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Image thresholding segmentation based on a novel beta differential evolution approach
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: DOI: Reference:
S0957-4174(14)00596-X http://dx.doi.org/10.1016/j.eswa.2014.09.043 ESWA 9582
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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: [email protected], [email protected], [email protected], [email protected]
* 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:
zi (t + 1) = xr1 (t ) + F ⋅[ xr2 (t ) − xr3 (t )]
(1)
In the above equation, the indices i, r1, r2 , r3 ∈ {1,..., NP} represent the individuals’ index of population, randomly selected such that they are distinct; t is used to indicate the generation (time or iteration);
[
]
xi (t ) = xi (t ), xi (t ), ..., xi (t ) T stands for the position of the i-th individual of population of NP real1 2 n
[
]
valued n-dimensional vectors. The vector zi (t ) = zi (t ), zi (t ), ..., zi (t ) T stands for the position of the 1 2 n 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 ui (t + 1) 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
⎧u (t + 1) if f (ui (t + 1)) > f ( xi (t )), xi (t + 1) = ⎨ i ⎩ xi (t ) otherwise,
(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 x r2 (t ) − x r3 (t ) , determines how far from point xr (t ) the offspring will be generated. In the original DE (Storn & Price, 1 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
f (v ) =
Γ(a + b) a −1 (1 − v )b −1,0 ≤ v ≤ 1,a,b > 0 v Γ(a)Γ(b)
(4)
with mean equal to a /(a + b) and variance given by a /( a + b) 2 (a + b + 1) . 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 f ( x, y) with intensities varying from [0, L − 1] , where ( x, y) are the Cartesian coordinates of a pixel and L − 1 is the maximum intensity value, is given by a discrete function h(rk ) = nk where rk
90
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frequenc y
frequency
denotes the k-th intensity value and nk represents the number of pixels with intensity rk .
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(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 g ( x, y) the resulting segmented image from f ( x, y) . If we consider NT threshold values, the resulting image will be given by L − 1, if f ( x, y ) > TNT ⎧ ⎪ g ( x, y ) = ⎨(Tni −1 + Tni ) / 2, if Tni −1 < f ( x, y ) < Tni ⎪ 0, if f ( x, y ) < T1 ⎩
with ni = 2, … , NT – 1.
(5)
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
pi = ni N , N = ∑ ni .
(6)
i =1
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
ω 0 = Pr(C0 ) = ∑ pi , ω1 = Pr(C1 ) = i =1
L
∑p
i
(7)
i = k +1
and their mean values are given by k
k
i =1
i =1
L
L
μ 0 = ∑ i Pr(i | C0 ) = ∑ μ1 =
∑ i Pr(i | C1 ) =
i = k +1
ipi
ω0 ipi
∑ω
i = k +1
,
(8)
.
(9)
1
One can see that for every value of k, the following conditions are satisfied
ω0 μ 0 + ω1 μ1 = μT , ω0 + ω1 = 1
(10)
L
where μT = ∑ ipi represents the total average level of the image. The variances of the classes C0 and C1 i =1
are given by k
σ 02 = ∑ (i − μ 0 ) 2 i =1
pi
ω0
, σ 12 =
L
∑ (i − μ ) 1
i = k +1
2
pi
ω1
In order to evaluate the efficiency of the level k, three metrics where stated [22]
(11)
λ=
σ B2 σ T2 σ B2 , ; κ = ; η = σ W2 σ W2 σ T2
(16)
where σ B2 , σ W2 and σ T2 are respectively inter-class , intra-class and total variances defined as
σ B2 = ω 0ω1 + ( μ 0 − μ1 ) 2 ,
(17)
σ W2 = ω 0σ 02 + ω1σ 12 ,
(18)
L
σ T2 = ∑ (i − μT ) 2 pi .
(19)
i =1
The problem of determining a threshold is summarized the definition of a threshold k* such that one of the metrics ( λ; κ ;η ) 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
σ B2 (k * ) = max σ B2 , in an exhaustive search procedure. In multilevel thresholding with n levels, it is also 1≤ k < L
possible to proceed in the same way such that σ B2 (k1* , k 2* ,..., k n*−1 ) =
max
1≤ k1* < k 2* <...,< k n*−1 < L
σ B2 (k1* , k 2* ,..., k n* ) . In this
approach however the number of possibilities may be too big according to the number of levels required, as n( L − n + 1) n−1 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
f obj = σ B2 .
(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.
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(a) (b) Figure 2. (a) Image of a polymeric insulator used in the simulations and (b) its histogram
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(a) (b) Figure 3. (a) Image of an infrared thermography of the derivation of a transformer used in the simulations and (b) its histogram.
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-particleswarm-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 2 3 4 5 6
52 45 44 33 34
FODPSO 103 78 114 75 107 134 53 80 107 133 59 59 73 93 119
BDE/rand/1/exp 53 104 2 68 91 1 91 91 256 1 1 68 68 91 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 -13 4 G 288.2 288.2 288.2 1⋅10 545.1 545.2 545.2 3⋅10-2 4 B 4862.8 4864.9 4864.8 0.38 7042.3 7042.3 7042.3 0 -2 5 R 1808.5 1808.7 1808.6 4⋅10 2588.4 2597.3 2594.3 3.2 -5 5 G 289.0 289.0 289.0 785.5 785.5 785.5 8⋅10 3⋅10-5 5 B 4872.9 4873.9 4873.9 0.23 8191.9 8192.1 8192.1 6⋅10-2 15.9 6 R 1821.1 1821.3 1821.2 4⋅10-2 2883.1 2929.8 2907.4 -2 6 G 289.3 289.5 289.5 804.9 806.5 806.1 0.3 5⋅10 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 2 R 37 129 2 G 50 170 2 B 49 122 3 R 23 61 133 3 G 44 123 204 3 B 43 109 141 4 R 23 60 104 182 4 G 17 64 127 205 4 B 43 109 141 198 5 R 20 54 77 119 195 5 G 17 65 121 173 223 5 B 29 79 117 142 198
BDE/rand/1/exp 38 130 51 171 49 123 41 66 256 110 111 256 75 123 256 38 66 130 256 53 111 171 256 1 75 79 256 38 62 131 134 256 110 112 112 256 256 75 123 123 256 256
6 6 6
2
R G B
20 53 74 92 132 201 12 43 80 122 175 223 86 118 146 146 155 217
3
4
39 66 67 131 256 256 52 111 112 169 256 256 1 75 79 123 256 256
5
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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).
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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. REFERENCES Ali, M. M. (2007). Synthesis of the β-distribution as an aid to stochastic global optimization. Computational Statistics & Data Analysis, 52, 133-149. Bertelsen, A., Garin-Muga, A., Echeverría, M., Gómez, E., & Borro, D. (2014). Distortion correction and calibration of intra-operative spine X-ray images using a constrained DLT algorithm. Computerized Medical Imaging and Graphics (in press). Bertrand, M. M., Macri, F., Mazars, R., Droupy, S., Beregi, J. P., & Prudhomme, M. (2014). MRIbased 3D pelvic autonomous innervation: a first step towards image-guided pelvic surgery. European Radiology (in press). Bhandari, A. K., Singh, V. K., Kumar, A., & Singh, G. K. (2014). Cuckoo search algorithm and wind driven optimization based study of satellite image segmentation for multilevel thresholding using Kapur’s entropy. Expert Systems with Applications, 41(7), 3538-3560. Coelho, L. S., Mariani, V. C., & Leite, J. V. (2012). Solution of Jiles-Atherton vector hysteresis parameters estimation by modified differential evolution approaches. Expert Systems with Applications, 39(2), 2021-2025. Cuevas, E., Zaldivar, D., & Pérez-Cisneros, M. (2010). A novel multi-threshold segmentation approach based on differential evolution optimization. Expert Systems with Applications, 37(7), 5265-5271. Das, S., & Suganthan, P. N. (2011). Differential evolution: a survey of the state-of-the-art. IEEE Transactions on Evolutionary Computation, 15(1), 4-31. Ghamisi, P., Couceiro, M. S., Benediktsson, J. A., & Ferreira, N. M. F. (2012). An efficient method for segmentation of images based on fractional calculus and natural selection. Expert Systems with Applications, 39(16), 12407-12417. Horng, M. -H. (2010). Multilevel minimum cross entropy threshold selection based on the honey bee mating optimization. Expert Systems with Applications, 37(6), 4580-4592. Horng, M. -H. (2011). Multilevel thresholding selection based on the artificial bee colony algorithm for image segmentation. Expert Systems with Applications, 38(11), 13785-13791. Horng, M. -H., & Liou, R. -J. (2011). Multilevel minimum cross entropy threshold selection based on the firefly algorithm. Expert Systems with Applications, 38(2), 14805-14811. Huang, P., Cao, H., & Luo, S. (2008). An artificial ant colonies approach to medical image segmentation. Computer Methods and Programs in Biomedicine, 92(3), 267-273.
Johnson, N., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions, 2nd edition. John Wiley and Sons, New York, USA. Kaelo, P., & Ali, M. M. (2006). A numerical study of some modified differential evolution algorithm. European Journal of Operations Research, 169(3), 1176-1184. Kulkarni, R. V., & Venayagamoorthy, G. K. (2010). Bio-inspired algorithms for autonomous deployment and localization of sensor nodes. IEEE Transactions on Systems, Man, and Cybernetics, 40(6), 663-675. Lee, Y.-J., & Song, J. -B. (201). Autonomous salient feature detection through salient cues in an hsv color space for visual indoor simultaneous localization and mapping. Advanced Robotics, 24(11), 1595-1613. Manikandan, S., Ramar, K., Iruthayarajan, M. W., & Srinivasagan, K. G. (2014). Multilevel thresholding for segmentation of medical brain images using real coded genetic algorithm. Measurement, 47, 558-568, 2014. Mendes, R., & Kennedy, J. (2007). Stochastic barycenters and beta distribution for Gaussian particle swarm, EPIA 2007, LNAI 4874, J. Neves, M. Santos and J. Machado (eds.), Springer-Verlag, Berlin, Germany (pp. 259-270). Osuna-Enciso, V., Cuevas, E., & Sossa, H. (2013). A comparison of nature inspired algorithms for multi-threshold image segmentation. Expert Systems with Applications, 40(4), 1213-1219. Otsu, N. (1979). A threshold selection method from gray-level histograms. IEEE Transactions on Systems, Man, and Cybernetics, 9(1), 62-66. Pal, N. R., & Pal, S. K. (2003). A review on image segmentation techniques. Pattern Recognition, 26(9), 1277-1294. Panda, R., Agrawal, S., & Bhuyan, S. (2013). Edge magnitude based multilevel thresholding using cuckoo search technique. Expert Systems with Applications, 40(18), 7617-7628. Romero, A., & Cazorla, M. (2012). Topological visual mapping in robotics. Cognitive Processing, 13(1), S305-S308. Sakamoto, T. (2014). Roles of universal three-dimensional image analysis devices that assist surgical operations. Journal of Hepato-Biliary-Pancreatic Sciences, 21(4), 230-234. Sathya, P. D., & Kayalvizhi, R. (2011). Modified bacterial foraging algorithm based multilevel thresholding for image segmentation. Engineering Applications of Artificial Intelligence, 24(4), 595615. Storn, R., & Price, k. (1995). Differential evolution - a simple and efficient adaptive scheme for global optimization over continuous spaces. Technical Report TR-95-012, ICSI, USA, March, ftp.icsi.berkeley.edu. Storn, R., & Price, k. (1997). Differential evolution ⎯a simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimization, 11(4), 341-359. Zhou, A., Qu, B. -Y., Li, H., Zhao, S. -Z., & Suganthan, P. N. (2011). Multiobjective evolutionary algorithms: a survey of the state of the art. Swarm and Evolutionary Computation, 1(1), 32-49.
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.
S0957-4174(14)00596-X http://dx.doi.org/10.1016/j.eswa.2014.09.043 ESWA 9582
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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: [email protected], [email protected], [email protected], [email protected]
* 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:
zi (t + 1) = xr1 (t ) + F ⋅[ xr2 (t ) − xr3 (t )]
(1)
In the above equation, the indices i, r1, r2 , r3 ∈ {1,..., NP} represent the individuals’ index of population, randomly selected such that they are distinct; t is used to indicate the generation (time or iteration);
[
]
xi (t ) = xi (t ), xi (t ), ..., xi (t ) T stands for the position of the i-th individual of population of NP real1 2 n
[
]
valued n-dimensional vectors. The vector zi (t ) = zi (t ), zi (t ), ..., zi (t ) T stands for the position of the 1 2 n 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 ui (t + 1) 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
⎧u (t + 1) if f (ui (t + 1)) > f ( xi (t )), xi (t + 1) = ⎨ i ⎩ xi (t ) otherwise,
(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 x r2 (t ) − x r3 (t ) , determines how far from point xr (t ) the offspring will be generated. In the original DE (Storn & Price, 1 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
f (v ) =
Γ(a + b) a −1 (1 − v )b −1,0 ≤ v ≤ 1,a,b > 0 v Γ(a)Γ(b)
(4)
with mean equal to a /(a + b) and variance given by a /( a + b) 2 (a + b + 1) . 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 f ( x, y) with intensities varying from [0, L − 1] , where ( x, y) are the Cartesian coordinates of a pixel and L − 1 is the maximum intensity value, is given by a discrete function h(rk ) = nk where rk
90
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frequenc y
frequency
denotes the k-th intensity value and nk represents the number of pixels with intensity rk .
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(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 g ( x, y) the resulting segmented image from f ( x, y) . If we consider NT threshold values, the resulting image will be given by L − 1, if f ( x, y ) > TNT ⎧ ⎪ g ( x, y ) = ⎨(Tni −1 + Tni ) / 2, if Tni −1 < f ( x, y ) < Tni ⎪ 0, if f ( x, y ) < T1 ⎩
with ni = 2, … , NT – 1.
(5)
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
pi = ni N , N = ∑ ni .
(6)
i =1
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
ω 0 = Pr(C0 ) = ∑ pi , ω1 = Pr(C1 ) = i =1
L
∑p
i
(7)
i = k +1
and their mean values are given by k
k
i =1
i =1
L
L
μ 0 = ∑ i Pr(i | C0 ) = ∑ μ1 =
∑ i Pr(i | C1 ) =
i = k +1
ipi
ω0 ipi
∑ω
i = k +1
,
(8)
.
(9)
1
One can see that for every value of k, the following conditions are satisfied
ω0 μ 0 + ω1 μ1 = μT , ω0 + ω1 = 1
(10)
L
where μT = ∑ ipi represents the total average level of the image. The variances of the classes C0 and C1 i =1
are given by k
σ 02 = ∑ (i − μ 0 ) 2 i =1
pi
ω0
, σ 12 =
L
∑ (i − μ ) 1
i = k +1
2
pi
ω1
In order to evaluate the efficiency of the level k, three metrics where stated [22]
(11)
λ=
σ B2 σ T2 σ B2 , ; κ = ; η = σ W2 σ W2 σ T2
(16)
where σ B2 , σ W2 and σ T2 are respectively inter-class , intra-class and total variances defined as
σ B2 = ω 0ω1 + ( μ 0 − μ1 ) 2 ,
(17)
σ W2 = ω 0σ 02 + ω1σ 12 ,
(18)
L
σ T2 = ∑ (i − μT ) 2 pi .
(19)
i =1
The problem of determining a threshold is summarized the definition of a threshold k* such that one of the metrics ( λ; κ ;η ) 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
σ B2 (k * ) = max σ B2 , in an exhaustive search procedure. In multilevel thresholding with n levels, it is also 1≤ k < L
possible to proceed in the same way such that σ B2 (k1* , k 2* ,..., k n*−1 ) =
max
1≤ k1* < k 2* <...,< k n*−1 < L
σ B2 (k1* , k 2* ,..., k n* ) . In this
approach however the number of possibilities may be too big according to the number of levels required, as n( L − n + 1) n−1 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
f obj = σ B2 .
(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.
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(a) (b) Figure 2. (a) Image of a polymeric insulator used in the simulations and (b) its histogram
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(a) (b) Figure 3. (a) Image of an infrared thermography of the derivation of a transformer used in the simulations and (b) its histogram.
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-particleswarm-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 2 3 4 5 6
52 45 44 33 34
FODPSO 103 78 114 75 107 134 53 80 107 133 59 59 73 93 119
BDE/rand/1/exp 53 104 2 68 91 1 91 91 256 1 1 68 68 91 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 -13 4 G 288.2 288.2 288.2 1⋅10 545.1 545.2 545.2 3⋅10-2 4 B 4862.8 4864.9 4864.8 0.38 7042.3 7042.3 7042.3 0 -2 5 R 1808.5 1808.7 1808.6 4⋅10 2588.4 2597.3 2594.3 3.2 -5 5 G 289.0 289.0 289.0 785.5 785.5 785.5 8⋅10 3⋅10-5 5 B 4872.9 4873.9 4873.9 0.23 8191.9 8192.1 8192.1 6⋅10-2 15.9 6 R 1821.1 1821.3 1821.2 4⋅10-2 2883.1 2929.8 2907.4 -2 6 G 289.3 289.5 289.5 804.9 806.5 806.1 0.3 5⋅10 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 2 R 37 129 2 G 50 170 2 B 49 122 3 R 23 61 133 3 G 44 123 204 3 B 43 109 141 4 R 23 60 104 182 4 G 17 64 127 205 4 B 43 109 141 198 5 R 20 54 77 119 195 5 G 17 65 121 173 223 5 B 29 79 117 142 198
BDE/rand/1/exp 38 130 51 171 49 123 41 66 256 110 111 256 75 123 256 38 66 130 256 53 111 171 256 1 75 79 256 38 62 131 134 256 110 112 112 256 256 75 123 123 256 256
6 6 6
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R G B
20 53 74 92 132 201 12 43 80 122 175 223 86 118 146 146 155 217
3
4
39 66 67 131 256 256 52 111 112 169 256 256 1 75 79 123 256 256
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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).
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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.
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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.