Cross entropy based thresholding for magnetic resonance brain images using Crow Search Algorithm

Accepted Manuscript

Cross entropy based thresholding for magnetic resonance brain images using Crow Search Algorithm Diego Oliva , Salvador Hinojosa , Erik Cuevas , Gonzalo Pajares , Omar Avalos , Jorge Galvez ´ PII: DOI: Reference:

S0957-4174(17)30132-X 10.1016/j.eswa.2017.02.042 ESWA 11150

To appear in:

Expert Systems With Applications

Please cite this article as: Diego Oliva , Salvador Hinojosa , Erik Cuevas , Gonzalo Pajares , Omar Avalos , Jorge Galvez , Cross entropy based thresholding for magnetic resonance ´ brain images using Crow Search Algorithm, Expert Systems With Applications (2017), doi: 10.1016/j.eswa.2017.02.042

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We use an evolutionary mechanism to improve the image segmentation process. We optimize the minimum cross entropy with an evolutionary method for image segmentation. We test the approach in multidimensional spaces. An alternative method for MR brain image segmentation is proposed. Comparisons and non-parametric test support the experimental results.

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Cross entropy based thresholding for magnetic resonance brain images using Crow Search Algorithm 1

Diego Oliva a, c, Salvador Hinojosa b, c, Erik Cuevas c, Gonzalo Pajares b, Omar Avalos c, d and Jorge Gálvez c, d 

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Departamento de Ciencias Computacionales, Tecnológico de Monterrey, Campus Guadalajara, Av. Gral. Ramón Corona 2514, Zapopan, Jal, México 1 [email protected] b

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Dpto. Ingeniería del Software e Inteligencia Artificial, Facultad Informática, Universidad Complutense de Madrid, 28040 Madrid, Spain {salvahin, pajares}@ucm.es c

Departamento de Electrónica, Universidad de Guadalajara, CUCEI, Av. Revolución 1500, Guadalajara, Jal, México {1diego.oliva, erik.cuevas}@cucei.udg.mx d

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Departamento de Ingenierías, Universidad de Guadalajara, CUTONALA, Sede Provisional Casa de la Cultura - Administración: Calle Morelos 180, Tonalá, Jalisco {omar.avalos, salvador.hinojosa, jorge.galvez}@cutonala.udg.mx

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Corresponding author, Tel +52 33 1378 5900, ext. 7715, E-mail: [email protected] 2

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Abstract

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Segmentation is considered the central part of an image processing system due to its high influence on the posterior image analysis. In recent years, the segmentation of magnetic resonance (MR) images has attracted the attention of the scientific community with the objective of assisting the diagnosis in different brain diseases. From several techniques, thresholding represents one of the most popular methods for image segmentation. Currently, an extensive amount of contributions has been proposed in the literature, where thresholding values are obtained by optimizing relevant criteria such as the cross entropy. However, most of such approaches are computationally expensive, since they conduct an exhaustive search strategy for obtaining the optimal thresholding values. This paper presents a general method for image segmentation. To estimate the thresholding values, the proposed approach uses the recently published evolutionary method called the Crow Search Algorithm (CSA) which is based on the behavior in flocks of crows. Different to other optimization techniques used for segmentation proposes, CSA presents a better performance, avoiding critical flaws such as the premature convergence to sub-optimal solutions and the limited exploration-exploitation balance in the search strategy. Although the proposed method can be used as a generic segmentation algorithm, its characteristics allow obtaining excellent results in the automatic segmentation of complex MR images. Under such circumstances, our approach has been evaluated using two sets of benchmark images; the first set is composed of general images commonly used in the image processing literature, while the second set corresponds to MR brain images. Experimental results, statistically validated, demonstrate that the proposed technique obtains better results in terms of quality and consistency.

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Keywords: Magnetic Resonance Images; Evolutionary Algorithms; Minimum Cross Entropy; Crow Search Algorithm.

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1.Introduction

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Image segmentation is one of the most common preprocessing techniques used in image analysis. It is part of several applications such as object tracking, detection of regions of interest, medical imaging, among others (Ji, Liu, Cao, Sun, & Chen, 2014; Jia, Qi, Li, & Lu, 2016; Lee, Jang, Baek, & Shim, 2014). The main objective of segmentation is the partition of the image into homogeneous classes, where the elements of each class share common properties such as intensity or texture. Currently, an extensive amount of contributions has been proposed in the literature, which can be found in the following surveys (Osuna-Enciso, Cuevas, & Sossa, 2013; Zaitoun & Aqel, 2015; Y. J. Zhang, 1996). One of the simplest techniques for segmentation is Image Thresholding (TH). Image Thresholding uses the gray-scale histogram to select threshold intensity values wich are meant to separate classes. Bi-level thresholding is the simplest case and only uses one threshold value to create two classes. Under this approach, it is extracted an object from its background. While bi-level thresholding is easy to implement, Multilevel Thresholding (MTH) presents more complications, since it aims to find several classes. Segmentation methods based on threshold can be divided into parametric and nonparametric (Akay, 2013; Hammouche, Diaf, & Siarry, 2010; Liao, Chen, & Chung, 2001). Parametric approaches estimate parameters of a probability density function to describe each class, but this approach is computationally expensive. By contrast, nonparametric approaches use criteria such as between-class variance, entropy, and error rate (Kapur, Sahoo, & Wong, 1985; Kittler & Illingworth, 1986; Otsu, 1979). These criteria are optimized to find the optimal threshold value providing robust and accurate methods(Sankur, 2004).

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The segmentation of magnetic resonance (MR) brain images has attracted the researchers' attention due to their potential number of applications. MR images are used for medical diagnosis of tumors and other neurological diseases, such as multiple sclerosis, dementia, and squizophrenia (Kamber, Shinghal, Collins, Francis, & Evans, 1995). Commonly, the analysis of MR brain images is performed by experts during visual ratings based on their experiences and skills. However, human visual inspection is limited and timeconsuming. Those restrictions have lead to the development of computer-aided techniques intended to extract the anatomical structures (Moeskops et al., 2016; Ortiz, Gorriz, Ramírez, Salas-González, & Llamas-Elvira, 2013; Suzuki & Toriwaki, 1991), where many methodologies are segmentation based.

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Otsu proposed a popular method wich can find thresholds by maximizing the between class variance of intensity levels of the foreground and background (Otsu, 1979). Otsu´s method is considered one of the top threshold selection methods for real world images (Sahoo, Soltani, & Wong, 1988). Nevertheless, the formulation of between class variance is inefficient in case of multilevel thresholding. As the number of levels grows, the computational time scales exponentially, and its accuracy decreases with each new threshold point (Sathya & Kayalvizhi, 2011). Another widely used method has been proposed by Kapur (Kapur et al., 1985), and it maximizes the entropy to measure the homogeneity of each class. Both approaches have been evaluated in several contexts, demonstrating their efficiency and accuracy (Sathya & Kayalvizhi, 2011). The development of the information theory has been adopted to explore the use of several entropies for segmentation purposes. Under such approaches, the objective is to find efficient entropy criteria that with their optimization allow to separate objects and background. Some examples include, Kapur entropy (Kapur et al., 1985), Tsallis entropy (Portes de Albuquerque, Esquef, & Gesualdi Mello, 2004), and cross entropy (Li & Tam, 1998) to list some. One of the most important entropy measurements is the Minimum Cross Entropy Thresholding (MCET) criterion (Li & Lee, 1993). It has been widely used in the literature to segment images. This criteria was introduced by Li and Lee (Li & Lee, 1993) in segmentation algorithm that identifies thresholds by minimizing the cross entropy between the original and segmented images. On the other hand, 4

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Yin proposed a recursive programming technique wich reduce the magnitude of computing the objective function of the MCET (Yin, 2007).

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As an alternative to parametric techniques, the problem of MTH has also been handled through evolutionary computation techniques (ECT). Such approaches produced several MTH applications by selecting different evolutionary computation techniques and optimizing various criteria; such as Particle Swarm Optimization (PSO), Firefly (FF) and Cross-Entropy (Horng & Liou, 2011), Electromagnetism-Like Optimization (EMO) and Tsallis´entropy (Diego Oliva et al., 2015), whose results have been individually reported. Recently, significant contributions to medical image segmentation using ECT were published such as an evaluation of PSO in the segmentation of biomedical images (Lahmiri & Boukadoum, 2014) and the glioma detection on MR images using segmentation techniques and ECTs (Lahmiri, 2017). Table 1. Provides a summary table comparing significant approaches in terms of pros and cons. The works listed are directly related to the main contribution of this article. Although these approaches produce acceptable results, they have a significant limitation: They usually find sub-optimal solutions as a result of their limited balance between exploration and exploitation in their optimization strategies. This limitation is associated with their evolutionary operations. In such methods, during their execution, the position of each candidate solution at the next iteration is modified producing an attraction towards the location of the best element seen so-far or towards other promising solutions. Therefore, as the algorithm operates, such behaviors cause that the complete set rapidly concentrates around the best elements, producing the premature convergence and damaging the adequate exploration of the search space (Chen, Low, & Yang, 2009; Tan, Chiam, Mamun, & Goh, 2009). Characteristics

Pros

QPSO MCET (Horng & Liou, 2011)

Minimum cross entropy. Quantum Particle Search Optimization. Minimum cross entropy. Firefly Algorithm.

One of the first approaches to MTH with ECTs.

Tsallis entropy. Electromagnetismlike optimization.

ABC-Kapur PSO-Kapur (Akay, 2013)

Kapur Artificial Bee Colony and Particle Search Optimization Tsallis entropy. Modified Bacterial Foraging Algorithm

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TSEMO (Diego Oliva et al., 2015)

Eleven benchmark images. Three quality metrics (PSNR, FSIM, SSIM). Use of significance proof.

Cons

Focused on bi-level thresholding. Evaluated for small thresholds nt={2,3,4} Only one image as a benchmark. Does not evaluate the quality of the segmented image. Evaluated for small thresholds nt={2,3,4,5} Only five benchmark images. Only two quality metrics (PSNR, RMSE). Four parameters to be tuned. The absence of significant difference between the analyzed algorithms. Evaluated for small thresholds nt={2,3,4,5} Emo performs a local search wich might increase computational time. Four parameters to be tuned.

Twelve benchmark images Use of significance proof.

Evaluated for small thresholds nt={2,3,4,5} Only two quality metrics (PSNR, SSIM) Multiple parameters to be tuned

Proposal of a modification of BFA applied to MTH.

Evaluated for small thresholds nt={2,3,4,5}

Table 1. Comparison of related methodologies.

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MBFO MTH (Tang et al., 2017)

Faster implementation via native code.

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FF-based MCET (Horng & Liou, 2011)

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ECT are often designed to meet specific requirements of particular problems because no single optimization algorithm can solve all problems competitively (Wolpert & Macready, 1997). Every new algorithm proposed should be properly evaluated. Common comparisons (Shilane, Martikainen, Dudoit, & Ovaska, 2008) involve synthetic numerical benchmark problems with statistical analysis to determinate if a particular algorithm outperforms others. Only a few comparative studies of various ECT consider the application context. Thus, it is important to evaluate the performance of ECT methods from an application perspective.

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Crow Search Algorithm (CSA) is a recently proposed ECT (Askarzadeh, 2016) based on the behavior that crows exhibit. In CSA, the population emulates a flock of crows which behave based on the thievery behaviors of crows. Different to other optimization techniques used for segmentation purposes, CSA presents a better performance, avoiding critical flaws such as the premature convergence to sub-optimal solutions and the limited exploration-exploitation balance in the search strategy. The CSA properties have demonstrated that CSA performs better than other ECT such as the Genetic Algorithm (GA), Harmony Search (HS), and PSO (Askarzadeh, 2016).

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In this paper, a new method for image segmentation is introduced. The approach, called MCET-CSA, integrates the Minimum Cross Entropy Thresholding (MCET) criterion with the CSA method. Under the proposed method, CSA is used to minimize the cross entropy among classes. Therefore, at each generation, CSA encodes a set of candidate threshold points into a solution. The objective function that evaluates the cross entropy determinates the quality of the proposed solution. Guided by the values of the objective function, new candidate solutions are generated using the predefined operators of CSA while the the segmentation quality is improved as the process evolves. Although the proposed method can be used as a generic segmentation algorithm, its characteristics allow obtaining excellent results in the automatic segmentation of complex MR images. Under such circumstances, our approach has been evaluated using two sets of benchmark images; the first set is composed of general images commonly used in the image processing literature, while the second set corresponds to MR brain images. In order to evaluate the performance of MCET-CSA, its results are compared with those produced by two popular ECT such as the Differential Evolution (DE) and Harmony Search (HS) (Loganathan, 2001; Storn & Price, n.d.).

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The remainder of the paper is organized as follows: Section 2 describes cross entropy. In section 3 the standard version of CSA is introduced. Section 4 describes the implementation of the proposed algorithm. Section 5 discusses the experimental results after testing the MCET-CSA on two sets of benchmark images, and a statistical analysis is presented. Lastly, the work is concluded in Section 6.

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2. Entropy-based Multilevel Thresholding (MTH)

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Thermodynamics defines entropy as a metric to measure the order of irreversibility in the universe. In physical terms, entropy expresses the amount of disorder of a system. In the context of information theory, various formulations of entropy are used in order to measure homogeneity of data.

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2.1. Cross entropy

Kullback proposed the cross entropy in (Kullback, 1968). Let J { j1, j2 ,..., jN } and G{g1, g2 ,..., g N } be two

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probability distributions on the same set. The cross entropy between F and G is an information theoretic distance between the two distributions and it is defined by: N j D ( J ,G )  ji log i g i i 1

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The minimum cross entropy thresholding (MCET) algorithm (Li & Lee, 1993) selects the threshold by minimizing the cross entropy between the thresholded version and its original image. The original image is I and hGr (i ) , i= 1, 2, …, L, is the corresponding histogram with L being the number of gray levels. Then the thresholded image, denoted by It using th as the threshold value is constructed by: 6

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  (1,th), I ( x, y )th, I ( x, y )   t  (th, L1), I ( x, y )th,

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where b1 b1  ( a, b)   ihGr (i )/  hGr (i ) i a i a

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Since Eq. 2 generates a thresholded image instead of an entropy value, the cross entropy is rewritten as an objective function: th1  i  L Gr   i fCross (th)  ihGr (i )log   ih (i )log   (1, th )  ( th , L  1)     i 1 i th

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The objective function considers a single threshold value for a bilevel thresholding. Eq. 4 can be extended to a multilevel approach. First Eq. 4 can be expressed as: L th1 L fCross (th)  ihGr (i )log i   ihGr (i )log  (1,th)   ihGr (i )log  (th, L 1)  i 1 i 1 i th

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The multilevel approach is based on the use of the vector thth1,th2 ,...,thnt  wich contains nt different

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thresholds values.

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L nt fCross (th )  ihGr (i )log i   Hi i 1 i 1

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Where k is the number of threshold and entropies to calculate

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th11 H1  ihGr (i )log  (1,th1)  i 1

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thk 1 H k   ihGr (i )log  (thk 1,thk )  , 1k nt i thk 1 L H nt   ihGr (i )log  (thnt , L 1)  i thnt

It must be noticed that the MCET process can be easily implemented for a color image. For this purpose each channel of the picture is treated as a single gray level image, allowing to compute the cross entropy as explained in this section.

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3. Crow Search Algorithm (CSA) The Crow Search Algorithm (CSA) proposed by Askarzadeh (Askarzadeh, 2016), is an ECT inspired by the intelligent behavior of crows. Crows are considered to be among the world´s most intelligent animals (Black, 2013). Thus, their behaviors can provide interesting heuristics. CSA is inspired by the thievery behavior that crows exhibit; such behavior can be summarized as follows: crows memorize the position where they hide excedent food, a crow can follow another one to do thievery on their caches, in order to protect their food hideouts, crows can mislead thieves by moving randomly. From a computational point of view, in the CSA

c , c ,..., c  k 1

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of N individuals (crows) is evolved from initial point  k  0  to a total

number of iterations  k  gen  . Each crow

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, cik,2 ,..., cik, N  where each dimension corresponds to a decision variable of the problem to be solved. In

CSA, a new population Ck 1 is generated considering two states: the first is when the crow is aware that is being followed and the second is when the crow is not aware. An awareness probability factor APi k determines wich state is selected. Each new element can be computed as follows:

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ck  r  fl   m kj  cik  rj  APi k cik 1   i i otherwise  random position

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where ri and rj are random numbers drawn from a uniform distribution between 0 and 1, fl is a parameter that controls the flight length. m kj is the memory of the crow j at iteration k; it stores the best position found so

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far by the crow j. CSA only require the configuration of two parameters, fl and APi k . Besides, the update expression is quite simple, providing a user-friendly yet powerful implementation.

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The required steps of CSA are presented in the flowchart of Fig. 1. The first stage is the initialization of the problem (dimensions and limits) and the parameters of the CSA (fl, AP, stop criteria). The next step consists in randomly initialize the crow positions and evaluate them using the objective function. The positions are uniformly distributed in the search space. After that, the new positions are generated according to Eq. 8 and their feasibility is verified. All the new positions are evaluated in the objective function, and the memory is updated. Finally, the stop criteria is verified in order to terminate or continue with the iterative process. The stop criteria depend on the implementation of the CSA, but two common rules are used: 1) the use of a predefined number of iterations and 2) when the objective function value of the best crow converges and does not change along the iterations.

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Figure 1. Crow Search Algorithm.

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4. MCET-CSA

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In this paper, a new methodology called MCET-CSA is presented for image thresholding. Multilevel cross entropy thresholding partitions the image into a finite number of classes by the determination of threshold values, where every new threshold adds complexity to the problem by increasing the restrictions and the modality of the search space, especially when the histogram of the image presents irregularities. CSA is used to minimize the cross entropy among classes. For every generation, CSA encodes a set of candidate threshold points into a solution. The objective function uses the cross entropy to determinate the quality of the proposed solution. Following the rules of the CSA and the value of the objective function, new candidate solutions are generated using the predefined operators of CSA while improving the segmentation quality as the process evolves.

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Since multilevel thresholding can be treated as an optimization problem, the objective function for the cross entropy criterion is stated as:

arg min th subject to

fCross  th  th  X

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where fCross (th) is the cross entropy function (Eq. 6) and X th nt |0thi 255,i 1,2,...,nt is the constrained feasible region, bounded by the intensity values of the interval 0-255. Thus, The CSA is used to find threshold values that solve Eq. 6.

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The proposed MCET-CSA takes advantage of the user-friendly scheme proposed by Askarzadeh where a single equation provides enough search capabilities. Besides, most ECTs suffer from a time-consuming parameter tuning step, while CSA only requires the selection of two parameters named awareness probability AP and flight length fl. The non-greedy nature of CSA encourages diverse solutions through the search space. To ensure the convergence of the algorithm each crow retains on its memory as a food cache the best solution found by that particular crow. The mixture of such mechanisms provides an efficient and robust search algorithm. MCET-CSA also benefits from the definition of Li´s minimum cross entropy wich is faster and more accurate than traditional approaches such as Otsu´s between class variance. Nevertheless, the recent publication of CSA provides only a handful of publications that use and evaluate such algorithm. Thus, the real potential and limitations of CSA require a thorough analysis. In this paper, it was observed that the fl parameter does not drastically modify the general performance of the algorithm, while the correct selection of the AP value significantly improves results. 4.1. Solution representation

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Every threshold th is incorporated into the optimization algorithm as a decision variable at each element of the population. Thus, the population is represented as: Spt [th1,th2 ,...,th N ],thi [th1,th2 ,...,thnt ]T

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where t is the iteration number, N is the size of the population, T refers to the transpose operator and nt is the number of thresholds applied to the image.

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4.2. CSA implementation

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In this paper, a MCET-CSA implementation considers cross entropy as the objective function. The implementation of the MCET-CSA can be summarized into the following steps: Read the image I and store it into I Gr .

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Calculate histogram hGr of I Gr .

Step 3:

Initialize CSA parameters: Awareness probability APi k and flight length fl .

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Initialize a population Spt of N random particles with nt dimensions.

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Evaluate objective function

 fCross  for each element of

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Spt . Eq. 6.

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Generate a random number r j

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k 1 k k k If rj  AP update solution via ci  ci  ri  fl  m j  ci .

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If rj  AP update solution via cik 1  random position

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Step 9: Step 10: Step 11:

Verify if all the N crows are feasible solutions. If the stop criteria are not satisfied jump to step 5. Generate segmented image I s with g and Eq. 12

4.4. Multilevel thresholding Once the CSA stops iterating, the best threshold values that minimize the cross entropy are selected to segment the image pixels. One of the most used segmentation rules for two thresholds is expressed as follows: 10

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IGr ( r ,c ) if  I S ( r ,c ) th1 if I ( r ,c ) if  Gr

IGr ( r ,c )th1 th1IGr ( r ,c )th2 IGr ( r ,c ) th2

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where IGr ( r ,c) is the gray value of the segmented image both in the pixel position r, c. th1 and th2 are the threshold values. Since this paper presents a multilevel approach Eq. 23 is extended to nt levels. IGr ( r ,c )th1 thi 1IGr ( r ,c )thi ,i 2,3,...,nt 1 IGr ( r ,c )thnt

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IGr ( r ,c ) if  I ( r , c )   thi 1 if S I ( r ,c ) if  Gr

5.Experimental results

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The proposed methodology has been validated using two sets of benchmark images; the first set is composed of general images commonly used in the image processing literature, while the second set is made out of MR brain images. The first benchmark set is used to analyze the search capabilities of CSA. For this purpose, a high-dimensional framework is used to demonstrate the search capabilities of the algorithm. Most related literature focuses on the search for optimal threshold values in an up to 5-dimensional scheme (5 threshold values), while in this paper, the search is conducted in an up to 32-dimensional space. The number of selected

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thresholds for the first benchmark nt 2n ,n1,2,...,5 provides a broader comparison range. Most related work only takes into account the results of segmented images with 2, 3, 4 and five thresholds wich leads to a rather small difference between the compared approaches. The superiority of an algorithm against other approaches regarding quality and speed becomes noticeable as the number of thresholds increases. The limit of 32 thresholds is established since not many applications require a larger amount of classes. This set is composed by ten general purpose benchmark images. Most of the selected images are widely used in the image processing literature to test different methods (Cameraman, Lena, Peppers, Baboon, among others). All images have the same size (512 x 512 pixels), and they are in TIFF format.

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The second set of benchmark images is used to determine the performance of MCET-CSA when applied specifically to MR brain images. A set of 8 images extracted from Brainweb database (http://brainweb.bic.mni.mcgill.ca/brainweb). Due to the nature of the images of the second benchmark it is possible to establish an approximate number of classes present on the images. Contrary to the first benchmark image set where the images varied from people, landscapes, and items, the second benchmark contains only brain images where the number of classes does not variate. This characteristic allows focusing the search on a maximum of five threshold values nt2,3,4,5 .

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For both benchmarks, MCET-CSA obtained results showing competitive segmentation quality and consistency. Moreover, results are validated within a statistically significant framework. All experiments were done using Matlab 8.3 on an i5-4210 CPU @ 2.3Ghz with 6GB of RAM. Each experiment of a population based algorithm, such as CSA and DE, has a stop criterion set to 2500 iterations, but the algorithm is terminated if the fitness value is not improved during 10% of the stop criterion. Since HS is based on a single particle, its stop criterion is set to the size of the population of CSA and DE multiplied by 2500. Using this criterion the HS is able to find the best value (Geem, Kim, & Loganathan, 2001). 11

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The first assessed metric is the stability of the algorithm at the end of each test; wich is verified by the standard deviation (STD) (Eq. 13). If the STD value increases the algorithm becomes more instable (Ghamisi, Couceiro, Benediktsson, & Ferreira, 2012).

STD 

(13)

Itermax     i  Ru i 1

 255  PSNR 20log10  ,(dB)  RMSE  co iro 1 j 1 IGr  i , j  Ith  i , j   roco

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RMSE 

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Moreover, the peak-to-signal ratio (PSNR) is used to compare the similarity of an image (image segmented) against a reference picture (original image) based on the mean square error (MSE) of each pixel (Agrawal, Panda, Bhuyan, & Panigrahi, 2013; Akay, 2013; Horng, 2011; Il-Seok Oh, Jin-Seon Lee, & Byung-Ro Moon, 2004). Both PSNR and RMSE are defined as: (14) (15)

Where IGr is the original image and Ith is the segmented image; the total numbers of rows is ro, and the total numbers of columns is co. The Structure Similarity Index (SSIM) is used to compare the structures of the original umbralized image (Wang, Bovik, Sheikh, & Simoncelli, 2004), and it is defined in Eq. 16. A better segmentation performances produce a higher SSIM. (16)

  2   2 C1  2  2 C1  I  I   Gr Ith  Gr Ith 

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1 N  IGri   IGr N 1i 1

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 2IGr Ith C1 2 IGr Ith C 2

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SSIM ( IGr , Ith )

 Ithi Ith 

(17)

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where  IGr and  Ith are the mean value of the original and the umbralized image respectively, for each image the values of  I and  I corresponds to the standard deviation. C1 and C2 are constants used to Gr th

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avoid the instability when I2  I2 0 , experimentally in (Agrawal et al., 2013) both values are Gr th C1=C2=0.065. One more method used to determinate the quality of the segmented image is the Feature Similarity Index (FSIM) (D. Zhang, 2011). FSIM computes the similarity between two images, in this cases the original gray scale image and the segmented image (Eq.18). Just like PSNR and SSIM, a higher value is interpreted as a better performance of the thresholding method. S  w PCm  w  FSIM  w L  w PCm  w

where  represents the entire domain of the image:

12

(18)

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SL  wSPC  wSG  w S PC  w

(19)

2 PC1 w PC2  w T1 PC12  w PC22  wT1

(20)

2G  wG2  wT2 SG  w 1 G12  wG22  wT2

(21)

(22)

G  Gx2 G 2 y

PC is the phase congruence: E ( w)

 n An ( w) 

(23)

AN US

PC ( w)

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G is the gradient magnitude (GM) of an image and is defined as:

The magnitude of the response vector in w on n is E(w) and An ( w) is the local amplitude of scale n.  is a small positive number and PCm ( w)max PC1( w), PC2 ( w)  . 5.1. Parameter selection

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In this paper for the CSA implementation, the parameters used are the ones proposed on its definition (Askarzadeh, 2016). DE uses the parameters presented in (Erik, Pedersen, & Pedersen, 2010) while HS uses (D Oliva, Cuevas, & Pajares, 2013) from a similar application. Table 2 presents all parameter for every algorithm employed in this paper. DE Population N: 20

HS Harmony Memory: 100

Awareness probability AP:0.1

Crossover probability: 0.7455

Consideration Rate: 0.75

Flight length fl:2

Differential weight: 0.9362

Pitch adjusting rate: 0.5

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CSA Population N: 20

Distance Bandwith:0.5 Number of Improvisations:300 Table 2. Selected parameters for CSA, DE, and HS.

5.2. MCET for general purpose images This subsection analyzes the results provided by MCET implementations based on CSA, DE, and HS, after being applied to segment the first benchmark images. The distance between distributions, also known as the cross entropy fCross is selected as the objective function. The methodology proposed is applied to the complete benchmark image set. 13

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Table 3 reports the best threshold values obtained after testing CSA, DE, and HS with fCross as an objective function, evaluating five different number of thresholds

nt 2,4,8,16,32 .

Since most ECT work well on low-

2 4 8 16 32

Livingroom

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2 4 8 16

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32

Peppers

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2

HS 50 137 29 76 125 157 22 47 81 112 134 154 172 201 11 15 21 33 47 63 82 101 115 127 140 152 162 171 181 207

5 9 12 16 19 28 38 46 53 64 72 83 95 105 110 117 124 128 136 147 148 157 160 166 171 179 180 186 193 205 233 247 83 142 71 109 141 177 50 66 86 105 123 142 163 189 43 50 58 68 79 91 102 113 125 136 146 156 166 178 191 206

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DE 50 137 29 76 125 157 26 49 80 113 140 155 175 189 10 13 18 30 52 81 103 124 129 142 152 162 166 172 184 213 6 7 11 11 14 19 26 29 41 47 64 75 79 102 112 116 119 121 129 134 138 145 154 160 166 177 182 188 188 212 213 223 83 142 71 109 141 177 54 70 94 117 137 152 164 196 42 52 64 70 76 92 104 118 130 134 149 155 169 190 195 235 10 24 30 43 49 56 65 74 85 91 93 97 102 103 109 115 128 130 139 149 152 157 159 160 164 172 178 192 195 208 232 240 74 134 38 83 124 161 30 43 60 90 108 127 150 184 13 21 41 54 73 76 90 104 108 117 129 142 158 167 183 203 3 5 7 13 25 33 35 42 44 51 66 81 88 97 100 106 112 113 121 123 131 142 148 150 152 162 164 180 193 198 207 208 40 123 31 96 132 165 9 46 80 100 114 131 157 179 3 22 25 40 51 69 88 101 111 121 132 137 150 165 182 191 4 14 28 46 51 53 61 61 68 75 82 86 91 92 102 108 110 112 119 128 132 134 135 138 144 154 162 167 175 184 194 223 54 127 37 77 119 165 23 42 68 88 106 130 156 182 11 19 32 36 58 73 89 99 112 124 141 160 172 178 197 200 14 18 22 29 41 46 49 59 68 73 79 88 93 103 112 120 123 128 131 132 145 154 165 175 185 193 212 222 232 238 241 256 84 157

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CSA 50 137 29 77 126 158 13 25 50 83 114 136 156 176 11 15 21 32 46 63 82 99 113 125 137 149 160 170 180 205 4 8 11 13 16 20 25 31 38 46 54 62 71 81 94 104 109 116 121 127 134 140 148 154 160 166 172 177 183 193 202 226 83 142 71 109 141 177 52 69 90 111 130 147 166 191 43 51 60 70 81 92 102 112 122 132 141 151 161 173 188 204 37 42 46 50 54 59 64 69 75 81 86 92 98 104 109 114 120 126 131 136 141 146 152 158 163 169 175 183 191 199 208 216 73 134 38 83 124 161 21 43 66 89 111 131 152 181 10 20 33 46 59 72 86 98 110 120 132 142 152 165 181 202 2 6 11 16 23 30 36 42 50 56 63 68 75 83 90 96 102 108 115 121 126 131 136 142 147 155 162 170 180 190 201 215 38 122 31 96 132 165 11 49 77 95 117 138 157 177 5 22 42 59 74 83 93 104 116 128 138 148 158 168 178 188 2 5 12 20 31 40 50 59 68 74 80 85 90 95 101 106 113 120 125 130 136 141 146 150 156 161 167 173 179 186 191 199 54 127 37 77 119 165 22 43 68 89 109 134 158 183 14 22 32 45 58 71 83 93 104 117 131 145 158 171 185 200 7 12 16 21 26 32 40 48 54 61 68 75 81 87 93 99 105 113 121 128 135 141 147 152 158 164 170 178 186 192 199 209 84 156

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Image Cameraman

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dimensional search spaces, this article explores the behavior of MCET-CSA on a high dimensional framework by expanding the search up to 32 dimensions (32 thresholds). Threshold values obtained for most of the images with th2,4 show evidence that the three selected algorithms (CSA, DE, and HS) tend to converge to the same threshold values. As the number of thresholds increases the results grow apart due to the high-dimensionality of the problem´s search space. Under such circumstances, the algorithms can be intensely evaluated in a high modality environment.

14

11 15 38 46 48 54 62 68 74 86 93 99 105 109 114 125 129 139 146 147 152 157 163 173 175 184 193 205 212 212 229 241 73 134 38 83 124 161 21 42 64 86 108 129 151 180 10 21 34 47 59 72 85 99 110 121 131 141 151 163 180 203 5 11 18 26 29 40 46 50 59 62 69 72 79 88 101 107 114 124 130 133 141 147 153 161 164 172 177 188 201 216 219 244 38 122 31 96 132 165 11 49 77 95 117 138 157 177 5 20 39 59 74 83 94 107 120 132 143 152 161 170 180 189 5 11 19 36 51 59 71 73 79 80 83 86 95 103 117 122 124 131 140 145 152 158 163 169 177 182 186 190 193 226 236 240 54 127 37 77 119 165 22 43 68 89 109 133 158 183 15 24 34 47 60 73 85 95 106 119 134 147 160 173 186 200 7 13 19 26 35 39 50 55 57 68 71 80 86 94 95 99 102 113 123 133 141 152 158 159 168 178 183 190 201 210 228 244 84 156

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24 39 41 57 60 70 78 79 88 92 102 107 113 119 127 127 131 135 136 145 156 163 164 172 175 185 194 205 215 221 226 227 97 162 64 107 150 193 53 87 105 126 150 177 197 213 18 35 62 74 88 98 108 114 126 145 159 176 178 190 207 220 3 34 39 40 41 52 62 71 80 89 94 98 111 114 126 127 139 145 146 153 159 171 172 188 195 197 201 203 207 212 218 250 74 142 57 91 143 195 43 62 76 96 125 146 176 204 27 42 49 56 63 80 84 101 121 140 160 173 186 209 228 240 15 26 34 43 47 51 58 61 64 65 76 85 92 93 104 114 132 135 144 147 155 160 170 176 183 189 198 201 220 230 235 246

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14 24 31 38 44 50 56 62 68 73 79 85 91 96 100 105 111 117 122 127 131 136 140 146 153 158 165 171 178 184 191 200 97 162 64 107 150 193 54 84 106 128 153 179 198 211 38 55 71 83 95 106 117 129 143 158 171 183 194 202 209 216 3 28 36 45 54 62 70 77 83 90 97 103 109 115 122 129 136 143 151 159 166 173 179 185 190 194 199 203 207 211 215 220 74 142 57 91 143 195 40 54 69 89 114 144 173 202 30 39 47 55 64 74 85 98 113 130 148 165 179 190 202 217 1 12 23 29 35 40 44 48 52 57 62 67 73 80 87 94 101 109 117 128 138 148 156 164 172 179 185 192 200 207 215 223

14 34 44 47 49 51 54 59 65 66 72 77 83 91 95 99 110 120 131 142 145 158 165 172 184 190 196 208 216 223 235 250 68 144 35 80 127 186 25 47 66 85 102 126 159 201 20 33 45 55 65 76 86 96 105 113 124 140 157 177 198 221

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2 4 8 16

62 92 133 185 53 66 85 108 132 156 183 214 47 51 57 64 73 84 95 108 121 135 149 164 180 196 214 231

15 20 27 33 41 47 53 59 65 67 71 78 85 86 94 98 104 110 116 119 131 133 135 157 168 176 196 201 218 228 240 255 92 144 66 100 130 161 46 67 86 105 123 140 159 179 28 42 54 66 77 88 99 109 119 128 138 148 159 170 180 191 18 26 35 39 50 56 67 76 82 84 91 100 106 113 113 123 128 131 135 144 153 158 168 178 185 188 192 200 205 234 239 240 97 162 64 107 150 193 53 81 101 121 145 172 194 209 36 50 65 79 91 102 113 125 139 154 169 183 194 202 209 216

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Splash

62 92 133 185 52 67 87 106 133 160 183 213 50 54 61 69 80 91 98 112 127 131 144 159 183 205 231 239 9 47 52 54 60 61 66 68 75 83 89 93 93 101 106 118 120 121 125 130 135 147 156 170 181 181 195 207 214 218 224 235 67 142 35 80 127 186 25 51 68 89 102 128 159 208 23 32 41 44 58 72 85 101 114 117 131 142 164 174 202 203 16 21 33 39 40 43 44 46 53 62 65 74 79 88 94 98 100 102 111 118 136 145 168 173 174 179 188 188 196 216 224 236 92 144 66 100 130 161 49 65 86 102 116 134 151 173 43 56 58 69 84 90 97 104 114 131 142 151 163 174 178 181

M

32

62 92 133 185 53 67 86 110 134 158 185 215 48 54 61 69 79 90 101 114 127 140 153 167 182 197 213 231 42 47 50 54 58 62 67 72 77 81 88 94 99 105 113 119 125 133 140 146 153 161 168 176 183 190 197 204 213 222 228 240 68 144 35 80 127 186 26 48 67 86 103 127 159 201 19 32 44 55 65 76 86 95 104 111 121 136 155 175 195 219 15 20 26 33 39 44 49 55 61 67 73 79 84 89 93 97 101 105 109 113 118 124 132 141 150 157 166 177 188 203 217 229 92 144 66 100 130 161 46 67 86 105 122 139 158 178 29 44 56 67 78 89 99 109 118 127 136 146 157 168 179 191

ED

4 8 16

PT

Hair

29 32 40 48 49 57 63 71 79 87 89 95 100 112 122 132 143 146 157 158 168 174 185 190 197 203 205 207 212 216 220 253 74 142 56 89 142 195 39 53 68 88 113 143 173 202 28 37 45 52 60 69 80 93 108 125 143 161 176 189 202 216 12 25 34 39 45 49 52 59 61 68 71 82 93 94 101 113 126 137 144 146 153 161 169 180 186 195 207 209 210 221 226 243

Table 3. Best thresholds found by CSA, DE, and HS.

AC

Statistical data such as the mean and STD Eq. 13 is reported in Table 4. The mean reported is averaged from a 35 run experiment evaluating as an objective function the cross entropy Eq. 6. Since it is a minimization problem, the mean is expected to be as low as possible. From the three algorithms implemented (CSA, DE, and HS), CSA is able to find the best threshold values that minimize the objective function. CSA Image Cameraman

nt 2 4

Mean 1.4446 0.5629

DE

Std 0.0025 0.0032

15

HS

Mean

Std

1.4647 0.6007

0.0183 0.0262

Mean 1.5202 0.5901

Std 0.0059 0.0006

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Peppers

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Dark

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Jet

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Baboon

AC

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Sailboat on lake

0.2942 0.1280 0.0502 1.3690 0.5078 0.2264 0.0986 0.0396 1.9003 0.8110 0.3561 0.1429 0.0537 1.5343 0.5699 0.2438 0.0993 0.0372 1.8898 0.7781 0.3501 0.1450 0.0572 1.4417 0.5615 0.24 0.0981 0.0383 2.298 0.8664 0.3089 0.1263 0.0462 1.1072 0.4861 0.2288 0.0987 0.0382 0.822 0.3555 0.1511 0.0637 0.0244 1.4448 0.6700 0.2841 0.1196 0.0459

0.0155 0.0111 0.0039 0.0164 0.0231 0.0207 0.0096 0.0035 0.0196 0.0305 0.0216 0.0101 0.0028 0.011 0.0206 0.0201 0.0088 0.0031 0.0204 0.0384 0.0245 0.0119 0.0046 0.0149 0.0217 0.0197 0.0076 0.0037 0.0201 0.0378 0.0241 0.0096 0.0031 0.0175 0.0280 0.0200 0.0070 0.0029 0.0054 0.0124 0.0126 0.0050 0.0018 0.0133 0.0232 0.0257 0.0098 0.0031

0.2394 0.0817 0.0347 1.4212 0.4989 0.1670 0.0549 0.0288 1.9951 0.8092 0.2778 0.0953 0.0393 1.6025 0.5652 0.1813 0.0526 0.0254 1.9569 0.7553 0.2691 0.0837 0.0415 1.4952 0.5556 0.1753 0.0552 0.0261 2.3923 0.8533 0.2332 0.0702 0.0336 1.14460 0.47187 0.16863 0.05302 0.02908 0.86205 0.35847 0.11760 0.03507 0.01764 1.50937 0.66507 0.21420 0.0716 0.0347

0.0051 0.0115 0.0012 0.0048 0.0057 0.0004 0.0017 0.0016 0.0154 0.0068 0.0061 0.0038 0.0025 0.0034 0.0013 0.0033 0.0013 0.0001 0.0068 0.0008 0.0036 0.0018 0.0032 0.0056 0.0036 0.0024 0.0005 0.001 0.0005 0.0030 0.0044 0.0028 0.0006 0.0005 0.0076 0.0027 0.0010 0.0006 0.0067 0.0017 0.0051 0.0021 0.0005 0.0103 0.0036 0.0015 0.0059 0.0021

CR IP T

Blonde

0.0028 0.0032 0.0012 0.0025 0.0037 0.0019 0.0031 0.0011 0.0021 0.0040 0.0029 0.0021 0.0011 0.0017 0.0044 0.0027 0.0030 0.0010 0.0040 0.0054 0.0049 0.0032 0.0012 0.0037 0.0038 0.0028 0.0041 0.0013 0.0028 0.0031 0.0035 0.0028 0.0011 0.0027 0.0039 0.0022 0.0030 0.0009 0.0010 0.0019 0.0018 0.0021 0.0006 0.0029 0.0031 0.0029 0.0024 0.0009

AN US

Livingroom

0.2238 0.0722 0.0233 1.3503 0.4708 0.1584 0.0520 0.0163 1.8770 0.7665 0.2610 0.0794 0.0235 1.5224 0.5366 0.1719 0.0521 0.0160 1.8621 0.7161 0.2551 0.0783 0.0242 1.4233 0.5266 0.1680 0.0569 0.0185 2.2751 0.8114 0.2188 0.0668 0.0211 1.0885 0.4452 0.1585 0.0514 0.0159 0.8164 0.3382 0.1096 0.0343 0.0110 1.4304 0.6343 0.2060 0.0657 0.0209

M

Lena

8 16 32 2 4 8 16 32 2 4 8 16 32 2 4 8 16 32 2 4 8 16 32 2 4 8 16 32 2 4 8 16 32 2 4 8 16 32 2 4 8 16 32 2 4 8 16 32

Table 4. Results after applying MCET-CSA to the general benchmark images.

At the beginning of section 5, three quality metrics were introduced. Peak Signal Noise Ratio (PSRN), Structural Similarity Index (SSIM) and Feature Similarity Index (FSIM). The objective of these metrics is to assess the quality of the segmented image by comparing it with the original image. For the three metrics, a high value indicates an accurate representation of the original image. In order to establish a reference model to compare the results of the proposed approach, the classic Otsu´s method is implemented and evaluated for the entire benchmark dataset with three ECTs. Since the cross 16

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entropy function is minimized while Otsu´s between class variance is maximized the fitness values cannot be directly compared. To overcome this issue, the quality of the segmented images of each approach are contrasted regarding PSNR. Table 5 shows the PSNR generated by each experiment.

Livingroom

Blonde

Peppers

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Woman Dark Hair

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Baboon

Jet

AC

15.9977 21.4231 25.9145 31.4143 37.1093 15.6372 18.7857 23.3044 29.6854 37.2611 16.3067 20.4330 25.5823 31.1539 36.6436 14.7983 19.2065 26.4012 32.5018 38.2956 15.4950 20.2982 25.4441 30.9490 36.7179 14.0313 16.8290 19.7058 24.2454 35.4863 15.3878 19.4305 25.1993 30.6315 35.9632 15.8757 20.9663 26.6284 32.4899 38.2307 14.8736 21.2358 26.7901 32.4417 38.8118 14.2893 17.8982 24.3622 30.8055 36.7543

Sailboat on lake

Cross entropy DE 16.0037 21.2830 25.1436 29.0534 33.5438 15.6366 18.7711 22.6425 26.8279 32.9528 16.3067 20.4016 24.6698 28.9843 33.2211 14.796 19.1929 25.1298 29.8263 34.1692 15.4396 20.1879 24.5885 28.2993 32.6577 14.0322 16.8103 19.2465 24.2081 30.7002 15.3841 19.3874 24.3335 28.3716 33.0156 15.8774 20.8338 25.3969 29.5991 34.0164 14.8741 21.0921 25.6144 29.8470 34.7042 14.2771 17.9428 23.3960 27.9829 32.9337

HS

CSA

15.5178 20.7776 24.8912 30.4845 34.5856 15.1681 18.2182 22.6562 28.7519 33.2069 15.8168 19.8056 24.9327 30.2537 33.7799 14.3576 18.6226 25.6400 31.7121 34.6468 14.9817 19.6741 24.6655 30.0746 33.3549 13.6104 16.3435 19.1943 23.1113 31.2013 14.9262 18.8455 24.4594 29.7898 33.3499 15.3994 20.3373 25.8505 31.4009 34.3016 14.4274 20.5938 25.8937 31.5194 35.2961 13.8606 17.3603 23.5948 29.9086 33.7124

15.0362 20.6406 25.0075 30.8975 35.5327 12.5669 17.4860 21.3257 28.4563 31.6590 14.3946 18.3707 24.9349 30.9721 35.1318 12.5935 19.1572 24.6503 29.2286 28.9646 12.0578 18.6608 24.5579 28.4325 32.7740 12.0203 15.6595 19.1826 23.6972 26.3284 9.72490 18.3250 24.5988 29.2106 33.5241 10.9911 17.9615 24.9576 29.4283 32.9882 12.5989 19.2776 25.4857 29.9132 29.6218 12.1359 15.8598 22.7627 28.4566 33.2817

Otsu´s method DE 14.6311 21.0243 25.2883 30.3384 35.3635 12.6007 17.3036 21.4849 27.9376 29.4250 13.7776 18.4126 24.8798 30.4887 34.7629 12.2782 19.1915 24.6777 29.5547 29.5456 11.7910 18.5082 24.3902 29.4725 32.3887 12.0733 15.5805 18.4021 23.0617 27.5433 9.53860 17.0876 24.2083 29.2523 33.0466 10.8118 17.8907 24.1832 29.2898 32.6924 11.4996 19.3855 25.2497 29.3420 31.2678 11.7854 15.8959 22.2225 28.7639 32.1775

HS 14.5556 20.6871 25.1535 31.0723 35.5755 12.7133 17.4378 22.0128 27.8872 28.4686 13.8624 18.3967 24.9689 31.0989 35.0233 12.8562 19.1572 25.5613 29.112 28.7032 11.9489 18.6425 24.5958 29.0455 32.605 12.2958 15.6781 18.7115 25.8992 31.0324 9.72690 17.2665 24.7422 29.0773 33.4219 11.0492 17.9778 24.9269 28.9531 32.8527 11.9582 19.3045 25.2142 29.6200 32.9738 12.8466 15.8691 22.7197 28.4901 33.5117

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Lena

2 4 8 16 32 2 4 8 16 32 2 4 8 16 32 2 4 8 16 32 2 4 8 16 32 2 4 8 16 32 2 4 8 16 32 2 4 8 16 32 2 4 8 16 32 2 4 8 16 32

CSA

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Cameraman

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Table 5. Comparison of PSNR values obtained with CSA, DE and HS for both Cross entropy and Otsu´s method.

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Algorithms based on evolutionary methods are, in general, complex processes with several elements and stochastic operations. Under such conditions, it is difficult to conduct a complexity analysis from a deterministic perspective. Therefore, the computational time is commonly used to evaluate their computational effort. In order to assess the computational effort, an experiment is conducted. In the experiment, the computational time is evaluated and compared for each algorithm when they operate over a set of representative images. Figure 2 depicts the computational time required for the evaluation of both cross entropy and Otsu´s method using CSA as the optimization algorithm. On the horizontal axis, the names of the general purpose benchmark set are listed with groups of five bars for each image. Every bar represents the required time to evaluate CSA with the corresponding threshold technique. The bars are grouped by the number of thresholds for 2, 4, 8, 16, and 32 for each image from left to right. It can be easily noticed that cross entropy requires less computational time in comparison to Otsu´s method, especially for a high number of thresholds.

ED

Figure 2. Time comparison Otsu vs. Cross entropy using CSA.

CE

PT

Table 6 reports SSIM, and FSIM from the evaluation of the segmented images. As described in section 4.4, segmented images are generated using Eq. 12 and the threshold values calculated by an ECT. In this case, CSA, DE, and HS are used considering cross entropy as the objective function. It must be noticed that the results reported are averaged after a 35 run experiment. CSA outperforms DE and HS according to the three quality metrics. Image

AC

Cameraman

Lena

Livingroom

nt 2 4 8 16 32 2 4 8 16 32 2 4 8

CSA SSIM 0.5891 0.6688 0.7635 0.9148 0.9716 0.5601 0.6499 0.7809 0.9034 0.9727 0.5454 0.7147 0.8512

DE

FSIM 0.7951 0.8849 0.9485 0.9771 0.9944 0.7674 0.8549 0.9087 0.9661 0.9948 0.7581 0.8820 0.9559

18

SSIM 0.5891 0.6685 0.7440 0.8578 0.9414 0.5599 0.6500 0.7640 0.8542 0.9345 0.5454 0.7137 0.8278

FSIM 0.7952 0.8829 0.9373 0.9632 0.9835 0.7674 0.8546 0.9014 0.9396 0.9815 0.7581 0.8808 0.9422

HS SSIM 0.5890 0.6699 0.7616 0.8909 0.9602 0.5600 0.6497 0.7819 0.9008 0.9486 0.5454 0.7147 0.8501

FSIM 0.7951 0.8851 0.9460 0.9759 0.9904 0.7674 0.8549 0.9079 0.9668 0.9867 0.7581 0.8819 0.9583

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0.9073 0.9536 0.5904 0.7053 0.8266 0.9046 0.9517 0.5891 0.6716 0.7728 0.8532 0.9252 0.3354 0.4479 0.5589 0.7614 0.8957 0.5774 0.6969 0.7736 0.8477 0.9187 0.6036 0.7797 0.8792 0.9372 0.9709 0.7460 0.8033 0.8519 0.9138 0.9583 0.5279 0.6306 0.8011 0.8933 0.9493

0.9759 0.9899 0.7145 0.8731 0.9427 0.9757 0.9896 0.7522 0.8471 0.9210 0.9597 0.9840 0.7167 0.7860 0.8409 0.9006 0.9578 0.7526 0.8384 0.9039 0.9438 0.9727 0.8174 0.9045 0.9560 0.9821 0.9937 0.8051 0.8862 0.9455 0.9728 0.9902 0.8347 0.8819 0.9364 0.9719 0.9894

0.9395 0.9669 0.5885 0.7047 0.8441 0.9433 0.9647 0.5891 0.6722 0.7873 0.9037 0.9451 0.3353 0.4506 0.5857 0.7654 0.9137 0.5772 0.6975 0.7902 0.8837 0.932 0.6035 0.7844 0.9020 0.9655 0.9786 0.7460 0.8065 0.8675 0.9442 0.9694 0.5281 0.6288 0.8274 0.9343 0.9644

0.9892 0.9942 0.7156 0.8743 0.9591 0.9918 0.9940 0.7522 0.8490 0.9326 0.9825 0.9907 0.7166 0.7867 0.8554 0.9033 0.966 0.7524 0.8393 0.9167 0.9696 0.9786 0.8174 0.906 0.9688 0.9937 0.9966 0.8051 0.8881 0.9617 0.9884 0.9937 0.835 0.8821 0.9461 0.9862 0.9938

CR IP T

Woman Dark Hair

0.9880 0.9963 0.7144 0.8743 0.9589 0.9912 0.9971 0.7522 0.8490 0.9328 0.9822 0.9954 0.7166 0.7873 0.8564 0.9153 0.9850 0.7524 0.8400 0.9169 0.9702 0.9855 0.8174 0.9060 0.9698 0.9937 0.9986 0.8051 0.8880 0.9630 0.9885 0.9972 0.8350 0.8820 0.9466 0.9860 0.9969

AN US

Peppers

0.9405 0.9789 0.5907 0.7055 0.8436 0.9408 0.9795 0.5891 0.6716 0.7875 0.9048 0.9686 0.3353 0.4493 0.5805 0.7745 0.9511 0.5772 0.6975 0.7909 0.8875 0.9543 0.6035 0.7844 0.9028 0.9665 0.9892 0.7460 0.8061 0.8686 0.9462 0.9822 0.5281 0.6293 0.8287 0.9336 0.9772

M

Blonde

16 32 2 4 8 16 32 2 4 8 16 32 2 4 8 16 32 2 4 8 16 32 2 4 8 16 32 2 4 8 16 32 2 4 8 16 32

Table 6. Comparison of SSIM, and FSIM values on general benchmark images using cross entropy.

AC

CE

PT

Table 7 presents the results visually after applying the MCET-CSA to the whole set of general purpose benchmark images. For every picture, the first row presents the segmented image. The images were segmented using Eq. 12 and the best threshold values found by the MCET-CSA. From left to right 2, 4, 8, 16 and 32 thresholds were used in order to demonstrate the search capabilities of the CSA on a high dimensional search space. The second row presents the histogram of the image and their corresponding threshold values.

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Livingroom

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Woman dark hair

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Jet

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Baboon

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Table 7. Results after applying the MCET-CSA to the general purpose benchmark images.

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5.2. MR brain images This subsection analyzes the results provided by MCET implementations based on CSA, DE, and HS, after being used to segment the medical benchmark images. The MCET-CSA is applied to the complete benchmark image set. The benchmark images are extracted from the z planes of the MR with values of 1, 2, 5, 10, 36, 72, 108 and 144 for de z-axis. Those values are selected in order to acquire representative pictures of different sections of the brain. For MR brain images nt 2,3,4,5 .

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

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CSA Mean STD 0 3.4002 1.6976 0 0.9924 0 0.7055 0.0068 3.3614 0 1.6694 0 0.9882 0 0.7054 0.0020 3.4281 0 1.5929 0 0.9991 0 0.7004 0.0089 3.3689 0 1.6002 0 1.1008 0 0.7062 0 3.2285 0 1.7369 0.0042 1.1412 0 0.6853 0 2.0598 0 1.1428 0 0.6606 0 0.4812 0.0203 2.0561 0 1.1331 0 0.6292 0 0.4879 0.0116 1.8075 0 1.1050 0 0.6919 0.0083 0.4128 0

Mean

STD

Mean

3.454 1.7636 1.0929 0.8297 3.4037 1.7382 1.0736 0.825 3.4595 1.6708 1.0856 0.8103 3.4244 1.6981 1.2047 0.8439 3.2796 1.8092 1.2263 0.8386 2.1231 1.2526 0.7989 0.607 2.1156 1.2286 0.7691 0.6162 1.8649 1.1639 0.7774 0.5201

0.0422 0.0401 0.061 0.0692 0.0296 0.0408 0.0472 0.0713 0.0269 0.0527 0.0369 0.0502 0.0364 0.0574 0.0476 0.0594 0.0337 0.0429 0.0413 0.0737 0.0584 0.0714 0.0737 0.0728 0.0485 0.053 0.0877 0.0589 0.0354 0.0239 0.0459 0.059

3.4209 1.7201 1.0142 0.7287 3.3783 1.6947 1.0106 0.7260 3.4482 1.6203 1.0312 0.7305 3.4009 1.6233 1.1232 0.7637 3.2572 1.7610 1.1697 0.7326 2.0790 1.1658 0.7078 0.5131 2.0797 1.1678 0.6762 0.5197 1.8325 1.1238 0.7721 0.4334

STD 0.0309 0.0142 0.0140 0.0105 0.0219 0.0181 0.0107 0.0090 0.0206 0.0142 0.0469 0.0150 0.0307 0.0149 0.0287 0.0463 0.0280 0.0131 0.0311 0.0227 0.0262 0.0220 0.0290 0.0183 0.0248 0.0514 0.0244 0.0170 0.0240 0.0125 0.0394 0.0114

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Table 8. Results after applying MCET-CSA to the MR brain benchmark images.

Similarly to Section 5.1, Table 8 reports the averaged mean and STD of the fitness function of MCET-CSA. Each experiment for every image and nt consist of 35 independent evaluations of the MCET-CSA. Since it is a minimization problem, the mean is expected to be as low as possible. From the three algorithms implemented (CSA, DE, and HS), CSA is able to find the best threshold values that minimize the objective function.

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Z5

Z10

Z36

Z72

Z108

FSIM 0.6076 0.7300 0.8134 0.8484 0.6059 0.7326 0.8138 0.8509 0.6114 0.7490 0.8221 0.8574 0.6642 0.7642 0.8205 0.8622 0.6807 0.7844 0.8414 0.8775 0.7123 0.8267 0.8699 0.9042 0.6913 0.7900 0.8538 0.8823 0.7616 0.8331 0.8675 0.9060

PSNR 14.1969 17.8876 19.7455 21.2603 14.2475 17.9703 19.7855 21.3264 14.5432 18.1763 19.8873 21.2333 16.1504 18.5934 20.1279 21.7720 15.8327 18.2945 20.0269 21.0218 15.6544 19.0740 20.4755 22.8000 16.0332 19.7081 21.2007 22.6305 18.2564 20.9497 22.2878 24.0175

DE SSIM 0.4938 0.6728 0.7322 0.7836 0.4916 0.6743 0.7300 0.7820 0.4943 0.6758 0.7262 0.7819 0.5830 0.6656 0.7023 0.7768 0.6306 0.7099 0.7480 0.8082 0.6648 0.7825 0.8416 0.8849 0.6831 0.8033 0.8692 0.8969 0.7814 0.8305 0.8871 0.9057

FSIM 0.6046 0.7257 0.8074 0.8364 0.6030 0.7291 0.8068 0.8389 0.6084 0.7439 0.8174 0.8472 0.6609 0.7596 0.8125 0.8548 0.6773 0.7798 0.8332 0.8655 0.7089 0.8211 0.861 0.8942 0.6881 0.7844 0.8441 0.8746 0.7575 0.8287 0.858 0.8966

PSNR 14.1942 17.9208 19.7858 21.6730 14.2472 17.9901 19.8458 21.7684 14.5432 18.1748 19.8860 21.8387 16.1597 18.6046 20.1848 21.7906 15.8327 18.1310 19.9601 21.1630 15.6575 19.1179 20.5580 22.9747 16.0210 19.6509 21.3288 22.0668 18.2555 20.9530 22.4511 24.1757

HS SSIM 0.4937 0.6737 0.7340 0.7897 0.4915 0.6739 0.7329 0.7878 0.4943 0.6769 0.7268 0.7811 0.5831 0.6668 0.7050 0.7711 0.6306 0.7065 0.7530 0.8168 0.6649 0.7843 0.8458 0.8947 0.6828 0.8029 0.8734 0.8861 0.7819 0.8308 0.8636 0.9087

FSIM 0.6046 0.7265 0.8094 0.8457 0.6029 0.7290 0.8098 0.8474 0.6084 0.7453 0.8169 0.8532 0.6609 0.7604 0.8158 0.8557 0.6773 0.7778 0.8349 0.8731 0.7088 0.8226 0.8656 0.9009 0.6879 0.7848 0.8496 0.8717 0.7578 0.829 0.8581 0.9015

ED

Z144

CSA SSIM 0.4962 0.6771 0.7377 0.7950 0.4940 0.6773 0.7366 0.7922 0.4968 0.6803 0.7305 0.7894 0.5860 0.6701 0.7085 0.7854 0.6338 0.7149 0.7559 0.8209 0.6682 0.7882 0.8500 0.8967 0.6862 0.8083 0.8778 0.8994 0.7858 0.8350 0.8968 0.9132

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

PSNR 14.2652 18.0139 19.8847 21.5104 14.3184 18.0801 19.9443 21.7553 14.6159 18.2657 19.9872 21.4894 16.2405 18.6976 20.2821 21.6823 15.9119 18.4363 20.1750 21.2671 15.7358 19.2135 20.6608 22.9666 16.1011 19.8096 21.4354 22.6324 18.3468 21.0578 22.5162 24.2984

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Table 9. Quality evaluation results after applying MCET-CSA to MR brain images where the number of the image title indicate de z plane of the analyzed image.

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Table 9 presents the averaged results of the quality metrics (PSNR, SSIM, and FSIM) of the segmented images. Each segmented images is generated using Eq. 12 and the threshold values calculated by an ECT. In this case, CSA, DE, and HS are used. From de quantitative analysis MCET-CSA provides evidence of highquality segmented images.

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Table 10. Qualitative results of MCET-CSA applied to MR brain images with nt=5.

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A qualitative comparison of segmented MR brain images is presented in Table 10. Four images of the medical benchmark set are selected to visually show the results of the implemented methods. For the sake of visibility, instead of displaying the images on grayscale the colormap is changed for the so-called jet. With this representation, the cold colors indicate low-intensity while the hotter stand for high-intensity values. The images segmented with MCET-CSA provide better contours and a more accurate representation.

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5.3. Comparisons

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To assess the results of the approach proposed in this paper, two different comparisons are carried on. The first comparison analyzes the performance metrics among CSA, DE, and HS when applied to the general purpose benchmark set. The second one is intended to evaluate the performance of the proposed methodology when applied to MR brain images. 5.3.1. Comparison general images To evaluate the performance of the proposed algorithm, MCET-CSA is compared with two state-of-the-art optimization techniques. The algorithms selected for comparison are Differential Evolution (DE) (Storn & Price, n.d.) and Harmony Search (HS) (Loganathan, 2001), both minimizing the cross entropy. 25

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All algorithms run 35 times over each selected image with the selected number of levels. The images chosen for this test are Cameraman, Lena, Livingroom, Blonde, Peppers, Woman Dark Hair, Splash, Baboon, Jet and Sailboat on a lake. On every test STD, PSNR, SSIM, FSIM and the mean of the objective function are reported.

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Results are presented in four tables, Table 4 shows the STD and mean values. Table 5, Table 6, and Table 7 presents quality metrics computed with the best threshold found on thresholded images. It should be noticed that all results on table Table 4, Table 5, Table 6 and Table 7 are averaged from a 35 run test. Table 3 lists the best threshold values found by each algorithm.

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2 4 8 16 32 2 4 8 16 32 2 4 8 16 32 2 4 8 16 32 2 4 8 16 32 2 4 8 16 32

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p-Values CSA vs. DE CSA vs. HS

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Wilcoxon's rank test (García, Molina, Lozano, & Herrera, 2008) is a non-parametric significance proof used to assess statistically results differences between two related methods. In this paper, Wilcoxon's rank test is conducted with 35 independent samples considering a 5% significance level over the best fitness (cross entropy) value corresponding to the five thresholds. Table 11 presents the p-values generated by Wilcoxon´s test for a pair-wise comparison of the fitness value between two groups formed by CSA vs. DE and CSA vs. HS. The test works assuming as a null hypothesis that there is no difference between the values of the two methods tested. An alternative hypothesis considered an actual difference between the values of both approaches. p-Values reported in Table 11 are less than 0.05 (5% significance level) wich is strong evidence refuting the null hypothesis, indicating that CSA fitness values did not occur by chance, and were evidently better.

5.32E-06 5.37E-11 6.53E-13 6.50E-13 6.52E-13 3.25E-04 3.47E-11 6.51E-13 6.53E-13 6.49E-13 1.28E-05 1.27E-10 6.54E-13 6.52E-13 6.48E-13 1.81E-05 1.13E-10 6.52E-13 6.51E-13 6.49E-13 2.20E-06 4.93E-09 6.53E-13 6.48E-13 6.52E-13 1.31E-05 1.13E-11 6.54E-13 6.53E-13 6.52E-13

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4.64E-03 1.04E-11 6.53E-13 6.53E-13 6.50E-13 1.83E-07 4.55E-12 6.52E-13 6.52E-13 1.29E-12 6.32E-08 7.07E-11 6.54E-13 6.52E-13 6.46E-13 4.99E-01 4.75E-12 6.53E-13 6.52E-13 6.47E-13 8.06E-06 2.33E-11 6.53E-13 6.52E-13 6.51E-13 1.21E-03 6.54E-13 6.52E-13 6.52E-13 7.66E-13

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Baboon

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on

1.90E-05 2.42E-11 6.52E-13 6.52E-13 6.52E-13 1.18E-05 2.17E-10 6.51E-13 6.49E-13 6.48E-13 7.26E-03 1.01E-09 6.53E-13 6.52E-13 6.42E-13 4.54E-04 6.79E-11 6.53E-13 6.53E-13 6.47E-13

4.81E-05 1.86E-10 6.54E-13 6.52E-13 6.52E-13 7.83E-06 1.53E-12 6.54E-13 6.51E-13 6.49E-13 5.72E-05 3.11E-12 6.49E-13 6.49E-13 6.70E-13 4.90E-03 1.90E-11 6.52E-13 6.53E-13 6.51E-13

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Sailboat lake

2 4 8 16 32 2 4 8 16 32 2 4 8 16 32 2 4 8 16 32

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Table 11. Wilcoxon p-values of the compared CSA vs. DE and CSA vs. HS on the general purpose benchmark.

5.3.2. Comparison of MR brain images

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As in Section 5.3.1, the performance of the proposed MCET-CSA is compared against DE and HS. In this case, the medical image benchmark set is selected. The objective of this experiment is to determinate whether or not MCET-CSA provide high quality segmented MR brain images. Since the previous comparisons provided evidence pointing out that MCET-CSA performs well on general images, this experiment focuses on analyzing the performance of this kind of pictures.

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Results are presented in three tables, Table shows the STD and mean. Table 9 presents quality metrics computed with the best threshold found on thresholded images and Table 10 provides a qualitative comparison of the segmented images. On Table 8, the consistency of the results of CSA is much better than the STD reported for DE and HS.

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Since the results are originated from similar experiments, the significance of the data must be assessed. For this purpose, Wilcoxon´s rank test (García et al., 2008) is conducted with 35 independent samples considering a 5% significance level considering the objective function´s value. Table 12 presents the p-values generated by Wilcoxon´s test for a pair-wise comparison of the fitness value between two groups formed by CSA vs. DE and CSA vs. HS when applied on MR brain images. The results indicate that the values generated by CSA did not occur randomly. Image Z1

Z2

Nt 2 3 4 5 2 3 4 5

p-Values CSA vs. DE 6,82E-13 9,19E-13 7,11E-13 1,29E-12 4,27E-13 1,18E-12 1,14E-12 1,41E-12

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CSA vs. HS 2,35E-12 3,09E-09 1,22E-08 2,35E-05 4,99E-12 8,87E-09 9,24E-09 8,49E-04

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

4,22E-13 1,29E-12 7,74E-13 8,82E-13 5,01E-13 9,96E-13 6,53E-13 6,90E-12 4,91E-13 3,00E-12 1,19E-12 4,38E-12 5,49E-13 1,41E-12 7,13E-13 2,55E-12 6,63E-13 7,11E-13 6,53E-13 2,77E-12 5,30E-13 7,76E-13 7,48E-12 1,41E-12

7,11E-13 1,01E-09 9,64E-08 1,69E-06 1,14E-12 5,24E-09 2,68E-05 7,67E-03 1,97E-11 5,54E-08 7,17E-04 1,14E-04 1,84E-10 9,97E-08 1,15E-05 1,39E-02 2,81E-12 1,68E-09 2,77E-06 5,53E-04 4,32E-11 1,45E-08 8,70E-10 2,98E-02

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Table 12 Wilcoxon p-values of the compared CSA vs. DE and CSA vs. HS on the MR brain image benchmark.

5.4 Discussion

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The experimental data obtained suggest that CSA has interesting search properties when applied to the MCET problem despite its simple operators. An important characteristic of the MCET-CSA implementation is that CSA only requires two parameters to be tuned apart from the maximum number of iterations and population size. MCET-CSA takes advantages of several proven mechanisms commonly used on ECTs. First, the poblational nature of CSA helps to explore the search space. Besides, MCET works with the information of the image´s histogram wich usually present multimodality and local sub-optima configurations. Under such scenario, single particle search algorithms such as HS might struggle to escape from stagnation leading to a premature convergence. Other population-based approaches involve costly update equations that can be competitively replaced by the single update equation of CSA with an additional random position generation. Since the exploitation and exploration of the CSA is regulated by the awareness probability AP it becomes the most sensitive part of the method. An incorrect selection of AP value might lead to sub-optimal segmentation results. It must be noticed that the experimental setup for the population-based algorithms named CSA and DE are set to 20. The results point out that the proposed MCET-CSA works with a relatively small population.

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The proposed approach proves to be an interesting alternative to traditional image thresholding techniques. Moreover, the MCET-CSA is suitable for enhancing MR brain images as the results demonstrate it. Since the medical image processing is crucial for the diagnosis of many diseases these topics have gained attention over the last years. The competitive results of MCET-CSA might encourage researchers to apply this technique to similar problematics. 6. Conclusions

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In this paper, a new methodology called MCET-CSA is presented for Magnetic Resonance (MR) brain image thresholding. MCET-CSA is based on the recently published Evolutionary Computation Technique (ECT). For this purpose, the multilevel thresholding problem is selected due to its high modality nature and its complexity. The proposed methodology considers the thresholding process as an optimization problem, where CSA searches the optimal threshold points considering cross entropy as objective function. The CSA uses particles to encode a set of candidate threshold points. Cross entropy is used as an objective function, evaluating the quality of the selected threshold points. Following the values of the objective function, the operators of CSA guide the evolution process while improving the segmentation of MR brain images.To assess the quality of the segmented images, STD, PSNR, SSIM, and FSIM are used. Those metrics consider the similarity between the original and the segmented images.

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The proposed methodology was validated using two sets of benchmark images; the first is composed of general images commonly used in the image processing literature, while the second is made out of images extracted from Brainweb database. The first aims to compare the performance of CSA against well-known alternatives. For this end, two other Evolutionary Computation Techniques are implemented, the Differential Evolution (DE) and Harmony Search (HS). The competence of the algorithms is assessed regarding PSNR, STD, SSIM, FISM and fitness values. Such comparisons evidence the convergence, accuracy and robustness of the CSA, in contrast with those of the DE and HS. Wilcoxon´s test is used to determinate that the results of the CSA are significantly different from the ones of DE and HS and did not occur by chance. The second comparison is intended to evaluate the performance of the proposed MCET-CSA when applied specifically to the segmentation of MR brain images. In this case, quantitative results indicate that MCET-CSA generates high quality segmented MRI regarding PSNR, SSIM, and FSIM. Besides, the qualitative analysis of the results after segmenting brain images on different depths shows well-delimited regions that are easier to distinguish in comparison to other techniques.

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Acknowledgments

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Despite that the results offer evidence that the proposed MCET-CSA method performs well on MR images, specifically on brain MRI, this paper is not aimed to devise a multilevel thresholding method able to outperform all currently available methods but to assess the CSA performance on real applications and not just benchmark problems. CSA proves to be an interesting methodology.

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The second author acknowledges The National Council of Science and Technology of Mexico (CONACyT) for the doctoral Grant number 298285 for supporting this research. References

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