new individuals is done with different random extreme values

most of the times for the control parameters at each iteration.

3. IMAGE SEGMENTATION

The present section deals with the image segmentation problem, treating the aspects it involves as

histogram, multilevel thresholding and Otsu’s optimal method for thresholding.

Image segmentation is the process of diving the whole region of a given image into a set of regions

which, when united, form the original image. One can adopt features based on histogram values in order

to make the segmentation process. As one can see, the histogram provides information about the

frequency of a given intensity among the pixels that form the image. Formally speaking, the histogram

of an image ),( yxf with intensities varying from ]1,0[ −L , where ),( yx are the Cartesian coordinates

of a pixel and 1−L is the maximum intensity value, is given by a discrete function kk nrh =)( where kr

denotes the k-th intensity value and kn represents the number of pixels with intensity kr .

(a) (b) Figure 1. (a) F and (b) CR values for a run generated using beta distribution in BDE.

One possibility for image segmentation based on histogram is to find threshold values so that each

range in the histogram corresponds to regions in the image. The simplest way to perform image

segmentation based on threshold values of the histogram is bilevel thresholding. On the other hand,

multilevel thresholding deals with the definition of threshold values so that more than two threshold

values are used to define ranges in the histogram space. Let us denote by ),( yxg the resulting

segmented image from ),( yxf . If we consider NT threshold values, the resulting image will be given by

⎪⎩

⎪⎨⎧

<<<+

>−= −−

1

11

),( if ,0

),( if ,2/)(

),( if ,1

),(

Tyxf

TyxfTTT

TyxfL

yxgiiii

T

nnnn

N

(5)

with ni = 2, … , NT – 1.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

F

freq

uenc

y

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

CR

freq

uenc

y

The definition of the threshold values Ti, i=1,…, NT, however, may not be straightforward for some

images. The success of the image segmentation task depends on the shape of its histogram and the

techniques utilized to define the threshold values. When analyzing the shape of the histogram, the

definition of thresholds may be difficult due to e.g. plain or wide valleys and peaks of different heights –

what results in valleys that may not be detected. In order to overcome these difficulties, the Otsu’s

method for image was proposed on the basis of the probabilities of the pixels to happen in the histogram

and the normal curve.

Consider an image whose pixels are separated by L levels in the range [1, …, L]. Let pi denote the

probability of the level i to occur in the image such that

∑=

==L

iiii nNNnp

1

, . (6)

Now consider two classes of pixels C0 and C1, which may refer respectively to background and

object, separated by a threshold of intensity k. Being so, C0 corresponds to the pixels with levels inside

the set [1, ..., k] and C1 to the range [k + 1, .., L]. The probability of the classes C0 and C1 are thus given

by

∑=

==k

iipC

100 )Pr(ω , ∑

+=

==L

kiipC

111 )Pr(ω (7)

and their mean values are given by

∑∑==

==k

i

ik

i

ipCii

1 0100 )|Pr(

ωμ , (8)

∑∑+=+=

==L

ki

iL

ki

ipCii

1 1111 )|Pr(

ωμ . (9)

One can see that for every value of k, the following conditions are satisfied

Tμμωμω =+ 1100 , 110 =+ ωω (10)

where ∑=

=L

iiT ip

1

μ represents the total average level of the image. The variances of the classes C0 and C1

are given by

01

20

20 )(

ωμσ i

k

i

pi∑

=

−= ,11

21

21 )(

ωμσ i

L

ki

pi∑

+=

−= (11)

In order to evaluate the efficiency of the level k, three metrics where stated [22]

2

2

2

2

2

2

;;T

B

W

T

W

B

σση

σσκ

σσλ === , (16)

where 22 , WB σσ and 2Tσ are respectively inter-class , intra-class and total variances defined as

21010

2 )( μμωωσ −+=B , (17)

211

200

2 σωσωσ +=W , (18)

∑=

−=L

iiTT pi

1

22 )( μσ . (19)

The problem of determining a threshold is summarized the definition of a threshold k* such that one

of the metrics ( ηκλ ;; ) are optimum. The basic idea of Otsu’s method (Otsu, 1979) is to divide the

pixels into two groups at a threshold and calculate the variance between them.

A simple way to find the optimal threshold k is the evaluation of all possible values of k, such that

2

1

*2 max)( BLk

B k σσ<≤

= , in an exhaustive search procedure. In multilevel thresholding with n levels, it is also

possible to proceed in the same way such that ),...,,(max),...,,( **2

*1

2

...,1

*1

*2

*1

2

*1

*2

*1

nBLkkk

nB kkkkkkn

σσ<<<<≤

−−

= . In this

approach however the number of possibilities may be too big according to the number of levels required,

as 1)1( −+− nnLn evaluations are needed for all sets of combinations possible (Kulkarni &

Venayagamoorthy, 2010).

On the basis of the Otsu’s criteria, the image segmentation task is faced as an optimization procedure.

In the present work we define the objective function as follows

2Bobjf σ= . (19)

Being so, the algorithm for image segmentation aims at the maximization of (19). This objective

function is adapted in a straightforward manner when RGB images are to be segmented. We proceed

three times to segment the image in each of the components.

4. CASE STUDIES AND RESULTS ANALYSIS

In the present section we introduce two benchmarks in order to test the optimization methodology

based on Otsu’s criteria to perform image segmentation.

4.1 Description of the benchmarks

The first image studied is given by Fig. 2a, while its histogram is drawn in Fig. 2b. The image was

taken by the radiography of a polymeric insulator used in electrical energy distribution networks. The

radiography is one of the oldest nondestructive testing methods, which are relevant in the evaluation of

the quality of the material and equipment. The second image consists of an infrared thermography of a

transformer, where it is possible to locate a problem in the plug connection as depicted in Fig. 3. The

image is property of the Electrophysics® company, specialized in solutions for infrared images. Note

that the studied examples have different histogram caracteristics.

(a) (b) Figure 2. (a) Image of a polymeric insulator used in the simulations and (b) its histogram

(a) (b) Figure 3. (a) Image of an infrared thermography of the derivation of a transformer used in the simulations and (b) its histogram.

0 50 100 150 200 2500

1000

2000

3000

4000

5000

6000

Instensity

Fre

quen

cy

0 50 100 150 200 25010

0

101

102

103

104

105

Instensity

Fre

quen

cy

R

G

B

4.2. Results analysis

In order to evaluate the proposed BDE approach, we applied it to the resolution of the image

segmentation of the examples described in the previous subsection. For sake of comparison, we solved

the same problems with the fractional order Darwinian PSO (FODPSO) (Ghamisi, Couceiro,

Benediktsson, & Ferreira, 2012). The source code of the FODPSO employed in Ghamisi, Couceiro,

Benediktsson, and Ferreira (2012) for Matlab environment is given in

http://www.mathworks.com/matlabcentral/fileexchange/33758-fractional-order-darwinian-particle-

swarm-optimization .

For BDE, we used 80 individuals and the termination criterion was set as 40 generations. In order to

cope with the integer decision variables, we rounded the solutions to their nearest integer.

In Tables 1-4 the results for both optimization algorithms are shown. Tables 1 and 3 show the values

of the objective function evaluations obtained through 30 runs with different initial conditions

respectively for the polymeric insulator and the infrared thermography of the derivation of a

transformer, considering the number of thresholds from 2 to 6. For the same range of thresholds, Tables

2 and 4 depict the best threshold values obtained for each study case. One can see from Tables 1 and 3

that the BDE approach has shown superior performance when considering more than 2 thresholds. This

shows that BDE has better performance in more complex problems, as the number of decision variables

increase with the number of thresholds. The proposed BDE optimization algorithm show superior results

when solving the image segmentation problem. Figures 4 and 5 presented the results to the two

benchmarks, respectively.

Table 1. Minimum (min), maximum (max), mean and standard deviation (std) of the objective function values (in 30 runs) for the polymeric insulator image.

Thresholds FODPSO BDE/rand/1/exp Min Max Mean Std Min Max Mean Std

2 1826.0 1826.0 1826.0 1⋅10-12 1826.0 1826.0 1826.0 1⋅10-12

3 1875.4 1875.6 1875.5 4⋅10-2 2588.5 2588.7 2588.7 3⋅10-2 4 1893.9 1900.1 1899.1 1.5 3283.6 3284.0 3284.0 7⋅10-2 5 1906.8 1916.9 1915.3 2.6 3461.3 3535.4 3513.0 2.2 6 1878.5 1921.9 1906.5 11.2 4208.0 4230.7 4224.1 6.5

Table 2. Best thresholds obtained in 30 runs for the polymeric insulator image.

Thresholds FODPSO BDE/rand/1/exp 2 52 103 53 104 3 45 78 114 2 68 91 4 44 75 107 134 1 91 91 256 5 33 53 80 107 133 1 1 68 68 91 6 34 59 59 73 93 119 1 3 70 90 100 198

Table 3. Minimum (min), maximum (max), mean and standard deviation (std) of the objective function values (in 30 runs) for the image of infrared thermography of the derivation of a transformer.

Thresholds Component FODPSO BDE/rand/1/exp Min Max Mean Std Min Max Mean Std 2 R 1654.7 1654.7 1654.7 9⋅10-13 1654.7 1654.7 1654.7 9⋅10-13 2 G 283.9 283.9 283.9 0 283.9 283.9 283.9 0 2 B 4823.3 4824.8 4824.8 0.27 4824.8 4824.8 4824.8 0 3 R 1727.8 1727.8 1727.8 9⋅10-13 1953.9 1953.9 1953.9 9⋅10-13 3 G 286.8 286.8 286.8 2⋅10-13 524.1 524.1 524.1 2⋅10-13 3 B 4845.7 4848.3 4848.2 0.57 6465.4 6465.4 6465.4 2⋅10-12 4 R 1775.8 1776.1 1776.0 9⋅10-2 2258.9 2292.4 2290.1 8.5 4 G 288.2 288.2 288.2 1⋅10-13 545.1 545.2 545.2 3⋅10-2 4 B 4862.8 4864.9 4864.8 0.38 7042.3 7042.3 7042.3 0 5 R 1808.5 1808.7 1808.6 4⋅10-2 2588.4 2597.3 2594.3 3.2 5 G 289.0 289.0 289.0 8⋅10-5 785.5 785.5 785.5 3⋅10-5 5 B 4872.9 4873.9 4873.9 0.23 8191.9 8192.1 8192.1 6⋅10-2 6 R 1821.1 1821.3 1821.2 4⋅10-2 2883.1 2929.8 2907.4 15.9 6 G 289.3 289.5 289.5 5⋅10-2 804.9 806.5 806.1 0.3 6 B 4881.7 6107.2 4918.6 224.5 8761.8 8769.1 8768.5 1.4

Table 4. Best thresholds of different segmentation algorithm (in 30 runs) for the image of an infrared thermography of the derivation of a transformer.

Thresholds Component FODPSO BDE/rand/1/exp 2 R 37 129 38 130 2 G 50 170 51 171 2 B 49 122 49 123 3 R 23 61 133 41 66 256 3 G 44 123 204 110 111 256 3 B 43 109 141 75 123 256 4 R 23 60 104 182 38 66 130 256 4 G 17 64 127 205 53 111 171 256 4 B 43 109 141 198 1 75 79 256 5 R 20 54 77 119 195 38 62 131 134 256 5 G 17 65 121 173 223 110 112 112 256 256 5 B 29 79 117 142 198 75 123 123 256 256

6 R 20 53 74 92 132 201 39 66 67 131 256 256 6 G 12 43 80 122 175 223 52 111 112 169 256 256 6 B 86 118 146 146 155 217 1 75 79 123 256 256

2 3 4 5 6

Figure 4. Results for the segmentation of the polymeric insulator image using 2, 3, 4, 5 and 6 thresholds,

with the best result obtained by BDE (upper) and FODPSO (lower).

2 3 4 5 6

Figure 5. Results for the segmentation of the infrared thermography of the derivation of a transformer image using 2, 3, 4, 5 and 6 thresholds, with the best result obtained by BDE (upper) and FODPSO

(lower).

5. CONCLUSION

The present paper dealt with image segmentation by the application of FODPSO and the novel BDE

according to Otsu’s criterion. Segmentation on greyscale and colored images is attained for the purpose

of better understanding and ease of interpretation for the user who needs to work with images. Thus, the

creation of automatic image segmentation algorithms and its successful application, as we did in the

present work, is an important topic of research in the image processing and evolutionary computation

fields.

The results indicate that the proposed BDE approach is more efficient than the FODPSO proposed in

Ghamisi, Couceiro, Benediktsson, and Ferreira (2012) when solving the image segmentation task based

on Otsu’s criteria and multilevel segmentation is considered. The proposed algorithm was able to

automatically define the thresholds for image segmentation and, according to Otsu’s criteria, to obtain

better results for the purpose of image segmentation in comparison with FODPSO. With respect to the

image segmentation algorithm, we can see that by increasing the number of thresholds did not

necessarily improve the result in terms of the objective function for the examples studied. Thus, it would

be also important to include in the optimization procedure the number of thresholds as a decision

variable. This would be very interesting to the user which would not need to set it beforehand further

reducing, thus, the number of control parameters as in BDE we also have less project parameters.

The successful application of the methodology based on Otsu’s criterion and improving the results by

making changes the optimization algorithm inspires the following research lines. The further

development on DE variants for this domain of application is encouraged, as we showed that by making

changes in the optimization algorithm we were able to perform better the image segmentation task. To

the best of our knowledge, the application of the DE algorithm based on Beta probability distribution

proposed here was still not applied to multiobjective optimization, what will be studied in future work in

comparison to other state-of-the-art algorithms (Zhou, Qu, Li, Zhao, Suganthan, & Zhang, 2011). We

focused on applications regarding electrical systems in the present work. Future research will address

different types of applications, such as in robotics (Romero & Cazorla, 2012; Lee & Song, 2010) and

biomedical (Sakamoto, 2014; Bertelsen, Garin-Muga, Echeverría, Gómez, & Borro, 2014; Bertrand,

Macri, Mazars, Droupy, Beregi, & Prudhomme, 2014) fields. As the setting of CR and F parameters in

DE are problem dependent, the study of the impact of the a and b parameters to generate them in BDE

will be studied so as to define a range of values which work well for most of the cases.

ACKNOWLEDGMENTS This study is supported by National Council of Scientific and Technologic Development of Brazil

(CNPq) under Grants 479764/2013-1, 307150/2012-7/PQ and 304783/2011-0/PQ. In addition, the

authors would like to thank Prof. Nuno Miguel Fonseca Ferreira (Politécnico de Coimbra, IPC

Instituto Superior de Engenharia de Coimbra, ISEC) by your collaboration related to FODPSO.

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Highlights

An improved differential evolution algorithm is proposed. The improved differential evolution algorithm is applied to select for the thresholds for segmenting the images. Simulation results of proposed algorithm demonstrate that the proposed differential evolution approach is superior to FODPSO method.