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1. 1057-7149 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIP.2020.2990346, IEEE Transactions on Image Processing IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. XX, NO. XX, XXX 2019 1 Segmentation of MR Brain Images Through Hidden Markov Random Field and Hybrid Metaheuristic Algorithm Thuy Xuan Pham, Patrick Siarry∗, Senior Member, IEEE, and Hamouche Oulhadj Abstract—Image segmentation is one of the most critical tasks in Magnetic Resonance (MR) images analysis. Since the performance of most current image segmentation methods is suffered by noise and intensity non-uniformity artifact (INU), a precise and artifact resistant method is desired. In this work, we propose a new segmentation method combining a new Hidden Markov Random Field (HMRF) model and a novel hybrid metaheuristic method based on Cuckoo search (CS) and Particle swarm optimization algorithms (PSO). The new model uses adaptive parameters to allow balancing between the segmented components of the model. In addition, to improve the quality of searching solutions in the Maximum a posteriori (MAP) estimation of the HMRF model, the hybrid metaheuristic algorithm is introduced. This algorithm takes into account both the advantages of CS and PSO algorithms in searching ability by cooperating them with the same population in a parallel way and with a solution selection mechanism. Since CS and PSO are performing exploration and exploitation in the search space, respectively, hybridizing them in an intelligent way can provide better solutions in terms of quality. Furthermore, initialization of the population is carefully taken into account to improve the performance of the proposed method. The whole algorithm is evaluated on benchmark images including both the simulated and real MR brain images. Experimental results show that the proposed method can achieve satisfactory performance for images with noise and intensity inhomogeneity, and provides better results than its considered competitors. Index Terms—Image segmentation, hidden Markov random field, Cuckoo search, particle swarm optimization I. INTRODUCTION IMAGE segmentation is one of the most important and challenging problems where pixels with similar features are grouped into homogeneous regions. Many high level processing tasks such as feature extraction, object recognition and medical diagnosis [1] depend heavily on the quality of the segmentation solutions. To obtain an appropriate segmentation, several criteria need to be satisfied, such as compactness, separation and overlapping. Sometimes, high level knowledge about the shape and appearance of the objects is required [2]. In many applications, however, such information is not available or impractical to use. In medical image analysis, magnetic resonance imaging (MRI) is now a popular way to get an image of the human brain with an increasingly high level of quality. However, the resulting images still contain some artifacts such as noise, T. X. Pham, P. Siarry and H. Oulhadj are with Laboratory Images, Signals, and Intelligent Systems (LiSSi) of University Paris-Est Creteil (UPEC), Paris, France. (*) indicates corresponding author (e-mails: thuy.pham@univ-paris- est.fr, {siarry,oulhadj}@u-pec.fr). partial-volume effect (PVE), and bias field effect due to various factors, such as spatial variations in illumination or radio frequency coil used in image acquisition [3]. Therefore, the automatic and accurate segmentation of MR images into different tissue classes, especially cerebrospinal fluid (CSF), gray matter (GM) and white matter (WM), remains a difficult task. Due to the importance of the identification of these structures in neuroscience applications, for example, clinical diagnosis of neuro-degenerative and psychiatric disorders, treatment evaluation, and surgical planning [4], many methods for segmentation of brain MR images have been proposed [5], [6]. However, there is no gold standard method and it still needs a significant amount of expert intervention for improving the performance. Among different approaches, the method that considers image segmentation problem as one of optimization problems solved by the gradient descent [7] or metaheuristics [8] are widely used. Since the fitting energy functions are non-convex and non-unique in nature and may have several local minimum points, the gradient descent technique faces the problem of getting stuck in local minima. In contrast, the metaheuristic approach is supposed to achieve the global solution efficiently. For instance, Maulik [9], Krishnan and Ramamoorthy [10], Vijay et al. [11], and Illunga-Mbuyamba et al. [12] proposed methods based on genetic algorithm (GA), ant colony optimization (ACO), particle swarm optimization (PSO), and cuckoo search (CS), respectively, for segmentation of MR brain images. In addition, there are different techniques to formulate the energy functions such as fuzzy concepts [13], tracking contours and shapes [14], or random fields [15]. There is no gold standard function while one criterion is usually dedicated to explore only a subset of search space such that it cannot model all the geometric properties of segmentation solutions. Therefore, simultaneous or sequential optimization of multiple segmentation criteria will lead to a higher quality solution and increase the robustness towards the different image properties. For example, in [16], to obtain a better trade- off between preserving image details and restraining noise for segmentation result, two criteria, namely fuzzy C-means (FCM) objective function and the other objective function designed by local information to restrain noise, were simultaneously optimized by using the multi-objective evolutionary algorithm with decomposition (MOEA/D). In our previous work [17], two criteria, which are based on fuzzy entropy clustering and adaptive weighting factors of global and local region-based active contours, were exploited by using multi- Authorized licensed use limited to: University of Canberra. Downloaded on May 03,2020 at 17:07:13 UTC from IEEE Xplore. Restrictions apply.

2. 1057-7149 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIP.2020.2990346, IEEE Transactions on Image Processing IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. XX, NO. XX, XXX 2019 2 objective particle swarm optimization (MOPSO) algorithm to achieve a better balance between two characteristics, named compactness and separation, in segmented regions. In the literature, Hidden Markov Random Field (HMRF) models, a generalization of Hidden Markov Model (HMM) [18], are widely used as a probabilistic robust-to-noise approach to model the joint probability distribution of the image pixels in terms of local spatial interactions [19]. Many methods based on HMRF have been proposed for MRI brain segmentation [15], [20]–[22]. The main idea underlying the segmentation process in this approach is that the image to segment (the observed image) and the segmented image (the hidden image) are considered as MRF. The segmented image is computed sequentially according to the MAP (Maximum A Posteriori) criterion, it leads to the minimization of an energy function [23]. There are two main advantages when using MRF models for image segmentation [24]: (1) The spatial relationship can be seamlessly integrated into a segmentation procedure through contextual constraints of neighbouring pixels; (2) Different types of image features can be utilized in the MRF-based segmentation model via the Bayesian framework. However, there still exist several problems which limit its performance [15]: (1) How to provide reasonably good initial values for mean and variance of each class (assuming that the classes are represented by Gaussian distributions); (2) How to deal with the problem of intensity inhomogeneities in the image; (3) What is the efficient algorithm for finding the optimal estimator to avoid the convergence to the first encountered minimum when including spatial coherence assumptions; (4) The basic model is inaccurate in nature. To address the aforementioned problems, many approaches have been proposed. For instance, for the first problem, Cuadra et al. [25] used GA algorithm to search initial values in predefined space. Recently, Krishnan et al. [22] introduced a method using Otsu’s criterion [26] combined with PSO algorithm to find the initialization of MRF. To overcome the second problem, Tohka et al. [27] proposed a method based on local image models where each models the image content in a subset of the image domain. Xie et al. [28] presented an interleaved method combined with a modified MRF classification and bias field estimation in an energy minimization framework. In order to find the global extremum solution in the optimization step, using metaheuristic algorithms has attracted a lot of attention in recent years. Yousefi et al. [20] introduced a combination method based on ACO algorithm and gossiping algorithm for the optimization step in MRF model. Guerrout et al. [29] proposed a method in which PSO algorithm was used. George et al. [30] used CS algorithm to obtain the optimum labels by minimizing the MAP values. For the last problem, which is a key in image segmentation, two main directions have been exploited. While one is to achieve a proper balance between the two components [31] in the standard model [32], the other is to use of spatial context or neighbourhood information efficiently [33], [34]. In recent years, machine learning-based segmentation algorithms have become also popular choices for MR brain image segmentation. Faced with the great challenges of this field of application (MICCAIs), they have been proved to be superior to the purely intensity or gradient driven conventional segmentation methods [35]. Especially, deep learning-based methods have made significant improvements and by now put them as a primary option for image segmentation. However, they also faced some difficulties. On the one hand, it is difficult to obtain a large number of training medical image data since generating highly accurate labels and finding sufficient preprocessed and representative data require a considerable amount of time. In addition, medical image annotation is carried out by experts, which is subjective and error-prone. Consequently, learning a model from a less accurate representation of training samples degenerates the algorithm accuracy and can cause an unstable segmentation network especially with small lesion or structure segmentation. On the other hand, among common problems for deep learning, it is difficult to train a deep network with original limited data without data augmentation or other optimized techniques [36]. Such approaches are beyond the scope of this paper. The goal of this paper is to present a new method combining MRF and metaheuristic algorithms that overcome the difficulties mentioned above and, thus, may provide superior performance. The work presented here is the continuation of [17]. To achieve this, first, results of iMOPSO-based image segmentation, which simultaneously optimizes two criteria, fuzzy entropy clustering and adaptive weighting factors of global and local region-based active contours, are used to provide inputs for the proposed algorithm. Particularly, we use segmented images and bias field correction of the original image, from which we can compute mean and variance of each region. Hence, we assume that the first two problems have been partially solved, and here, we only focus on the remaining ones. To deal with the third problem, we develop an efficient optimization engine based on two well-known algorithms: CS and PSO. While PSO algorithm is fast in finding solutions but may easily fall into local optima, CS algorithm is slow but is frequently capable to find the global optimal solution. By hybridizing them in an intelligent way, advantages of both algorithms are used. In addition, Dif- ferential Evolution (DE) schemes are also integrated in the algorithm to explore more untouched areas in the search space. To overcome the fourth problem, a new implementation scheme for MRF model is developed to construct the energy function by introducing adaptive weights for each class and a new potential function on classes of neighbours. The purpose of weighting components is to balance contributions of each class, as a result, small component such as CSF region is well considered in the optimization process. On the other hand, the proposed potential function, which takes into account both the spatial information and local intensity, is used to penalize dissimilarities between two neighbour pixels more effectively. To evaluate the effectiveness of the proposed algorithm, we examine it on both simulated and real MR images from the BrainWeb [37] and the IBSR database. The results are reported and compared with those from other recent segmentation methods in the literature. The rest of the paper is organized as follows. The general theory and development used in this work for MR image segmentation using MRF are presented in Section II. The hybrid Authorized licensed use limited to: University of Canberra. Downloaded on May 03,2020 at 17:07:13 UTC from IEEE Xplore. Restrictions apply.

3. 1057-7149 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIP.2020.2990346, IEEE Transactions on Image Processing IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. XX, NO. XX, XXX 2019 3 metaheuristic algorithm based on CS and PSO to enhance the searching ability and the quality of solutions is presented in Section III. The description of the proposed method, using the developed MRF model and the hybrid optimization algorithm, is given in Section IV. Experimental results and discussions are given in Section V which is illustrated by a set of figures and tables. Finally, in Section VI, we draw the conclusion of this work. II. MRF BASED SEGMENTATION MODEL A. The segmentation problem in Bayesian framework Let Ω = {s = (i, j)|1 6 i 6 H, 1 6 j 6 W} be the set of image lattice sites, where H and W are the image height and width in pixels. In the two-dimensional image lattice Ω, the pixel values x = {xs|s ∈ Ω} are a realization of random variables X = {Xs|s ∈ Ω}. The segmentation problem can be expressed in the Bayesian framework. Let us denote the observed image extracted from a random image (X = x) by F = f, where F and f are a random variable and its instance, respectively. Y = y stands for a segmented image based on the vector F = f, with the set of all possible configurations on Ω, ΩY . For the gray image, f takes its values in the space EF = {0, . . . , 255} and y takes its values in the discrete space EY = {1, . . . , C}, where C is the number of classes or homogeneous regions in the image. According to the Bayes rule, the segmentation problem is formulated as follows: P(Y = y | F = f) = P(F = f | Y = y)P(Y = y) P(F = f) (1) where P(Y = y | F = f) is the posteriori probability of Y = y conditioned on F = f. P(F = f | Y = y) denotes the probability distribution of F = f conditioned on Y = y, which is referred to as the feature modelling component. P(Y = y) is a priori probability of Y = y that describes the label distribution of a segmented result. In the literature, P(Y = y) is referred to as the region labelling component. P(F = f) is the probability distribution of F = f. Segmentation of an image can be considered as seeking the best realization y∗ by maximizing the probability P(Y = y | F = f), that is the Maximum a posteriori (MAP) estimate. As F = f is known, P(F = f) does not vary with respect to any solution Y = y and hence can be disregarded. y∗ = argmax y∈ΩY {P(F = f | Y = y)P(Y = y)} (2) We assume that each component of F = f be independent on the other components with respect to Y = y (conditional independence). Then, with M components in the feature vector f = {fm | m = 1, . . . , M}, P(F = f | Y = y) is transformed into: P(F = f | Y = y) = M Y m=1 P(fm | Y = y) (3) where P(fm | Y = y) is the probability distribution of the extracted data component fm conditioned on the segmented image Y = y. From Eqs. (2) and (3), the image segmentation problem becomes: y∗ = argmax y∈ΩY ( M Y m=1 P(fm | Y = y)P(Y = y) ) (4) Now, depending on how to formulate the feature modelling component and the region labelling component, we can get ex- act formulas for the image segmentation problem as described in the following sections. B. Intensity distribution model The form of P(fm | Y = y) may be different depending on which features are used. We assume that the distribution of all feature data is a Gaussian function with different means µm i and standard deviations σm i . That is: P(fm s | Ys = i) = 1 p 2π(σm i )2 exp − (fm s − µm i )2 2(σm i )2 = exp[−q(fm s , y)] (5) where q(fm s , y) = (fm s − µm i )2 2(σm i )2 + log q 2π(σm i )2 (6) where µm i and σm i are the mean and standard deviation for the ith class in the mth feature component. The feature modelling component, which is the product of all P(fm s | Ys = i), has the form: P(F = f | Y = y) = exp  − X s,Ys=i M X m=1 q(fm s , y)   (7) In this work, for the task of partitioning MR brain images, the intensity feature is used as the only image feature to extract different tissues (M = 1). Eq. (7) can be rewritten as follows: P(F = f | Y = y) = exp − C X i=1 X s∈Ωi q(fs, y) # (8) where µi and σi are the mean and standard deviation in terms of intensity for the ith region, Ωi = {s|ys = i}, of the image. They are defined as follows: µi = 1 |Ωi| P s∈Ωi fs σi = s 1 |Ωi| P s∈Ωi (fs − µi)2 (9) C. Spatial distribution model The other term in Eq. (4) is P (Y = y), which describes the prior knowledge about the spatial distribution of brain tissues in the image. For a given tissue class, one can consider the prior probability unchanged over the image, other authors suppose that the prior probability is varying at a given location, depending on the tissues found at the neighbouring locations. This can be done by using a MRF to model spatial interactions among tissue classes [38]. Definition 1. A random field Y is a MRF with respect to the neighbourhood system N = {Ns, s ∈ Ω}, if and only if: P(Y = y) 0, ∀y ∈ ΩY (10) Authorized licensed use limited to: University of Canberra. 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4. 1057-7149 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIP.2020.2990346, IEEE Transactions on Image Processing IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. XX, NO. XX, XXX 2019 4 and, with t 6= s P(Y = ys | Y = yt) = P(Y = ys | Y = yt, t ∈ Ns) (11) According to the Hammersley-Clifford theorem [39], a MRF can be equivalently characterized by a Gibbs distribution P(Y = y) = Z−1 exp{−U(y, β)} (12) where Z is a normalizing constant and U(y, β) is the energy function. β is a positive constant that controls the size of partitioning or interaction between the sites. Note that, the normalization factor of the Gibbs distribution is theoretically well-defined as follows: Z(U) = X y exp{−U(y, β)} (13) where the sum runs over all possible configurations of y, which is usually unknown [19]. Hence, Z is said to be computationally intractable. On the other hand, the choice of the energy function U(y, β) is arbitrary and there are several definitions of it in the framework of image segmentation [33], [40]. Using the multilevel logistic (MLL) model, a general expression of the energy function for pairwise interactions can be formulated by: U(y, β) = X ∀s∈Ω  V1(ys) + β X t∈Ns,t6=s V2(ys, yt)   (14) where V1(ys) is an external field that weights the relative importance of the different classes presenting in the image. This is usually unknown, hence, V1(ys) = 0 is used. V2(ys, yt) models the interactions between neighbours. Figure 1 illus- trates the second-order neighbourhood system with all possible pairwise cliques. Fig. 1: The second-order neighbourhood system with all two- points cliques. Here, we define V2(ys, yt) as follows: V2(ys, yt) = ( −(exp(−|Is − It|) + 1 2 ) 1 d(s,t) , if ys = yt +( 2 2−exp(−|Is−It|) − 1 2 ) 1 d(s,t) , otherwise (15) where |Is − It| and d(s, t) are the absolute difference of intensities and the Euclidean distance between the center pixel and one of its eight neighbours (Ns = 8), respectively. From Eqs. (12), (14) and (15), we can see that a pixel is classified like the majority of its neighbours. The more neighbours that have the same label as the center pixel, the more the P(Y = y) is increased. In addition, the proposed clique potential V2(ys, yt) involves the intensity information of the image, which makes the prior segmentation probability P(Y = y) adaptive to the local intensity. Particularly, the more neighbours have the same label as the center pixel (the more pairs of pixels comply with the first case in Eq. 15), the more the probability will be increased especially when the differences of intensity (|Is − It|) are small. In contrast, the probability will be less influenced when pairs of pixels with large differences of intensity comply with the second case in Eq. 15. D. A novel MRF based segmentation model From Eqs. (2), (8), (12) and (14), we obtain: P(Y = y | F = f) = Z0 exp[−Ψ (y, f)] (16) where Z0 is a constant and Ψ (y, f) is defined as follows: Ψ(y, f) = C X i=1 X s∈Ωi  q(fs, y) + β X t∈Ns,t6=s V2(ys, yt)   (17) Thus, maximizing the probability P(Y = y | F = f) is equivalent to minimizing the function Ψ(y, f). For that, we seek the segmentation result y∗ as follows: y∗ = argmin y∈ΩY {Ψ (y, f)} (18) The energy function Ψ(y, f) is then derived: Ψ(y, f) = C X i=1 λ(i) X s∈Ωi    q(fs, y) + β X t∈Ns t6=s V2(ys, yt)     = C X i=1 λ(i)E(i) (19) where E(i) is the energy of the ith region. λ(i) is a weighting parameter corresponding to the ith region, which determines how much the region contributes to the entire energy. In general, the energy weight parameter λ(i) in the energy function Ψ(y, f) is set to be 1.0 for all classes. However, the energy functional is affected by the energy weight parameters, which could reduce the segmentation accuracy. It is clear that E(i) is related to the number of pixels in associated region Ωi. Hence, it could be said that the energy term E(i) is positively correlated with the associated area of Ωi. The energy term of the largest target or background is the largest. As a result, the convergence of the energy functional would be controlled by the larger regions (GM and WM) and the smaller regions (CSF) would be covered. To overcome this problem, adaptive weight functions are proposed to configure λ(i) and adjust the contribution of each region. The weight functions are set as follows: λ(i) = 0.5 min{Area(Ω (k) j ) | j = 1, . . . , C} Area(Ω (k) i ) (20) where Area(Ω (k) i ) is the number of pixels in the kth iteration of the ith region. Furthermore, we can always compute y through µ = (µ1, . . . , µC) by classifying fs into the nearest mean µj i.e. ys = j if the nearest mean to fs is µj. Thus, with the same Authorized licensed use limited to: University of Canberra. Downloaded on May 03,2020 at 17:07:13 UTC from IEEE Xplore. Restrictions apply.

5. 1057-7149 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIP.2020.2990346, IEEE Transactions on Image Processing IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. XX, NO. XX, XXX 2019 5 approach as in [41], we seek for µ∗ instead of y∗ . Then, the image segmentation problem becomes: µ∗ = argmin µ∈Ωµ {Ψ(µ)} (21) where Ψ(µ) is defined: Ψ(µ) = C X i=1 λ(i) X s∈Ωi  q(µ) + β X t∈Ns,t6=s V2(ys, yt)   (22) where q(µ), V2(ys, yt) and λ(i) are defined in Eqs. (6), (15) and (20), respectively. III. HYBRID METAHEURISTIC ALGORITHM To solve the MR segmentation problem by using optimization techniques with the objective function given in Eq. (22), several approaches can be used such as direct search methods [21] or metaheuristic algorithms [20]. In this work, we attempt to exploit the second approach by introducing a new hybrid metaheuristic algorithm which benefits both advantages of the two well-known algorithms, named CS and PSO. The proposed approach is described below. A. Cuckoo search (CS) algorithm The Cuckoo Search (CS) algorithm, which is a population- based stochastic global search algorithm, is based on two behaviours of birds: the breeding behaviour of some cuckoo species and the Levy flight behaviour of some bird species [42]. Using CS has two main advantages: (1) The number of parameters which have to be configured in the initial search is very little; (2) The inexperienced user can easily interact with it. However, it is common to find that CS shows relatively slow convergence speed and low search accuracy because of the loss of diversity in the population, and readers are referred to [43] for comprehensive reviews. There are three rules that idealize behaviour of cuckoos in order to become appropriate for implementation as an optimization tool: (1) Each cuckoo lays one egg at a time and dumps it in a randomly chosen nest; (2) The best nests with high-quality of eggs will be carried further to the next generation; (3) The number of available host nests is fixed, and the egg laid by a cuckoo is discovered by the host bird with a probability of pa ∈ [0, 1]. In this case, the host bird can either get rid of the egg or simply abandon the nest and build a completely new nest. Basically, in the CS algorithm, each egg in a nest corresponds to a potential solution and each cuckoo’s egg corresponds to a new solution. The CS algorithm attempts to iteratively improve the candidate solutions (eggs in the nests) by replacing them with better generated solutions (cuckoo’s eggs) based on the fitness values. The algorithm consists of two different phases: a global Levy flight random walk (exploration) and a local random walk (exploitation), which are controlled by the switching parameter pa, from which the population is updated during the whole process. In the first phase, the global Levy flight random walk is used to generate new solutions around the best nest, gBest, in the current generation. Assuming that, the CS algorithm Algorithm 1: The improved CS Input: Randomly initialize a population X; determine gBest (the global best nest) Output: The optimal solution gBest repeat 1 for each nest do 1.1 Generate a cuckoo (say Xi) randomly, Eq. (25) 1.2 Evaluate the fitness value, fi 1.3 Randomly choose a nest among Np (say Xj) 1.4 if fi is better than fj then 1.4.1 Replace nest Xj by the new one, Xi else 1.4.2 Create a cuckoo (say X0 i) by using Eq. (26) 1.4.3 Replace nest Xj if an improvement is found 2 Abandon a fraction pa of worse nests and build new ones using Eq. (28) 3 Update the gBest and pass to the next generation until the stopping criteria are met processes with a population of Np eggs, X = {X1, . . . , XNp }, where each egg is composed of C decision variables, Xi = {xi1, . . . , xiC}. The step with the Levy flight is calculated based on Mantegna’s algorithm [44], [45], as below: stepsize (k) i = 0.05 uk |vk| 1 α (X (k) i − gBest) (23) where X (k) i is the ith egg of the population in the kth iteration. In our work, α is set to 1.5. u and v are normally distributed stochastic variables, u ∼ N(0, σ2 u) and v ∼ N(0, σ2 v). The standard deviation of the random matrix generated is defined as follows: σu(α) = Γ(1 + α) sin(πα 2 ) Γ((1+α) 2 )α2 (α−1) 2 # 1 α and σv = 1 (24) where Γ corresponds to the standard gamma function. Then, the new egg, X (k+1) i , can be obtained by the following equation: X (k+1) i = X (k) i + stepsize (k) i randn[C] (25) where randn[C] represents random scalars drawn from the standard normal distribution. In order to improve the global exploration abilities within a relatively small number of generations by discovering more untouched areas in the search space (enhance the diversity), we propose to add a further step of generating new eggs if the better ones cannot be found. Particularly, if a generated egg is not better than the old one in terms of fitness value, a Differential Evolution (DE) scheme proposed by Mohamed et al. [46], is used to lay another egg, X 0(k+1) i . The scheme is expressed as follows: X 0(k+1) i = X(k) r + Fm(X (k) best − X (k) worst) (26) where X (k) best and X (k) worst are the best and the worst solutions in terms of fitness value in the current population. X (k) r is a solution which has a rank of its fitness value in range of [round(εNp), Np−round(εNp)] with ε ∈ (0, 0.5). In our work, ε is set to 0.15 and Fm is generated uniformly within (0.1,1) in each iteration. Authorized licensed use limited to: University of Canberra. Downloaded on May 03,2020 at 17:07:13 UTC from IEEE Xplore. Restrictions apply.

6. 1057-7149 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIP.2020.2990346, IEEE Transactions on Image Processing IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. XX, NO. XX, XXX 2019 6 Using this scheme, there are two main benefits, which can be observed. First, the target vector is attracted towards good solutions, since it will always follow the same direction of the better ones. Second, the direction of the worst solution is avoided, which will force the direction of the search process towards the promising regions in the search space. In the second phase, CS continues to generate new eggs in terms of biased/selective random walk. By considering the probability of cuckoos of being discovered, a crossover operator is used to construct a new egg as follows: X 00(k+1) i = ( X (k) i + Fc(X (k) r1 − X (k) r2 ), if rand[0, 1] pa X (k) i , otherwise (27) where r1 and r2 are mutually different random integers; Fc denotes the scaling factor which is a uniformly distributed random number in the interval [0, 1]. Actually, the operator, Xk jump = X (k) i + Fc(X (k) r1 − X (k) r2 ) , aims to create a jump to avoid the local trap. However, if X (k) r1 and X (k) r2 are relatively close to each other, it is unable to explore any new prospective zone. To overcome this problem, in this work, instead of using Xk jump = X (k) i + Fc(X (k) r1 − X (k) r2 ), we deploy an operator proposed by Nguyen et al. [47], which is defined as follows: Xk jump = ( X (k) i + Fc(X (k) r1 − X (k) r2 ), if FDRi 10−3 X (k) i + Fc h (X (k) r1 − X (k) r2 ) + (X (k) r3 − X (k) r4 ) i , otherwise (28) where r1, r2, r3 and r4 are mutually different random integers; FDRi is the fitness difference ratio (FDR) of the ith solution in the current population, which is defined as follows: FDRi =

10. fi − fgBest fgBest

14. (29) where fi and fgBest are the fitness values of the ith solution and the best solution found so far, respectively. Furthermore, to increase the exploration of the search space in the beginning stage and the exploitation of the best solutions found so far towards the end of the algorithm, pa is updated as follows: p(k) a = pamax − (pamax − pamin ) · (k/Niter) (30) where p (k) a is the switching parameter pa at the kth iteration; [pamin , pamax ] is the range of pa, with pamin = 0.01 and pamax = 0.5; Niter is the maximum number of allowable iterations. Finally, by using the greedy strategy, the next generation solution is created. At the end of each iteration process, the best solution obtained so far is updated. The procedures of improved CS algorithm can be described as the pseudo code shown in Algorithm 1 B. Particle swarm optimization (PSO) algorithm The particle swarm optimization (PSO) algorithm is also a population-based stochastic optimization algorithm and re- garded as a global search strategy. This algorithm is inspired from the cooperation and communication of a swarm of Algorithm 2: The improved PSO Input: Initialize randomly a population X; set up the parameters: {c1, c2} and w; determine pBest and gBest Output: The optimal solution gBest repeat 1 for each particle (say Xi) do 1.1 Generate a particle (say X0 i), Eqs. (33) and (34) 1.2 Evaluate fitness value, f0 i 1.3 if f0 i is better then Replace Xi by the new one, X0 i 2 Update the population X 3 Update the inertia weight w using Eq. (35) 4 for each particle (say Xj) do 4.1 Update the velocity Vj using Eq. (31) 4.2 Update the position Xj using Eq. (32) 4.3 Evaluate fitness value, fj 5 Update the pBest and the gBest until the stopping criteria are met birds [48]. Due to the simple representation and relatively low number of adjustable parameters, PSO has become one of the most popular choices and is efficiently applicable to optimization problems. However, the major drawback of the PSO is that it may be trapped into a local optimal solution region. Comprehensive reviews can be found in [49]. In PSO, each individual (called particle), which represents a potential solution to the optimization, of a given population (called swarm), is updated according to its own experience and that of neighbours. The quality of a candidate solution is evaluated by the fitness value associated with it. Let us consider a swarm of Np particles, where each particle has a position vector, Xi = {xi1, . . . , xiC}, a velocity vector, Vi = {vi1, . . . , viC}, its own best position pBest found so far, and interacts with neighbouring particles via the best position gBest discovered in the neighbourhood so far. At iteration kth in the search process, particles are moved according to the following equations: V (k+1) i = w(k) V (k) i + c1r1 h pBest(k) − X (k) i i + c2r2 h gBest(k) − X (k) i i (31) X (k+1) i = X (k) i + V (k+1) i (32) where r1 and r2 are random variables, uniformly distributed in [0, 1] to provide stochastic weighting of the different components participating in the velocity. c1 and c2 are acceleration coefficients that scale the influence of the cognitive and social components, respectively, and w is inertia weight. In addition, the flying velocity is limited to a reasonable range [Vmin, Vmax]. In this work, Vmin = −Vmax = −3 is set to constraint particle movement. In order to improve the performance of the PSO algorithm, in this work, we use the same approach as in [50] that creates a high diversity population to provide a good guidance for particles. As a result, the problem of premature convergence can be partially avoided and the exploitation ability of the algorithm can be improved. Here, we employ a differential Authorized licensed use limited to: University of Canberra. Downloaded on May 03,2020 at 17:07:13 UTC from IEEE Xplore. Restrictions apply.

15. 1057-7149 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIP.2020.2990346, IEEE Transactions on Image Processing IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. XX, NO. XX, XXX 2019 7 mutation scheme proposed by Mohamed [51] for this purpose. The scheme is expressed as follows: X (k+1) i = X (k) i + Fp1(X (k) best − X (k) better) + Fp2(X (k) best − X (k) worst) + Fp3(X (k) better − X (k) worst) (33) where X (k) best , X (k) better and X (k) worst are the tournament best, better and worst three randomly selected particles in the current population, respectively. Fp1 , Fp2 and Fp3 are the mutation factors generated independently from a uniform distribution within (0, 1) from iteration to iteration. As a result, different sizes and shapes of triangles in the feasible region through the optimization process are created. And, X (k) i is a convex combination vector of the triangle, which is defined as follows: X (k) i = δ1X (k) best + δ2X (k) better + δ3X (k) worst (34) where the real weights δi, (i = 1, 2, 3) are given by δi = pi/ P3 i=1 pi. Here, p1, p2 and p3 are set representatively to 1, rand(0.75, 1) and rand(p2, 1), in which rand(a, b) returns a real number between a and b. From Eqs. (33) and (34), it can be observed that there are two main benefits. First, the landscape of different sub-regions around the best vectors can be explored by forming many different sizes and shapes of triangles through the optimization process, which significantly increases the global exploration ability. Second, the global solution can be easily reached if all vectors follow the direction of the best vectors since the convex combination X (k) i consists of the best vector with higher weight. Furthermore, to have a reasonable balance between exploration and exploitation during the optimization process (increase exploration ability in the beginning stage and increase exploitation towards the end), here, a parameter control strategy proposed by Yang et al. [52] is used. In the strategy, c1 = c2 = 2 and the inertia weight, w, updating scheme is expressed as follows: w(k) = wmax − (wmax − wmin) · (k/Niter) 1 π2 (35) where w(k) is the inertia weight at the kth iteration. [wmin, wmax] is the range of inertia weight, with wmin = 0.4 and wmax = 0.9. The procedures of improved PSO algorithm can be described as the pseudo code shown in Algorithm 2. C. Proposed hybrid algorithm based on ICS and IPSO There are no theoretical or experimental guarantees that any optimization algorithm can avoid getting stuck in suboptimal solutions. In order to increase the accuracy of the results and decrease the likelihood of trapping into local solution regions, the common approach that is hybridizing or combining different techniques [43], [49]. In this paper, an effective hybrid optimization algorithm, called hybrid ICS/IPSO, is proposed based on the strategies described above. The proposed algorithm is built based on two main steps. The first step is used for finding different promising regions in the search space by using both the improved CS and improved PSO algorithms. The second step is used for selecting and updating solutions by maintaining the merits of both algorithms. Hence, we can achieve fast Algorithm 3: The hybrid ICS/IPSO Input: Initialize randomly a population X and two external archives: pAic and pAip; set up parameters for both ICS and IPSO algorithms; determine gBestic, pBestip, gBestip and gBest Output: The optimal solution gBest repeat 1 for each potential solution in X (say Xi) do Perform from step (1.1) to step (1.4) in Algorithm 1 2 Perform step (2) in Algorithm 1 3 Sort and store new solutions in pAic 4 Perform from step (1) to step (4) in Algorithm 2 5 Update pBestip, sort and store solutions in pAip 6 Determine the current best by: argmin Xj ∈(pAic∪pAip) f(Xj) or argmax Xj ∈(pAic∪pAip) f(Xj) 7 Update gBest, and {gBestic, gBestip} ← gBest 8 Update X by selecting the best half of solutions in two external archives: pAic and pAip until the stopping criteria are met convergence and partially avoid the problem of getting stuck in suboptimal solutions. These two steps are repeated alternately until the termination criteria are satisfied. The main procedure of hybrid ICS/IPSO is given in Algorithm 3. There are several works, which hybridize CS and PSO algorithms [53], [54], for solving the global optimization problem. However, the proposed algorithm has its own specific characteristics, which make it different from others used in the literature. First, both CS and PSO algorithms are improved to explore more untouched areas in the search space. Second, since the ICS and IPSO are independently operated during the optimization process, the merits of both are maintained. Third, by using a simple effective mechanism of selection, potential solutions not only follow the direction of the best one, but also avoid the direction of the worst. In addition, sharing solutions can help both ICS and IPSO algorithms to compensate for their weaknesses. IV. PROPOSED METHOD This section describes the proposed MR brain segmentation method, which is based on MRF-based segmentation model and the hybrid metaheuristic algorithm (hybrid ICS/IPSO). The fitness function is calculated by using Eq. (22). To obtain the segmented images after achieving the optimal mean intensity of each region, two different steps need to be done: (1) Labelling each pixel in the image by classifying it to the nearest mean; (2) Filtering the labelled image by using Median filter with the structure of [3×3]. Note that, filtering is necessary for very noisy images such that the final segmented images can be achieved with high quality. The details of this algorithm are illustrated in the next section. A. Solution representation In this study, the potential solutions are made up of real numbers, which are nests in the ICS or positions in the IPSO. The potential solutions represent the mean intensities of various regions in the image. For Np solutions with C distinct elements, there are in total (Np.C) optimization variables that Authorized licensed use limited to: University of Canberra. Downloaded on May 03,2020 at 17:07:13 UTC from IEEE Xplore. Restrictions apply.

16. 1057-7149 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIP.2020.2990346, IEEE Transactions on Image Processing IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. XX, NO. XX, XXX 2019 8 Fig. 2: The block diagram of the proposed method. need to be encoded. For example, in the population X = X1, . . . , XNp , the ith potential solution can be described as: Xi = {xi1, . . . , xiC}, where xij(j = 1, . . . , C) represents the ith mean intensity of the jth region. In this way, the mean intensity of each region, µ = (µ1, . . . , µC), can be obtained by decoding Xi. B. Segmentation criterion In this research, the value of the fitness function, fi, corresponding to the ith solution, is calculated by evaluating Ψ (Xi) according to Eq. (22). fi = Ψ (Xi) (36) The minimization of fi is the same as the minimization of the objective function, Ψ (Xi), which can lead to an optimal partitioning of the MR image. C. The optimum search process This study proposes to use the hybrid ICS/IPSO algorithm described in Section III-C to do the optimization step. By taking both advantages of MRF-based segmentation model and the hybrid ICS/IPSO algorithm, the optimal solution for the MR image segmentation problem can be found. The framework for the problem is summarized in Algorithm 4. As aforementioned, in this work, we assume that the first two problems, named initialization and bias correction, have been partially solved. This can be done by using iMOPSO- based method [17], PSO-KFECSB method [8] or MICO method [55]. The aim is providing a good approximation Algorithm 4: ICS/IPSO-based image segmentation Input: Read the input image; set the number of regions C; set maximum number of allowable iterations Niter; get bias corrected image and initialize population X using results from iMOPSO algorithm; initialize parameters for ICS, IPSO and MRF-based segmentation model; determine gBestic, pBestip, gBestip and gBest; initialize two external archives: pAgBest and pACJV . Output: The optimal region centers, gBest∗ for (l = 1; l ≤ L; l = l + 1) do /* Hybrid ICS/IPSO algorithm */ /* Compute fitness values, Eq. (36) */ /* argmin is used in step (6) */ 1 repeat Perform from step (1) to step (8) in Algorithm 3 until the stopping criteria are met /* Get solution */ 2 Store gBest in pAgBest 3 Calculate CJV using Eq. (37) and store it in pACJV /* Increase β */ 4 β ← β + 0.1 × l 5 Get the optimal region centers by: gBest∗ ← argmin gBest∈pAgBest {pACJV (gBest)} solution, called starting point, so that the proposed algorithm can converge quickly and be able to obtain the global optimal solution. In current work, we use the iMOPSO-based method. The main reason is that the iMOPSO-based method provides a set of solutions instead of one solution only. As a result, we can have a flexible choice for setting up the starting point of the proposed algorithm. Here, we use coefficient of joint variation (CJV ) between WM and GM regions [56], which is a well- known criterion to evaluate bias field correction methods, as a metric to determine the point. From that, the initial population is created by randomly generating solutions around it. The CJV is defined as follows: CJV = σ(GM) + σ(WM) |µ(GM) − µ(WM)| (37) where σ(.) and µ(.) denote the standard deviation and the mean intensity. A smaller CJV value corresponds to the better performance. In addition, to ensure that all solutions are moving within the search space and avoiding divergent behaviour, the bound- ary conditions for the ith potential solution are limited as follows: x (k) ij =      xmax, if x (k) ij xmax xmin, if x (k) ij xmin +x (k) ij , otherwise (38) v (k) ij =      vmax, if v (k) ij vmax vmin, if v (k) ij vmin +v (k) ij , otherwise (39) where vmin and vmax are the smallest and largest allowable step sizes in any dimension (vmin = −vmax = −3 is set in this paper); and {xmin, xmax} are the bounds of the search Authorized licensed use limited to: University of Canberra. Downloaded on May 03,2020 at 17:07:13 UTC from IEEE Xplore. Restrictions apply.

17. 1057-7149 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIP.2020.2990346, IEEE Transactions on Image Processing IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. XX, NO. XX, XXX 2019 9 Fig. 3: Qualitative segmentation results: (a) original image; (b) ground truth; (c) FRFCM results; (d) MICO results; (e) PSO- MRF results; (f) iMOPSO results; (g) the proposed method results. space in each dimension. Actually, they are the minimum and maximum of the intensity of the input image. Furthermore, to stop the phase of hybrid ICS/IPSO algorithm efficiently, two criteria are set: the maximum number of the allowable iterations and the maximum number of non- significant improvements of the fitness value, fgBest. Particu- larly, if (iter Niter) is reached or (|fgBest,new −fgBest,old| 10−4 ) is completed (0.1 × Niter) times, this phase is imme- diately stopped. Finally, to get rid of the problem of selecting parameter β, which is described in Section V-A, we use CJV criterion to estimate the quality of solutions [28] so that the final solution can be found. The main steps of the proposed method are described in Figure 2. V. RESULTS AND DISCUSSIONS In this section, we empirically evaluate the performance of the proposed method. Both qualitative and quantitative evaluations are involved in this study. A. Experimental setup To validate the effectiveness, the performance metric values of the proposed method have been evaluated and compared with five state-of-the-art algorithms in the literature. These algorithms are: FCM algorithm based on morphological reconstruction and membership filtering (FRFCM) [57], the hidden Markov random field model and its expectation-maximization algorithm (HMRF-EM) [58], the multiplicative intrinsic component optimization (MICO) [55], the integrating metaheuristic multilevel threshold with Markov random field (PSO-MRF) [22] and our previous work, called iMOPSO-based method [17]. The algorithms’ parameters are set as default values in their works. Experiments are conducted on MATLAB 2014b running on an Intel Core i7 1.8 GHz CPU and 8 GB RAM. To perform experiments, the parameters of the proposed algorithm are set as in Table I. Notice that, the values of those parameters are set based on both suggestions from the literature [43], [49], [54] and experiments. Varying those values can affect the final result. How to tune those parameters so as to achieve the best performance is out of scope for this work. Furthermore, the parameter β in Eq. (22), which controls the balance between two components: feature modelling and region labelling, has an important effect on the performance of the algorithm. Particularly, it tunes the degree of homogeneity TABLE I: Specific values for given parameters used in the proposed algorithm. Parts used Parameters Values All Population size 30 Maximum iterations 100 Number of clusters 3 (4) IPSO Range of inertia weight [0.4, 0.9] Acceleration coefficients c1 = c2 = 2 Scaling factors: Fp1, Fp2, Fp3 rand(0, 1) Range of flying velocity [-3, 3] ICS Switching parameter pa ∈ [0.01, 0.5] Scaling factor, Fc 0.9 Scaling factor, Fm rand(0.1, 1) {α, ε} {1.5, 0.15} of each region in the segmented image. If β makes the feature modelling component dominant (small value of β), spatial relationship information would be ignored. On the other hand, if β makes the region labelling component dominant (large value of β), the values of estimated solutions may deviate con- siderably and the segmented result is not consistent (excessive smoothing of boundaries may occur). Unfortunately, there is no closed-form definition for β as the normalizing constant Z(U) is intractable (Eqs. (13), (14) and (15)). In addition, different input images can have different spatial organizations. However, from experiments we found that β ∈ [0.95, 1.45] can produce satisfactory results in most cases. Hence, in this work, β is determined empirically as proposed in [59] by gradually increasing its value, from 0.95 to 1.45 with step of 0.1 (L = 6), through the algorithm loops. All the other parameters are set as in the previous Sections. B. Datasets The MR images used in this study include both T1-weighted simulated and real 2D MRI brain images. For simulated MR images, they are downloaded from a well-known database: the BrainWeb from a McConnell Brain Imaging Center [37], which can be reached in (https://brainweb.bic.mni.mcgill.ca/ brainweb/). This dataset includes different noise levels: from 0 to 9%, and different INU levels: 0, 20%, and 40%, and five different slice thicknesses. Images with size of 181 × 217 and thickness of 1 mm are used in this work. On the other hand, real MR images are taken in the 20-normal MR brain data sets and MRBrainS18 data sets. The 20-normal MR brain data sets contain manual segmentations by an expert Authorized licensed use limited to: University of Canberra. Downloaded on May 03,2020 at 17:07:13 UTC from IEEE Xplore. Restrictions apply.

18. 1057-7149 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIP.2020.2990346, IEEE Transactions on Image Processing IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. XX, NO. XX, XXX 2019 10 Fig. 4: Qualitative segmentation results: (a) original images; (b) ground truth images; (c) segmentation results. technician, provided by the Center for Morphometric Anal- ysis at Massachusetts General Hospital. The data sets are available at (https://www.nitrc.org/projects/ibsr/). Images with characteristics of size 135×142 and 1.171751 mm thickness, are used in our experiments. The MRBrainS18 data sets are provided by the Grand Challenge on MR Brain Segmentation 2018 (https://mrbrains18.isi.uu.nl/). This data consists of 7 sets of brain MR images (T1, T1 inversion recovery, and T2- FLAIR) with manual segmentations of ten brain structures. C. Performance measures Since the ground truth images are available in the datasets, for quantitatively comparing the performance, three criteria are involved, which are the Dice Similarity Coefficient (DICE), the Hausdorff distance (HD), and the Accuracy (AC) [60]– [62]. These are defined below. 1) Dice coefficient: The Dice coefficient [60] (DICE) is an overlap-based metric which directly compares a segmented image (Iseg) with a ground truth image (Itr) by measuring similarity between them. The highest value of DICE indicates the best performance. This metric is one of the most used measures in validating medical volume segmentation. Given an input image with N pixels I = (I1, I2, . . . , IN ), and its two partitions, Iseg = (Iseg1, Iseg2, · · · , IsegN ) (the segmented image) and Itr = (Itr1, Itr2, · · · , ItrN ) (the ground truth image), there are four common cardinalities that reflect the overlap between the two partitions, named the true positives (TP), the false positives (FP), the true negatives (TN), and the false negatives(FN). Then, the pair-wise overlap of the repeated segmentations is calculated using the DICE, which is defined by: DICE (Iseg, Itr) = 2.|Iseg T Itr| |Iseg| + |Itr| = 2.TP 2.TP + FP + FN (40) 2) Hausdorff distance: The Hausdorff distance [61] (HD) is a distance-based metric which measures the dissimilarity between the segmented image (Iseg) and the ground truth image (Itr). The lowest value of HD indicates the best performance. This is also a widely used metric, defined as follows: HD (Iseg, Itr) = max {h (Iseg, Itr) , h (Itr, Iseg)} (41) where h (Iseg, Itr) is called the directed Hausdorff distance given by: h (Iseg, Itr) = max Isegi∈Iseg min Itri∈Itr kIsegi − Itrik (42) where kIsegi − Itrik is the Euclidean distance between the intensity values of the Isegi pixel and the Itri pixel in the segmented and ground truth images, respectively. 3) Accuracy: This criterion (AC) determines how much the segmentation algorithm results match with the ground truth. The highest value of AC indicates the best performance. It is defined as below: AC (Iseg, Itr) = TP + TN TP + TN + FP + FN (43) D. Simulated MR brain images In this section, simulated MR brain images from the Brain- Web are used for the purpose of performance evaluation. The experiment is conducted on a data set of 9 images (slices: 85, 87, 89, 91, 93, 95, 97, 99 and 101) with the ”worst case” (low contrast and relatively large spatial inhomogeneities); the images have the characteristics of 9% noise and 40% INU artifact. Each image is segmented into four classes: cerebro spinal fluid (CSF), gray matter (GM), white matter (WM) and the background. Figure 3 shows the qualitative results of the segmentation of a T1-weighted image (slice 101) provided by the considered algorithms. This figure reveals that though the iMOPSO-based method and the proposed algorithm produce better results among those, the proposed method reserves the best in details of the image. Figure 4 shows the qualitative results of the segmentation of 8 T1-weighted images (slices: 85, 87, 89, 91, 93, 95, 97, and 99) produced by the proposed algorithm. In spite of existing artifact, as can be seen from the figure, the segmented image is almost close to the ground reference data, hence, our method achieves a high performance when segmenting MR brain images. Thus, it can be concluded that the proposed method qualitatively provides satisfactory results. In order to compare more clearly the performance of the considered methods, quantitative evaluation is also taken into Authorized licensed use limited to: University of Canberra. Downloaded on May 03,2020 at 17:07:13 UTC from IEEE Xplore. Restrictions apply.

19. 1057-7149 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIP.2020.2990346, IEEE Transactions on Image Processing IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. XX, NO. XX, XXX 2019 11 Fig. 5: Qualitative segmentation results: (a) original image; (b) ground truth; (c) FRFCM results; (d) HMRF-EM results; (e) MICO results; (f) PSO-MRF results; (g) iMOPSO results; (h) proposed method results. TABLE II: Mean and standard deviation of DICE and AC results for the considered algorithms calculated over the set of 9 simulated MR brain images from the BrainWeb Methods Metrics Regions CSF GM WM Mean Std. Dev. Mean Std. Dev. Mean Std. Dev. FRFCM DICE 0.9791 0.0019 0.9333 0.0081 0.9532 0.0047 AC 0.9635 0.0033 0.9073 0.0081 0.9395 0.0067 MICO DICE 0.9815 0.0022 0.8896 0.0073 0.9040 0.0049 AC 0.9679 0.0036 0.8444 0.0072 0.8772 0.0061 PSO-MRF DICE 0.9790 0.0031 0.9219 0.0102 0.9451 0.0050 AC 0.9639 0.0053 0.8905 0.0107 0.9288 0.0072 iMOPSO DICE 0.9847 0.0015 0.9436 0.0060 0.9633 0.0023 AC 0.9739 0.0028 0.9222 0.0062 0.9530 0.0035 Proposed DICE 0.9865 0.0015 0.9469 0.0071 0.9661 0.0033 AC 0.9764 0.0026 0.9261 0.0074 0.9566 0.0045 *The values in bold indicate the best performance. account. Two supervised metrics, named DICE and AC, are involved. Note that a higher value indicates a better corre- spondence to the ground-truth. Figures 6 and 7 show the comparison results. The results are summarized in Table II. As can be seen from the figures and table, the proposed method generally gives the best scores. Even though the iMOPSO-based method provides a little more consistent GM and WM results, the proposed method achieves better performance. This confirms that our method performs more efficiently on simulated MR brain images compared to the others. E. Real MR brain images We have also examined the performance of our method on real MR brain images. The first experiment is conducted on a data set of 9 images in the 20-normal T1-weighted real MR brain data (slices: 24, 26, 28, 30, 32, 34, 36, 38 and 40). It has been used in a variety of volumetric studies in the literature as it contains varying levels of difficulty, with the worst scans consisting of low contrast and relatively large spatial inhomogeneities. The second experiment is conducted on a data set of 7 T1-IR images (without bias correction) in the MRBrainS18 data (slice 25 in volumes: 1, 4, 5, 7, 14, 70 and 148). The number of regions in the segmentation processes are set to three and four, respectively. Note that, the background pixels are ignored in the computation when doing the first experiment. Figure 5 shows the qualitative results of the segmentation of a T1-weighted image (slice 30) provided by the considered algorithms. This figure reveals that the proposed method provides superior results. It may be worth mentioning here that though the iMOPSO-based method is comparable with the proposed method on simulated MR images, current method achieves much higher performance on real MR brain images. Figures 11 12 show the qualitative results of the segmentation of different images in the 20-normal T1-weighted data and the MRBrainS18 data, respectively. As can be seen from the figures, our method accomplishes appropriately segmentation of real MR brain images. TABLE III: Mean and standard deviation of DICE, AC and HD results for the considered algorithms calculated over the set of 9 real MR brain images from the 20-normal T1-weighted real MR brain data Methods Metrics Regions CSF GM WM Mean Std. Dev. Mean Std. Dev. Mean Std. Dev. FRFCM DICE 0.9913 0.0042 0.8581 0.0423 0.8640 0.0491 AC 0.9831 0.0082 0.7990 0.0583 0.8105 0.0563 HD 1815.7 0399.6 0199.0 0205.6 0157.5 0078.4 HMRF-EM DICE 0.8481 0.0112 0.8195 0.0409 0.8657 0.0458 AC 0.7394 0.0162 0.7464 0.0517 0.8063 0.0547 HD 2422.7 0434.7 0186.4 0157.3 0248.4 0140.4 MICO DICE 0.9914 0.0042 0.8713 0.0228 0.9014 0.0270 AC 0.9831 0.0081 0.8169 0.0294 0.8587 0.0323 HD 1815.7 0399.6 0188.0 0208.7 0126.8 0055.2 PSO-MRF DICE 0.9138 0.0111 0.8205 0.0355 0.9023 0.0438 AC 0.8445 0.0180 0.7432 0.0449 0.8547 0.0582 HD 1914.7 0375.3 0198.8 0152.5 0400.0 0244.7 iMOPSO DICE 0.9910 0.0031 0.8921 0.0288 0.9132 0.0298 AC 0.9824 0.0060 0.8462 0.0376 0.8755 0.0322 HD 1803.6 0397.4 0191.1 0211.3 0111.5 0072.1 Proposed DICE 0.9915 0.0025 0.9184 0.0240 0.9352 0.0253 AC 0.9834 0.0048 0.8909 0.0306 0.9044 0.0298 HD 1788.3 0391.2 0184.8 0207.9 0093.0 0066.9 *The values in bold indicate the best performance. To evaluate quantitatively, in this experiment, the surface distance-based metric, Hausdorff distance, is also calculated along with DICE and AC. Figures 8, 9, 10, 13, 14 and 15 show the comparison results. It is noticeable that there are fluctuations in the quantitative values since artifacts such as bias field, contrast and noise existing in the input images are different. The results are summarized in Tables III and V. From the figures and tables, it can be seen that the proposed method outperforms the others. Again, the results presented here confirm the efficiency of the proposed method and demonstrate its superiority over the others. Authorized licensed use limited to: University of Canberra. Downloaded on May 03,2020 at 17:07:13 UTC from IEEE Xplore. Restrictions apply.

20. 1057-7149 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIP.2020.2990346, IEEE Transactions on Image Processing IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. XX, NO. XX, XXX 2019 12 Fig. 6: Dice coefficient for tissues (brainWeb dataset): (a) CSF; (b) GM; (c) WM. Fig. 7: Accuracy for tissues (brainWeb dataset): (a) CSF; (b) GM; (c) WM. Fig. 8: Dice coefficient for tissues (IBSR dataset): (a) CSF; (b) GM; (c) WM. Fig. 9: Accuracy for tissues (IBSR dataset): (a) CSF; (b) GM; (c) WM. Fig. 10: Hausdorff distance for tissues (IBSR dataset): (a) CSF; (b) GM; (c) WM. Authorized licensed use limited to: University of Canberra. Downloaded on May 03,2020 at 17:07:13 UTC from IEEE Xplore. Restrictions apply.

21. 1057-7149 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIP.2020.2990346, IEEE Transactions on Image Processing IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. XX, NO. XX, XXX 2019 13 Fig. 11: Qualitative segmentation results (IBSR dataset): (a) original images; (b) ground truth images; (c) segmentation results. Fig. 12: Qualitative segmentation results (MRBrainS18 dataset): (a) original images; (b) ground truth images; (c) segmentation results. TABLE IV: Mean ± standard deviation of time cost for each for-loop iteration. Dimension Datasets Number of images Image size Time/loop (s) 2D BrainWeb 9 181 x 217 302.72 ± 45.06 IBSR 9 135 x 142 90.46 ± 15.43 F. Analysis of using undesirable initialization In practice, the proposed method performs poorly when images exhibit strong bias field, blurry object boundaries and unexpected initializations. Figure. 16 shows an example when the proposed method performed with FCM and iMOPSO initializations. It can be seen that large parts from the real objects cannot be classified correctly with FCM initialization case. However, when the proposed method is provided with ”good” initialization, iMOPSO initialization case, the segmentation result is satisfactory. There are two main reasons leading to this situation. First, when the boundaries of objects are blurry and the bias field is significant, the difference approximations will inevitably introduce large error during the evolution. Second, due to the initialization which aims to provide the starting point for Markov random field methods, the direction of the search engine falls to local optimal area and cannot jump out (natural drawback of MRF-based methods). G. Computational analysis The proposed method benefits both advantages of the Markov random field based model and the hybrid metaheuristic algorithm to satisfy requirements of image segmentation problem. However, to solve the problem of selecting appropriate value of β, the core of the proposed method has to be repeated several times (L), hence the total time required to end increases in proportion to (L). To analyse the computational complexity of the proposed algorithm, we calculated the running time when segmenting brain MR images. Both simulated and real brain MR image datasets are involved in our experiments. Since the eventual computational cost will be the multiplication of cost for each for-loop iteration and the number of for-loop iterations (see Algorithm 4), the average time cost of each for-loop iteration is recorded. The mean Authorized licensed use limited to: University of Canberra. Downloaded on May 03,2020 at 17:07:13 UTC from IEEE Xplore. Restrictions apply.

22. 1057-7149 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIP.2020.2990346, IEEE Transactions on Image Processing IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. XX, NO. XX, XXX 2019 14 Fig. 13: Dice coefficient for tissues (MRBrainS18 dataset): (a) CSF; (b) GM; (c) WM. Fig. 14: Accuracy for tissues (MRBrainS18 dataset): (a) CSF; (b) GM; (c) WM. Fig. 15: Hausdorff distance for tissues (MRBrainS18 dataset): (a) CSF; (b) GM; (c) WM. Fig. 16: Qualitative segmentation results using different initializations: (a) original image; (b) ground truth; (c) initial label image using FCM algorithm; (d) result with FCM; (e) initial label image using iMOPSO algorithm; (f) result with iMOPSO. and standard deviation of the time cost (Intel Core i7 1.8 GHz CPU, 8GB RAM and Matlab 2014b) are listed in Table V. Note that, the computational time for initialization step (using iMOPSO algorithm consists in doing bias correction and providing initial label image) is not considered here. From our previous work, it takes around 900(s) for brainWeb dataset and 400(s) for IBSR dataset. VI. CONCLUSION In this paper, a new method, which takes advantages of hidden Markov random field approach and searching ability of hybrid metaheuristic algorithm, for the segmentation of MR brain images has been proposed. To achieve satisfactory segmentation results, first, a novel Markov random field model is derived. By using an adaptive weight mechanism and a new potential function, not only balancing contributions of components in the optimization process is achieved but also spatial information and local intensity are utilized, the exploitation of local spatial interactions, to deal with artifacts existing in images. Secondly, a new hybrid metaheuristic algorithm, which is based on two well-known algorithms, Cuckoo search (CS) and Particle swarm optimization (PSO), is also proposed. This algorithm cooperates the two algorithms, by working on the same population in a parallel way and with a solution selection mechanism. Therefore, the algorithm can produce better results in terms of quality in a shorter time. Furthermore, to enhance the efficiency of searching solutions, iMOPSO- based method is utilized to provide a starting point for gen- Authorized licensed use limited to: University of Canberra. Downloaded on May 03,2020 at 17:07:13 UTC from IEEE Xplore. Restrictions apply.

23. 1057-7149 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIP.2020.2990346, IEEE Transactions on Image Processing IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. XX, NO. XX, XXX 2019 15 TABLE V: Mean and standard deviation of DICE, AC and HD results for the considered algorithms calculated over the set of 7 real MR brain images from the MRBrainS18 data Methods Metrics Regions CSF GM WM Mean Std. Dev. Mean Std. Dev. Mean Std. Dev. FRFCM DICE 0.9089 0.0174 0.8644 0.0324 0.9043 0.0333 AC 0.8526 0.0235 0.8031 0.0447 0.8616 0.0475 HD 0766.4 0798.9 0188.1 0062.9 0171.1 0044.9 HMRF-EM DICE 0.9342 0.0169 0.8475 0.0379 0.8777 0.0425 AC 0.8952 0.0233 0.7754 0.0545 0.8275 0.0581 HD 0278.7 0208.3 0204.2 0066.4 0183.2 0034.4 MICO DICE 0.9208 0.0167 0.8177 0.0178 0.8425 0.0191 AC 0.8703 0.0226 0.7304 0.0231 0.7822 0.0285 HD 0378.8 0322.9 0218.2 0073.1 0191.0 0040.4 PSO-MRF DICE 0.9345 0.0132 0.8694 0.0310 0.9050 0.0321 AC 0.8980 0.0179 0.8091 0.0430 0.8621 0.0460 HD 0168.1 0061.4 0195.4 0073.8 0178.5 0038.2 iMOPSO DICE 0.9013 0.0157 0.8651 0.0201 0.9057 0.0225 AC 0.8423 0.0220 0.8027 0.0248 0.8624 0.0332 HD 0183.8 0056.7 0162.2 0057.9 0172.1 0074.4 Proposed DICE 0.9374 0.0114 0.8744 0.0245 0.9200 0.0223 AC 0.9025 0.0153 0.8214 0.0306 0.8809 0.0336 HD 0169.2 0061.1 0160.8 0056.8 0169.7 0061.2 *The values in bold indicate the best performance. erating initial population. To confirm the effectiveness of the proposed method, it has been examined on both simulated and real MR images, then compared to four recent segmentation methods in the literature. The experimental results show that our method can produce better segmentation results and is able to handle high levels of noise and INU artifact contained in input images. In particular, both qualitative and quantitative results of segmentation show the better performance compared to the considered algorithms. However, in this method, the computational cost is high because of the problem of selecting appropriate value of parameter β as well as the running time for both ICS and IPSO algorithms. In addition, the method may not be suitable for images produced by functional imaging technique, such as PET; since the initialization obtained from the iMOPSO-based method relied on assumptions about mathematical image model for structural imaging techniques. Furthermore, tuning a number of parameters (including both CS and PSO algorithms) may also be a difficulty for using the proposed method. As a part of future work, the problem of selecting β will be solved by maximum likelihood estimation within the optimization process. In addition, the possibility of extending the proposed algorithm to other image modalities used in medical imaging can also be worked out. REFERENCES [1] F. Zhao and X. Xie, “An overview of interactive medical image segmentation,” Annals of the BMVA, vol. 2013, no. 7, pp. 1–22, 2013. [2] D. Cremers, F. Tischhäuser, J. Weickert, and C. Schnörr, “Diffusion snakes: Introducing statistical shape knowledge into the Mumford-Shah functional,” Int J Comput Vis, vol. 50, no. 3, pp. 295–313, 2002. [3] A. Simmons, P. S. Tofts, G. J. Barker, and S. R. Arridge, “Sources of intensity nonuniformity in spin echo images at 1.5 T,” Magn Reson Med, vol. 32, no. 1, pp. 121–128, 1994. [4] X. Han and B. Fischl, “Atlas renormalization for improved brain MR image segmentation across scanner platforms,” IEEE Trans. Med. Imag., vol. 26, no. 4, pp. 479–486, 2007. [5] N. Gordillo, E. Montseny, and P. Sobrevilla, “State of the art survey on MRI brain tumor segmentation,” Magn Reson Imaging, vol. 31, no. 8, pp. 1426–1438, 2013. [6] L. Dora, S. Agrawal, R. Panda, and A. Abraham, “State-of-the-art methods for brain tissue segmentation: A review,” IEEE Rev Biomed Eng, vol. 10, pp. 235–249, 2017. [7] H. Zhou, X. Li, G. Schaefer, M. E. Celebi, and P. Miller, “Mean shift based gradient vector flow for image segmentation,” Comput Vis Image Underst, vol. 117, no. 9, pp. 1004–1016, 2013. [8] T. X. Pham, P. Siarry, and H. Oulhadj, “Integrating fuzzy entropy clustering with an improved PSO for MRI brain image segmentation,” Appl Soft Comput, vol. 65, pp. 230–242, 2018. [9] U. Maulik, “Medical image segmentation using genetic algorithms,” IEEE Trans Inf Technol Biomed, vol. 13, no. 2, pp. 166–173, 2009. [10] P. Krishnan and P. Ramamoorthy, “Fuzzy clustering based ant colony optimization algorithm for MR brain image segmentation,” JATIT, vol. 65, no. 3, pp. 644–649, 2014. [11] V. Vijay, A. Kavitha, and S. R. Rebecca, “Automated brain tumor segmentation and detection in MRI using enhanced darwinian particle swarm optimization (EDPSO),” Procedia Computer Science, vol. 92, pp. 475–480, 2016. [12] E. Ilunga-Mbuyamba, J. M. Cruz-Duarte, J. G. Avina-Cervantes, C. R. Correa-Cely, D. Lindner, and C. Chalopin, “Active contours driven by Cuckoo search strategy for brain tumour images segmentation,” Expert Syst Appl, vol. 56, pp. 59–68, 2016. [13] M. N. Ahmed, S. M. Yamany, N. Mohamed, A. A. Farag, and T. Mori- arty, “A modified fuzzy c-means algorithm for bias field estimation and segmentation of MRI data,” IEEE Trans. Med. Imag., vol. 21, no. 3, pp. 193–199, 2002. [14] C. Li, R. Huang, Z. Ding, J. C. Gatenby, D. N. Metaxas, and J. C. Gore, “A level set method for image segmentation in the presence of intensity inhomogeneities with application to MRI,” IEEE Trans. Image Process., vol. 20, no. 7, pp. 2007–2016, 2011. [15] J. L. Marroquin, B. C. Vemuri, S. Botello, E. Calderon, and A. Fernandez-Bouzas, “An accurate and efficient Bayesian method for automatic segmentation of brain MRI,” IEEE Trans. Med. Imag., vol. 21, no. 8, pp. 934–945, 2002. [16] M. Zhang, L. Jiao, W. Ma, J. Ma, and M. Gong, “Multi-objective evolutionary fuzzy clustering for image segmentation with MOEA/D,” Appl Soft Comput, vol. 48, pp. 621–637, 2016. [17] T. X. Pham, P. Siarry, and H. Oulhadj, “A multi-objective optimization approach for brain MRI segmentation using fuzzy entropy clustering and region-based active contour methods,” Magn Reson Imaging, vol. 61, pp. 41–65, 2019. [18] L. E. Baum and T. Petrie, “Statistical inference for probabilistic functions of finite state Markov chains,” Ann. Math. Stat., vol. 37, no. 6, pp. 1554–1563, 1966. [19] S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” in Readings in Computer Vision: Issues, Problem, Principles, and Paradigms. Morgan Kaufmann, 1987, pp. 564 – 584. [20] S. Yousefi, R. Azmi, and M. Zahedi, “Brain tissue segmentation in MR images based on a hybrid of MRF and social algorithms,” Med Image Anal, vol. 16, no. 4, pp. 840–848, 2012. [21] E. Guerrout, S. Ait-Aoudia, D. Michelucci, and R. Mahiou, “Hidden Markov random fields and direct search methods for medical image segmentation,” in ICPRAM, 2016, pp. 154–161. [22] P. T Krishnan, P. Balasubramanian, and C. Krishnan, “Segmentation of brain regions by integrating meta heuristic multilevel threshold with Markov random field,” Curr Med Imaging Rev, vol. 12, no. 1, pp. 4–12, 2016. [23] Y. Zhang, M. Brady, and S. Smith, “Segmentation of brain MR images through a hidden Markov random field model and the Expectation- Maximization algorithm,” IEEE Trans. Med. Imag., vol. 20, no. 1, pp. 45–57, 2001. [24] Huawu Deng and D. A. Clausi, “Unsupervised segmentation of synthetic aperture radar sea ice imagery using a novel Markov random field model,” IEEE Trans. Geosci. Remote Sens, vol. 43, no. 3, pp. 528–538, March 2005. [25] M. B. Cuadra, L. Cammoun, T. Butz, O. Cuisenaire, and J. . Thiran, “Comparison and validation of tissue modelization and statistical classification methods in T1-weighted MR brain images,” IEEE Trans. Med. Imag., vol. 24, no. 12, pp. 1548–1565, Dec 2005. [26] N. Otsu, “A threshold selection method from gray-level histograms,” IEEE Trans. Syst., Man, Cybern. A, Syst., Humans, vol. 9, no. 1, pp. 62–66, 1979. Authorized licensed use limited to: University of Canberra. Downloaded on May 03,2020 at 17:07:13 UTC from IEEE Xplore. Restrictions apply.

24. 1057-7149 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIP.2020.2990346, IEEE Transactions on Image Processing IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. XX, NO. XX, XXX 2019 16 [27] J. Tohka, I. D. Dinov, D. W. Shattuck, and A. W. Togab, “Brain MRI tissue classification based on local Markov random fields,” Magn Reson Imaging, vol. 28, no. 4, pp. 557—-573, 2011. [28] M. Xie, J. Gao, C. Zhu, and Y. Zhou, “A modified method for MRF segmentation and bias correction of MR image with intensity inhomogeneity,” Med Biol Eng Comput, vol. 53, no. 1, pp. 23–35, Jan 2015. [29] E.-H. Guerrout, R. Mahiou, and S. Ait-Aoudia, “Hidden Markov random fields and particle swarm combination for brain magnetic resonance image segmentation,” Int Arab J Inf Techn, vol. 15, no. 3, 2017. [30] E. Ben George, G. J. Rosline, and D. G. Rajesh, “Brain tumor segmentation using Cuckoo search optimization for magnetic resonance images,” in 2015 IEEE 8th GCC Conference Exhibition, Feb 2015, pp. 1–6. [31] H. Deng and D. A. Clausi, “Unsupervised image segmentation using a simple MRF model with a new implementation scheme,” Pattern Recognit, vol. 37, no. 12, pp. 2323–2335, 2004. [32] S. Z. Li, Markov Random Field Modeling in Computer Vision. New York: Springer-Verlag, 2001. [33] X. Yang, X. Gao, D. Tao, X. Li, and J. Li, “An efficient MRF embedded level set method for image segmentation,” IEEE Trans. Image Process., vol. 24, no. 1, pp. 9–21, Jan 2015. [34] M. Chen, Q. Yan, and M. Qin, “A segmentation of brain MRI images utilizing intensity and contextual information by Markov random field,” Computer Assisted Surgery, vol. 22, no. sup1, pp. 200–211, 2017. [35] I. Despotović, B. Goossens, and W. Philips, “MRI segmentation of the human brain: challenges, methods, and applications,” Comput Math Method M, vol. 2015, 2015. [36] T. Zhou, S. Ruan, and S. Canu, “A review: Deep learning for medical image segmentation using multi-modality fusion,” Array, p. 100004, 2019. [37] R.-S. Kwan, A. C. Evans, and G. B. Pike, “MRI simulation-based evaluation of image-processing and classification methods,” IEEE Trans. Med. Imag., vol. 18, no. 11, pp. 1085–1097, 1999. [38] J. Zhang, “The mean field theory in EM procedures for Markov random fields,” IEEE Trans. Image Process., vol. 40, no. 10, pp. 2570–2583, 1992. [39] J. Besag, “Spatial interaction and the statistical analysis of lattice systems,” J R Stat Soc Series B Stat Methodol, vol. 36, no. 2, pp. 192– 225, 1974. [40] T. Chen, T. S. Huang, and Z.-P. Liang, “Segmentation of brain MR images using hidden Markov random field model with weighting neighbourhood system,” in IEEE Symposium Conference Record Nuclear Science 2004., vol. 5. IEEE, 2004, pp. 3209–3212. [41] E.-H. Guerrout, S. Ait-Aoudia, D. Michelucci, and R. Mahiou, “Hidden Markov random field model and Broyden–Fletcher–Goldfarb–Shanno algorithm for brain image segmentation,” J Exp Theor Artif Intell, vol. 30, no. 3, pp. 415–427, 2018. [42] X.-S. Yang and S. Deb, “Cuckoo search via lévy flights,” in 2009 World Congress on Nature Biologically Inspired Computing (NaBIC). IEEE, 2009, pp. 210–214. [43] I. Fister Jr, D. Fister, and I. Fister, “A comprehensive review of cuckoo search: variants and hybrids,” International Journal of Mathematical Modelling and Numerical Optimisation, vol. 4, no. 4, pp. 387–409, 2013. [44] R. N. Mantegna, “Fast, accurate algorithm for numerical simulation of levy stable stochastic processes,” Physical Review E, vol. 49, no. 5, p. 4677, 1994. [45] A. J. Siswantoro, “Soft computing applications and intelligent systems,” 2013. [46] A. W. Mohamed, A. K. Mohamed, E. Z. Elfeky, and M. Saleh, “En- hanced directed differential evolution algorithm for solving constrained engineering optimization problems,” International Journal of Applied Metaheuristic Computing (IJAMC), vol. 10, no. 1, pp. 1–28, 2019. [47] T. T. Nguyen, D. N. Vo, and B. H. Dinh, “An effectively adaptive selective cuckoo search algorithm for solving three complicated short- term hydrothermal scheduling problems,” Energy, vol. 155, pp. 930–956, 2018. [48] R. Eberhart and J. Kennedy, “Particle swarm optimization,” in Proceed- ings of the IEEE International Conference on Neural networks, vol. 4. Citeseer, 1995, pp. 1942–1948. [49] S. Sengupta, S. Basak, and R. Peters, “Particle swarm optimization: A survey of historical and recent developments with hybridization perspectives,” Mach Learn Knowl Extr, vol. 1, no. 1, pp. 157–191, 2018. [50] Y. Chen, L. Li, H. Peng, J. Xiao, Y. Yang, and Y. Shi, “Particle swarm optimizer with two differential mutation,” Appl Soft Comput, vol. 61, pp. 314 – 330, 2017. [51] A. W. Mohamed, “A novel differential evolution algorithm for solving constrained engineering optimization problems,” J Intell Manuf, vol. 29, no. 3, pp. 659–692, Mar 2018. [52] C. Yang, W. Gao, N. Liu, and C. Song, “Low-discrepancy sequence ini- tialized particle swarm optimization algorithm with high-order nonlinear time-varying inertia weight,” Appl Soft Comput, vol. 29, pp. 386–394, 2015. [53] L. Wang, Z. Yiwen, and Y. Yilong, “A hybrid cooperative cuckoo search algorithm with particle swarm optimisation,” International Journal of Computing Science and Mathematics, vol. 6, no. 1, pp. 18–29, 2015. [54] A. Bouyer and A. Hatamlou, “An efficient hybrid clustering method based on improved cuckoo optimization and modified particle swarm optimization algorithms,” Appl Soft Comput, vol. 67, pp. 172 – 182, 2018. [55] C. Li, J. C. Gore, and C. Davatzikos, “Multiplicative intrinsic component optimization (MICO) for MRI bias field estimation and tissue segmentation,” Magn Reson Imaging, vol. 32, no. 7, pp. 913–923, 2014. [56] Z. Y. Chua, W. Zheng, M. W. Chee, and V. Zagorodnov, “Evaluation of performance metrics for bias field correction in MR brain images,” JMRI-J Magn Reson Im, vol. 29, no. 6, pp. 1271–1279, 2009. [57] T. Lei, X. Jia, Y. Zhang, L. He, H. Meng, and A. K. Nandi, “Signif- icantly fast and robust fuzzy C-means clustering algorithm based on morphological reconstruction and membership filtering,” IEEE Trans. Fuzzy Syst., vol. 26, no. 5, pp. 3027–3041, 2018. [58] Q. Wang, “HMRF-EM-image: implementation of the hidden Markov random field model and its expectation-maximization algorithm,” arXiv preprint arXiv:1207.3510, 2012. [59] J. Besag, “On the statistical analysis of dirty pictures,” J R Stat Soc Series B Stat Methodol, vol. 48, no. 3, pp. 259–279, 1986. [60] L. R. Dice, “Measures of the amount of ecologic association between species,” Ecology, vol. 26, no. 3, pp. 297–302, 1945. [61] M. Beauchemin, K. P. Thomson, and G. Edwards, “On the Hausdorff distance used for the evaluation of segmentation results,” Can J Remote Sens, vol. 24, no. 1, pp. 3–8, 1998. [62] A. A. Taha and A. Hanbury, “Metrics for evaluating 3D medical image segmentation: analysis, selection, and tool,” BMC Med Imaging, vol. 15, no. 1, p. 29, 2015. Thuy Xuan Pham received the B.Eng. degree in Automatic control from Le Quy Don Technical University, Vietnam, in 2006, the M.Sc. degree in industrial sciences, electronic engineering from GroupT International University College Leuven, Belgium, in 2013. He is currently pursuing the Ph.D. degree in signal, image, and automation with the doctoral School of MSTIC of Université Paris-Est, France. His current research interests include image processing and metaheuristics optimization. Patrick Siarry was born in France in 1952. He received the PhD degree from the University Paris 6, in 1986 and the Doctorate of Sciences (Habilitation) from the University Paris 11, in 1994. He was first involved in the development of analog and digital models of nuclear power plants at Electricité de France (E.D.F.). Since 1995 he is a professor in automatics and informatics. His main research interests are computer-aided design of electronic cir- cuits, and the applications of new stochastic global optimization heuristics to various engineering fields. He is also interested in the fitting of process models to experimental data, the learning of fuzzy rule bases, and of neural networks. Hamouche Oulhadj was born in Algeria in 1956. He is a teacher-researcher at the University of Paris- Est Créteil, since September 1993. He received the Engineer Graduation in Electrical Engineering at the Polytechnic School of Algiers in 1984, the Postgraduate Diploma in Biomedical Engineering and the Doctorate of Sciences from the University Paris 12, in 1985 and 1990 respectively. His main research interests are optimization and biomedical image segmentation. Authorized licensed use limited to: University of Canberra. Downloaded on May 03,2020 at 17:07:13 UTC from IEEE Xplore. Restrictions apply.

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