Fully automatic grayscale image segmentation based fuzzy C‑means with firefly mate algorithm

(1)

https://doi.org/10.1007/s12652-021-03430-3 ORIGINAL RESEARCH

Fully automatic grayscale image segmentation based fuzzy C‑means with firefly mate algorithm

Waleed Alomoush¹ · Ayat Alrosan¹ · Yazan M. Alomari² · Alaa A. Alomoush³ · Ammar Almomani⁴ · Hammoudeh S. Alamri³

Received: 27 June 2020 / Accepted: 5 August 2021

Abstract

Image segmentation is the method of dividing an image into many segments, comprising groups of pixels. It is a process used to determine objects within the image. Fuzzy c-means (FCM) technique has been popularly employed as grayscale image segmentation method. Meanwhile, the conventional FCM suffers from some drawbacks including easy fall into local optimal solution resulting from inappropriate selection of the initial cluster center values and optimal number of clusters (regions) for each image without a prior knowledge or input by the operator. To solve FCM issues, the paper proposes a new fully automatic segmentation method for grayscale images based on fuzzy c-means with firefly mate algorithm (AUTO-FCM-FMA). This approach utilizes the mate list (M) mechanism with firefly algorithm (FMA) to search for the near-optimal number clusters, the location of centroids by exploring the search space and void stuck in local optimum, and the best outcomes from FMA as input for FCM. To evaluate its effectiveness, the proposed algorithm was tested on different types of images. These images can be categorized into simulated MRI images (normal and MSL), synthetic images and natural images. All these images cover different domains and levels of difficulty (e.g. clusters overlapping). The results of validation experiments were encouraging, especially when the performance of proposed algorithm outcomes was compared to that of other state-of-the-art algorithms.

Keywords FCM · MRI image · Fuzzy clustering · Fully automatic images segmentation · Metaheuristic search algorithms and firefly mate algorithm

1 Introduction

The image segmentation process outcome comprises a set of regions covering every object within a given image. The pixels (objects) inside a given region all share similarity in terms of several computed property or features (e.g., texture, intensity and color) (Alrosan and Norwawi 2017;

Bose and Mali 2016; Houssein et al. 2021). The success of image segmentation can be observed in in many aspects, as can be observed in segmentation of medical images for the extraction of tumors as well that of other pathologies (W.

Alomoush et al. 2021a, b; Alrosan et al. 2021a). The role played by image segmentation is vital in several types of operations, therapy estimate, surgical emulation, treatment

* Waleed Alomoush

[email protected] Ayat Alrosan

[email protected] Yazan M. Alomari

[email protected] Alaa A. Alomoush [email protected] Ammar Almomani [email protected] Hammoudeh S. Alamri [email protected]

1 School of Information Technology, Skyline University College, Sharjah, P.O. Box 1797, United Arab Emirates

2 Department of Management Information Systems, College of Applied Studies and Community Services, Imam Abdulrahman Bin Faisal University, Dammam, Saudi Arabia

3 IBM Center of Excellence, Faculty of Computer Systems and Software Engineering, Universiti Malaysia Pahang, 26300 Kuantan, Pahang, Malaysia

4 Department of Information Technology, Al-Huson University College, Al-Balqa Applied University, Irbid, Jordan

(2)

strategy, and in the review of anatomical structure as well (Alia et al. 2011; W. Alomoush et al. 2014a, b).

In many real image applications, limitations such as problems in spatial resolution, poor contrast, overlapping between boundaries and intensity inhomogeneities have made methods of segmentation a difficult task. As a solution, fuzzy set theory was presented, with the idea of partial membership described by a membership objective function (Alrosan et al. 2021b). Fuzzy c-means technique (FCM) has been the popularly employed grayscale image segmentation method (Alrosan et al. 2021a; Balafar 2014; Bose and Mali 2016; Ozturk et al. 2015b). It has robust characteristics for ambiguity and can retain much more information than hard segmentation methods (Alia et al. 2011; W. Alomoush and Alrosan 2018; W. Alomoush et al. 2018; W. K. Alomoush et al. 2014a, b; Bose and Mali 2016).

The research was primarily motivated by two reasons.

Firstly, the application of clustering based approach in image segmentation is still rather imperfect, as the approach is still unable to obtain a completely automatic segmentation, owing to unknown prior knowledge, location centroid and easy fall into local optimal solution. Additionally, in each image, experts or operators need to know the optimal clusters, meaning that, this is a semi-automatic segmentation method. But, the need for human experts makes this approach time consuming. Meanwhile, efforts in developing a clustering approach that can automatically decide the suitable number of clusters without prior knowledge are ongo- ing. Accordingly, in resolving the fuzzy clustering based problems, a fuzzy clustering technique (FCM) utilizing optimization search algorithms appears to be appropriate.

Among these optimization algorithms, several are inspired from intelligent behaviors of the swarm, such as particle swarm optimization (PSO) (Kennedy and Eberhart 1995), ant colony optimization (ACO) (Dorigo et al. 2006), artificial bee colony (ABC) (Karaboga and Basturk 2007), cuckoo search (CS) (W. Alomoush 2019; Yang and Deb 2010), Bat algorithm (BA) (Yang 2010c), harmony search (HS) (A. A. Alomoush et al. 2019, 2021a, b; Geem et al.

2001) and firefly (FA) (Yang 2008). These algorithms are naturally inspired by, among others, social behaviors of bird flocking, ant colonies, bee colonies, etc. Among these algorithms, FA is a comparatively recent swarm intelligence algorithm and it was proposed by Yang (2008).

Yang (2008) reported that a group of experimental results on benchmark functions shows the competence of FA algorithm against several traditional biologically inspired optimization algorithms such as GA and PSO. FA is reliable and entails easy steps, and these attributes encourage researchers from different fields to use this algorithm in segmentation of images (Dey et al. 2020). Also, this algorithm has been successfully applied in various optimization fields such as image watermarking (Jagatheesan et al. 2020), economic

dispatch (Banumalar et al. 2017), Power systems (Albert and Stonier 2020), human recognition (Sánchez et al. 2017), scheduling problem (Aldeeb et al. 2019), social networks (Langari et al. 2020) and others fields (Alauthman et al.

2019; Almomani et al. 2021). Based on the advantages and various applications mentioned above, FA algorithm has been considered a well-performing optimization algorithm.

However, FA has some limitations such as the tendency to be stuck into several local optima. Also, it does not memorize the history of any situation during iteration time. There- fore, some researchers use mate list (M) mechanism with firefly algorithm in order to solve the local optima problem (W. Alomoush et al. 2020). Also, in this research the mate list (M) mechanism with firefly algorithm was used to prevent entrapment into into local optima. This paper proposes a new fully automatic segmentation method for grayscale images based on fuzzy c-means with firefly mate algorithm (AUTO-FCM-FMA). This approach utilizes mate list (M) mechanism with firefly algorithm (FMA) to search for near optimal number of clusters and location of centroids by exploring the search space, with no inclination to easily fall into local optimal solution, which could solve the main issues associated with FCM segmentation method. The rest of this paper is organized as follows: Sect. 2 discusses the conventional fuzzy c-means algorithm, Sect. 3 presents FCM based optimization algorithms and related works, Sect. 4 discusses firefly algorithm based mate list (FMA), Sect. 5 presents automatic grayscale image segmentation based fuzzy C-Means with firefly mate algorithm AUTO-FCM-FMA, Sect. 6 displays the experimental results, and the last section (Sect. 7) presents the conclusion of this paper.

2 Fuzzy C‑means algorithm

As an unsupervised learning approach, FCM could divide undistinguishable data elements into groups using their similarity level. As a result, the similarity between objects within a region is increased, while the similarity in several region is decreased (Bezdek et al. 1981; Nayak et al. 2015).

With the application of a clustering algorithm, fuzzy data are classed using a group of n objects or pixels. However, FCM algorithm is significantly limited in terms of sensitive to noise because it it possesses no information about spatial context. So, the AUTO-FCM-FMA algorithm employs the reformulated objective function as in Eq. (1) (Bose and Mali 2016; Rhee and Hwang 2001),

where is Ji is the value of objective function of ith cluster, the value of membership is hkj which is different from

(1) J_i=

∑n j=1

∑

k∈C_j

h²_kj∗D_kj

(3)

standard FCM membership value as in objective function;

it is used to reduce noise sensitivity. The new membership value is used as in Eq. (2)

where: the membership value is μ_kj of pixel k to the jth cluster, where the distance between clusters center c_j and pixel k is Dk j, also, Eq. (3) is employed to update centroids location value vkj.

These processes repeat until the number of iterations (t) is completed or termination criteria are performed as in equation.

3 FCM based optimization search algorithms and related works

In the image segmentation clustering based FCM method, several efforts have been reported during the last several years focusing on the development of segmentation approaches based on FCM clustering algorithm. In this regard, FCM based optimization search algorithms were considered a preferable choice to determine the location of centroids and optimal number of clusters (regions) for each image without a prior knowledge or input by the operator.

Accordingly. The present section presents a review of related works on FCM based optimization search algorithms.

W. Alomoush et al. (2014a, b) proposed a hybrid standard firefly algorithm and fuzzy c-means algorithm namely FFCM. The authors evaluated the algorithm using simulated brain dataset and actual MRI brain image, and the results were promising. However, the main limitation of FFCM is that it has sensitivity to noise. Ghosh et al. (2018) proposed FCM clustering based Firefly algorithm with a chaotic map, which is called C-FAFCM for medical images application.

C-FAFCM was tested to several real T1-weighted and simulated of MRI brain images and it generated satisfactory outcomes when compared to other state of the art methods algorithms.

Chinta et al. (2018) proposed a new hybrid algorithm called IFCMFA which combines intuitionistic fuzzy c-means. IFCMFA Experimental analysis shows superior results to those obtained through FCM and IFCM. Pant et al. (2019) combined two fuzzy clustering algorithms namely fuzzy C-means and intuitionistic fuzzy C-means (2) h_kj= μ_kj−1− μ_kj

2

(3) v_kj =

∑n

k=1h_kj. x_kj

∑n k=1h_kj

(4)

‖‖v_new−v_old‖

‖< 𝜀 Where𝜀 <0.001

with a metaheuristic algorithm called fuzzy firefly algorithm. The authors used these hybrid clustering algorithms (FCMFFA and IFCMFFA) in image segmentation. Kumar et al. (2019) proposed a segmentation method for CT/MR medical image using firefly search with FCM algorithm.

Meanwhile, in removing artifacts and in denoising, the authors utilized Nonlinear Tensor Diffusion (NLTD) fil- ter. The authors reported that outcomes of this approach were superior to those generated by other state of the art approaches.

The utilization of dynamic fuzzy clustering algorithm (DC) with hybrid harmony search (HS) algorithm and FCM was proposed in Alia et al. (2011). This algorithm generates automatic segment method known as DCHS for MRI brain images, both real (IBSR) and simulated ones (SBD).

The authors then performed evaluation on DCHS using both types of MRI brain images (real and stimulated ones), and affirmed the ability of DCHS approach in identifying the right number of clusters (regions) for both types of images.

Wan et al. (2018) Proposed a self-adaptive multi-objective HS based FCM namely SAMOHSFC. This method uses harmony vector and optimizes multiple objectives to encode several cluster centers. SAMOHSFC was tested on synthetic images and two real images. In the experiment, the outcome of SAMOHSFC was analyzed based of different kinds of spatial information.

Ouadfel and Meshoul (2012) proposed FCM based on a modified Artificial Bee Colony algorithm (MoABC) namely MoABC-FCM. The method was inspired by the Differential Evolution DE to enhance the process of exploitation. The outcomes showed that the proposed algorithm enhanced the effectiveness of FCM. The proposed algorithm was superior to other state-of-the-art methods. Salima et al. (2012) presented Artificial Bee Colony (ABC) with a new spatial fuzzy clustering algorithm called ABC-SFCM. The algorithm was tested on synthetic and real images, and based on the results of experiments; the authors reported the robustness of the proposed method against noise. The proposed algorithm proved its superiority to other state-of-the-art algorithms.

In Hancer et al. (2013), an image segmentation method was demonstrated with the application of ABC algorithm. The obtained results were compared with those of other algorithms such as FCM, K-means, and GA, and ABC.

Alrosan et al. (2014) proposed the combination of ABC algorithm and fuzzy c-means and named it ABC-FCM. The proposed method was tested using two types of MRI images namely simulated brain data and actual MRI images. Mean- while, FABC was proposed in Bose and Mali (2016). As an approach of image segmentation, FABC encompasses a combination of ABC and the original FCM. The authors employed FABC, GA, EM and PSO on numerous gray-scale images in their experiments, and they noted that it was chal- lenging to segment these images due to the low contrast and

(4)

noise. It was evident from the results that FABC had greater efficiency as opposed to other state-of-the-art methods.

The use of FCM based particle swarm optimization (DCPSO) which is an automatic hard clustering algorithm can be observed in Omran (2004) and Omran et al. (2005, 2006). The application of this algorithm is initiated by the partitioning of the dataset into a reasonably big number of clusters. This reduces the impact of initialization. Then, an optimal number of clusters were chosen using binary PSO and several indices of cluster validity. Multi-spectral, natural, and synthetic images can be segmented using DCPSO.

In Mekhmoukh and Mokrani (2015), an innovative image segmentation approach with the application of Particle Swarm algorithm and collaboration between level set and outlier rejection was proposed. MRI images were used in the experiment. Comparison was then made between the outcomes of the proposed method and those of other comparable methods, and the proposed method appears superior in terms of effectiveness and computational time. Dhanachan- dra and Chanu (2020) proposed dynamic particle swarm optimization (DPSO) and FCM for image segmentation algorithm. The experiments were carried out using sets of MRI and synthetic images. The outcomes of the proposed method showed better performance and lower sensitivity to noise when compared to FCM and other state-of-the-art methods.

Fuzzy variable string length genetic point symmetry or Fuzzy-VGAPS is an innovative dynamic fuzzy clustering algorithm proposed in some researches (Maulik and Saha 2009; Saha and Bandyopadhyay 2007b, 2009). It includes the application of new objective function with point symmetry index known as Sym-index adapted from the original PS-index (Chou et al. 2004). The outcomes obtained by VGAPS and those of other comparable methods were compared, particularly on fuzzy cluster techniques on four real and four synthetic datasets comprising MRI images of a brain with lesions from multiple sclerosis. The results generated by Fuzzy-VGAPS were promising.

Zhang et al. (2019) proposed hybrid biogeography-based optimization (BBO) versions with FCM approach for image segmentation. In this study, the experiments were carried out on a group of images and the proposed method outper- formed FCM and other state-of-the-art methods. Fred et al.

(2020) proposed the Crow Search (CS) and FCM for CT image segmentation. The Proposed approach showed better performance in terms of results when compared with other state-of-the-art methods.

Accordingly, the following Table 1 summarized the advantages and disadvantages of each reviewed method and the main limitations in all approaches.

It is clear from Table 1 that many of the presented approaches could not determine the number of clusters (semi- automatic segmentation), can be stuck in local optima, have

low convergence speed, have high computational time, cannot solve overlapping problem between regions in images segmented and are very sensitive to noise. Accordingly, the proposed method (AUTO-FCM-FMA) is developed to solve or reduce these limitations using mate list (M) mechanism with firefly algorithm (FMA) to search for the near-optimal number of clusters, the location of centroids by exploring the search space, while averting the problem of being stuck in local optimum. Also, to reduce sensitivity to noise, the AUTO-FCM- FMA algorithm employed the reformulated objective function as in Eq. (1) (Bose and Mali 2016; Rhee and Hwang 2001).

4 Firefly and firefly mate algorithm (FMA)

Firefly algorithm (FA) is one of the Swarm Intelligence (SI) methods developed by Yang (2008). In illustrating this algorithm; many kinds of insects have certain natural behavior to attract others, and fireflies are among such insects. Specifically, fireflies are able to send out flickering and glowing biological lights by illuminating themselves—this is a biological behavior of fireflies (Yang 2009, 2010a, b).

In general, there are three basic rules for fireflies based on their natural behavior.

1. Fireflies are unisexual, which means that any firefly can be attracted by other fireflies regardless of its gender.

2. There is a relation between brightness and attractiveness;

less bright fireflies go to brighter ones. Distance can also have effect on attractiveness; it decreases when the distance is increased. In this case, fireflies will randomly move when there is no brightness found.

3. There is a function called objective function, which is responsible in determining the brightness of the firefly (Yang 2009, 2010a, b).

4.1 Attractiveness and distance

At the beginning, all fireflies move randomly over the search space. At this moment, and based on the objective function, there are two stages for the firefly. During the first stage, every firefly with high/low intensity will entice a firefly with lower/

higher intensity; this is based on the association proposed between the light intensity and the objective function value (Yang 2008). Here, suppose that the number of firefly in search space is n, while the position of a firefly i is xi which represents the solution, and the objective function f(x) is used to evaluate x_i. In addition, f(x) is calculated based on the brightness I of each firefly, as seen in the equation below,

During stage two, fireflies move to the chosen firefly based on the attractive force between them. The amount of force is (5) I_i=f(x_i), 1≤i≤n .

(5)

proportionate with the amount of intensity of light from the nearby firefly (Yang 2008). Each firefly has a unique amount of attraction (β), and this represents the attraction amount of this firefly within the swarm, which is affected by the distance (rij) between two fireflies (i) and (j) at locations (xi) and (xj), respectively, as denoted by Eq. (6) below:

(6) r_ij=‖

‖‖x_i−x_j‖

‖‖ =

√√

∑d k=1

(x_i,k−x_j,k)²,

In Eq. (8), the amount of attraction (β) is calculated for each firefly as follows:

4.2 Movement

When the light observation coefficient is assigned as γ, and the amount of attractiveness of a firefly is assigned as 𝛽₀ at r = 0, then, the firefly i will be moved from position xi to (7) 𝛽(r) = 𝛽₀e^−𝛾r²,

Table 1 The summary of image segmentation by FCM based optimization algorithms

Approaches Advantages Disadvantages

1 FFCM (W. Alomoush et al. 2021a, b) FFCM has the ability to find centroid loca-

tion FFCM could be stucked in local optima,

could not determine the number of clusters (semi-automatic segmentation) and is very sensitive to noise

2 C-FAFCM (Ghosh et al. 2018) C-FAFCM is able to find centroid location

and reduce sensitivity to noise Inability in determining the number of clusters (semi-automatic segmentation) 3 IFCMFA (Chinta et al. 2018; Pant et al.

2019) IFCMFA reduced the initilaze sensetive

centriods and sensitivity to noise IFCMFA has low convergence speed, high computational time and works as semi- automatic method

4 Kumar et al. (2019) Able to find centroid location and reduce

sensitivity to noise Stuck in local optima, low convergence speed and could not determine the number of clusters

6 DCHS (Alia et al. 2011) Able to find centroid location and determine

the number of clusters DCHS has low convergence speed and is sensitive to noise

7 SAMOHSFC (Wan et al. 2018) SAMOHSFC is able to find centroid loca-

tion and reduce sensitivity to noise SAMOHSFC has high computational time and could not determine the number of clusters

8 MoABC-FCM (Ouadfel and Meshoul 2012) MoABC-FCM is able to find centroid loca-

tion MoABC-FCM is very sensitive to noise and

could not determine the number of clusters 9 ABC-SFCM (Salima et al. 2012) ABC-SFCM is able to find centroid location

and reduce sensitivity to noise ABC-SFCM has high computational time and could not determine the number of clusters 10 Hancer et al. (2013) It’s able to find centroid location and reduce

sensitivity to noise It cannot solve overlapping problem between regions in images segmented

11 ABC-FCM (Alrosan et al. 2014) ABC-FCM is able to find centroid location ABC-FCM is sensitive to noise and could not determine the number of clusters

12 FABC (Bose and Mali 2016) FABC is able to find centroid location and

reduce sensitivity to noise FABC has high computational time and could not determine the number of clusters 13 DCPSO (Omran 2004; Omran et al. 2005,

2006) DCPSO is able to determine the number of

clusters and reduce sensitivity to noise DCPSO cannot solve overlapping problem between regions in images segmented 14 DPSO (Dhanachandra and Chanu 2020) DPSO is able to find centroid location,

the number of clusters and could reduce sensitivity to noise

DPSO has low convergence speed and high computational time

15 Fuzzy-VGAPS (Maulik and Saha 2009;

Saha and Bandyopadhyay 2007b, 2009) Fuzzy-VGAPS is able to find centroid loca-

tion and the number of clusters Fuzzy-VGAPS can be stuck in local optima, has low convergence speed and is sensitive to noise

16 BBO-FCM (Zhang et al. 2019) BBO-FCM is able to find centroid location BBO-FCM is sensitive to noise and couldn’t determine the number of clusters

17 CS-FCM (Fred et al. 2020) CS-FCM is able to find centroid location CS-FCM can be stuck in local optima, has low convergence speed, is sensitive to noise and could not determine the number of clusters

(6)

another firefly j which is more attractive at position x_j. This is represented by the following:

The second part of the above equation represents the amount of attraction, while the third part involves randomization which includes randomization parameter 𝛼, and function (rand) that generates a number between [0, 1] randomly.

The observation coefficient parameter γ is responsible in determining the convergence speed. It also affects the behavior of FA algorithm owing to its effect on the amount of attractiveness, with its theoretical value of 𝛾 ∈ [0,∞).

4.3 Mate list

Original firefly algorithm (FA) has many limitations, including the inclination to be stuck into several local (8) x_i=x_i+ 𝛽₀e^−𝛾^r²^ij(x_j−x_i) + 𝛼

(

rand−1 2 )

,

optima. Also, it does not memorize the history of any situation during iteration time. Therefore, some researchers use mate list (M) mechanism with firefly algorithm in order to solve the local optima problem (W. Alomoush et al. 2020), However, in the process of a mate by female firefly, there, there is a huge amount of choices; this will assist the firefly in choosing the best firefly mate.

In firefly mate algorithm (FMA), firefly i will be moved from position x_i to another firefly j which is more attractive at position xj and (i mate j) ∉ {M}. This movement is stored in the M list, and the firefly i is not allowed to move or revisit this position again until this position is removed from mate list after a suitable number of movements, which depends on the mate list size. The movement process to new position will generate new solution, and this solution is then stored in a mate list M. Figure 1 represents the mate list content in FMA during iteration time.

Also, Fig. 2 shows the pseudo code which briefly explains the basic steps for firefly Mate algorithm (FMA).

Mt+1 Movements x_i^t⁺¹ x_i^t x_i^t⁻¹ x_i^t⁻² x_i^t⁻³ x_i^t⁻⁴ x_i^t⁻⁵ x_i^t⁻ⁿ

Solutions Solution(X1) Solution(X2) Solution(X3) … … … …. Solution(Xn)

Fig. 1 Mate list (M) content in FMA

Figure 2: Firefly Mate Algorithm (FMA)

1 Initialize the population of fireflies (N) and generate solutions X_i,i=1, 2,…, N;

2 Set the absorption coefficient (γ),t 3 While (t < MaxGeneration) do

4 For i= 1 to N do

5 For j= 1 to Ndo

6 if f(xj)< f(x_i)and (i mate j) {M} then

7 According to Eq. 8 Move firefly Xi toward X_j;

8 Update mate list {M}

9 End if

10 Attractiveness varies with distance r viaexp[−r2]

11 Compute the value of fitness for new solution;

12 end

13 end

14 Rank the fireflies and find the current best

15 t++;

16 end;

24 Rank the fireflies and return the best one Fig. 2 The pseudo code of FMA

(7)

5 Fully automatic grayscale image segmentation based fuzzy C‑means with firefly mate algorithm

AUTO‑FCM‑FMA

AUTO-FCM-FMA is a fully automatic segmentation approach which uses the capability of FMA to search for the optimum solution that comprises a set number of clusters and centroids’ location. While the steps of the proposed approach are explained as follows:

5.1 Initialization

The objective function f (x) is defined as a set of solutions implemented as X = (x₁, x₂, …, xn) which are evaluated based fitness value f(x) for each firefly n with current position xn. The value f(x) is calculated based on the brightness I of each firefly, as in Eq. (1). Then, the light absorption coefficient γ and mate list {M} are initialized, where each mate list {M}

vector encodes the clusters number and centroids locations of the given dataset. Each mate list {M} size can change based on the number of clusters unknown in this research.

Also, Initialize ClusMaxNum and ClustMinNum which are set based on the minimum and maximum number of clusters in dataset (image region number). In this research the mate list {M} size should be the equal maximum number of clusters ClusMaxNum, where the number of clusters is generated randomly as expressed in Eq. (10) below.

The ClusMaxNum and ClustMinNum are located based on the dataset (images). When the clusters number (ClustNo) is dynamic to change, it means that the mate list {M} size is allowed to change.

5.2 Attractiveness

During the second stage, fireflies move to the chosen firefly based on the best attractiveness between them. The best attractiveness between fireflies is calculated based on the amount of intensity of light from the nearby firefly I_i=f(x_i) . Each firefly has a unique amount of attraction (β). This represents the attraction amount of this firefly within the swarm, which is affected by the location of fireflies (i) and (j), and the distance between them.

ClustNo=ClustMinNum+rand().(ClustMaxNum−ClustMinNum)(9)

5.3 Movement

In this step, firefly i will be moved from positionxi to another firefly j which is more attractive at position xj and (i mate j) ∉ {M}. Here, the movement process is represented by Eq. (9). This movement is stored in the M list, and the firefly i is not allowed to move or revisit this position again until this position is removed from mate list after a suitable number of movements, which depends on the mate list size.

The movement process to new position will generate new solution (number of clusters and centroids location). This solution is then stored in a mate list M. The new position is x^t+1_i of firefly i when it moves towards j where t is the time iterations. In this regard, {

M_t+1}

is the new status of mate list after next movement and new solution are generated.

After the movement process, the mate list M is used as memory to store the number of clusters and centroids location, where the size of M represents the number of calculated clusters. Additionally, each vector length {M} must equal the maximum clusters number (ClusMaxNum). In some cases the clusters number ClustNo is smaller than ClusMax- Num. For this situation, vector{M} is filled by centroids and location randomly, where the rest of empty elements in vector are represented as do not care elements ‘#’ sign (Alia et al. 2011; Saha and Bandyopadhyay 2009). Figure 3 represents the mate list content in FMA during iteration time.

When the Mate list M vector is produced, the number of cluster and centers location will be determine based on the produced new vector. If Mate list M is less than the Clust- MinNum minimum number of clusters, the new vector M will be rejected. Else, the new vector M will be accepted and the objective function is computed using Eq. (1). Then, the new M vector will be compared with the worst Mate list M vector. If the new one is better, it will be included in the Mate list M vectors, while the worst M is excluded.

5.4 Segmentation

All the steps of AUTO-FCM-FMA are repeated until reached to the maximum number of iterations (NI) is reached. Finally, the solution M vector has the best fitness value and it is selected and considered as the number of clusters and initial cluster centers’ values for FCM. In this case, the cluster centers’ values will change until the distance between the pixels in the same cluster (inter-cluster) reaches the minimum value, and the number of clusters will

Mt+1 Movements x_i^t⁺¹ x_i^t x_i^t⁻¹ x^t_i⁻² x_i^t⁻³ x_i^t⁻⁴ x_i^t⁻⁵

…

x_i^t⁻ⁿ Solutions Centroid1

# #

Centroid1 Centroid1

# #

… Centroid n

Fig. 3 Mate list content in AUTO-FCM-FMA

(8)

be updated until the distance between the clusters centroids (intra-cluster) reaches the maximum value. The Pseudo-code and flowchart of AUTO-FCM-FMA Algorithm is represented in Figs. 4 and 5.

6 Experiments and results

The present section details the evaluation of the proposed AUTO-FCM-FMA solution. This section is divided into five parts whereby the first part involves the parameters setting of AUTO-FCM-FMA, while the second part relates to the quality measurement (Pakhira et al. 2004) cluster validity (PBMF) index. The third part of the proposed AUTO- FCM-FMA involves the segmentation of a set of normal brain images and MSLs brain images, which are simulated brain MRI images obtained from Brain Web (BainWeb 2016). These simulated images are full 3D data volumes which were developed using three sequences namely T1WI,

T2WI, and PD, from several different values of slice thick- ness, noise levels, and non-uniform intensity levels (intensity non-homogeneity). The fourth part comprises experiments involving synthetic images produced by an automatic image generating tool known as SIGT (Salman et al. 2005), while the fifth part of the proposed AUTO-FCM-FMA involves segmentation on a set of natural images.

6.1 AUTO‑FCM‑FMA parameters setting

In order to obtain the best outcomes from any optimization algorithm, suitable parameters have to be selected; these parameters are important in the algorithm’s performance and accuracy. In view of that, the values of parameters of AUTO-FCM-FMA include: amount of attraction (β), light observation coefficient (γ), randomization parameter (α), number of firefly in search space (n), the weighted exponent m, maximum clusters number (ClusMaxNum), minimum clusters number (ClustMinNum), the length of mate

Figure 4: Automatic Fuzzy C-Means based Firefly Mate Algorithm (AUTO-FCM-FMA) 1 Initialize the population of fireflies (N) and generate solutions Xi, i=1, 2,…, N;

2 Set the absorption coefficient (γ) 3 Initialize {M} as mate list.

4 Initialize ClusMaxNum and ClustMinNum 5 Set number of clusters ClustNo using Eq 9

6 Compute Fuzzy c-means objective function of each firefly using Eq 1 7 t =0;

8 While (t < MaxGeneration) do

9 For i= 1 to N do

10 For j= 1 to Ndo

11 iff(xj)< f(xi)and (i mate j) ∉∉ {M} then o

t g n i d r o c c A 2

1 Eq. 8 Move firefly Xi toward X j;

13 Update mate list {M} /* Number of Cluster and centroids value

14 End if

[ p x e a i v r e c n a t s i d h ti w s e ir a v s s e n e v it c a rt t A 5

1 −r2]

g n i s u e u l a v p i h s r e b m e m e h t e t u p m o C 6

1 Eq 2

g n i s u n o it u l o s w e n r o f s s e n ti f f o e u l a v e h t e t u p m o C 7

1 Eq 1;

e t u p m o C 8

1 ClustNo using Eq 9

19 end

20 end

21 Rank the fireflies and find the current best

22 t++;

23 end;

24 Rank the best number of cluster and centroids values

25 Do the segmentation image by the best number of cluster and centroids value for FCM.

Fig. 4 The pseudo code of AUTO-FCM-FMA

(9)

AUTO-FCM-FMA Initialization

Population of fireflies (N), Mate list {M}, ClustNo, ClusMaxNum, ClustMinNum, number of iterations (NI)

Generate initial solutions Xi, i=1, 2,…, N;

Compute the light intensity Ii at xi is determined by f(xi) Eq 1.

Set light absorption coefficient (γ)

For i= 1 to N do For j= 1 to Ndo

iff(xj)< f(xi)and (i mate j) ∉∉ {M} then

According to Eq. 8 Move firefly Xi toward X j;

Update mate list {M} /* Number of Cluster and centroids value

End if

Attractiveness varies with distance r viaexp[−r2]

Compute the membership value using Eq 2

Compute the value of fitness for new solution using Eq 1;

Compute ClustNo using Eq 9 End End

Rank the fireflies and find the current best NI< MaxGeneration

NI=NI+1

The best number of cluster and centroids values obtained by FMA

Set the number of cluster and centroids values generated by FMA as input of FCM algorithm

Execution of FCM algorithm

ε

<

− _old

new v

v ^t^new⁼^v^t^old⁻¹

Stop AUTO-FCM-FMA

No Yes

Quality and Quantity outcomes measurements Cluster validity index (PBMF), Minkowski Score (MS) and obtained number of clusters (OC)

Image segmented by AUTO-FCM-FMA

Fig. 5 Semantic representation of the AUTO-FCM-FMA proposed Method

(10)

list {M} and number of iterations(NI). The values of these parameters are set based on other state-of-the-art methods in all experiments. Accordingly, based on the parameters, analyses are obtained from studies of MRI brain segmentation via hybrid firefly search algorithm (FFCM) (W. Alo- moush et al. 2014a, b), Dynamic fuzzy c-mean based firefly photinus search algorithm for MRI brain tumor image segmentation (DCFPA) (W. Alomoush and Omar 2015) and a hybrid harmony search algorithm for MRI brain segmentation (DCHS) (Alia et al. 2011) with the values of β = 1, γ = 1, α = 0.2, n = 40 and NI = 30,000.

Secondly, each image in these experiments has different range of number of clusters. In normal simulated brain images, ten regions of brain tissues including background, CSF, grey matter, white matter, fat, muscle/skin, skin, skull, glial matter and connective tissues, were presented, representing the value of parameter of maximum clusters number (ClusMaxNum) = 10. In multiple sclerosis lesions (abnormal brain images), 11 regions of brain were presented. The regions are the same as in the normal case, except for the addition of MS lesion, representing the value of parameter of maximum clusters number (ClusMaxNum) = 11. In this regard, the minimum number of clusters (ClustMinNum) = 2 for both of normal and abnormal simulated brain images.

Also in the experiment of synthetic and Natural images, the range of cluster numbers is between ClustMinNum = 2 and ClusMaxNum = 10. Finally the values of parameter length of mate list {M} in all experiments is equal to the values of parameter maximum clusters number (ClusMaxNum). Then, the algorithm was developed and run using Matlab 2010 software, and a powerful Intel Core 3 Duo processor based computer with 2 GHz processing speed and 4 GB RAM was used to execute the calculations.

6.2 Cluster validity index

In this paper, the validity index used is PBMF. It is a special version of fuzzy PBM-index (Pakhira et al. 2004), and it is used to measure the quality of the resulting clustering.

PBMF is a developed index that exhibits a good comparison between computational concern and efficacy. The PBMF index function consists of three parameters and can be intro- duced as in Eq. (10) below:

where: c is the number of clusters

and;

(10) PBMF(c) =

(1 c×E₁

E₂ ×D_c )p

(11) E_c=

∑c j=1

∑n i=1

u^m_ij‖

‖‖x_i−v_j‖

‖‖

The total number of data points is represented by n for the given dataset. Meanwhile, clusters configurations were con- trolled using the variable p which was set at 2. Further, m is the fuzziness weighting exponent; E1 is a constant factor for a particular dataset which is used to prevent the index value from approaching zero and its value is dataset dependent;

Dc measures the maximum difference between two clusters for each possible couple of clusters; EC measures the minimum distances between elements in at the same cluster (i.e., compactness).

Table 1 illustrates the outcomes for comparative results (PBMF) between AUTO-FCM-FMA and Fuzzy Cluster- ing based Firefly Algorithm FCM-FA. As can be observed, the name of images is in the first column, while the second and third columns display the average (AVG) and standard deviation (SD) for PBMF index, which are obtained from the AUTO-FCM-FMA and FCM-FA FMA to find the best number of clusters. The outcomes based PBMF index are appeared the most MRI images show considerable improve- ment when applied the AUTO-FCM-FMA algorithm. In Table 2, the best values are the maximum values and they are presented in bold. As can be viewed in the Table 2, the outcomes values by AUTO-FCM-FMA are superior to those by FCM-FA.

6.3 Simulated brain data

In this section the AUTO-FCM-FMA algorithm is used to segment a set of normal brain images and MSLs brain images. These images are simulated brain MRI images obtained from Brain Web. The quantization index namely Minkowski Score (MS) is used as measurement of accuracy rate (Alia et al. 2011; W. Alomoush et al. 2014a, b; Saha and Bandyopadhyay 2009), which is determined by evaluating (12) D_c=max

il ‖

‖v_i−v_l‖

‖

Table 2 Comparative results (PBMF) between AUTO-FCM-FMA and FCM-FA

Images AUTO-FCM-FMA FCM-FA

AVG SD± AVG SD±

Volume(1_24) 69831342 2063 67043749 2516

Volume(8_4) 77265290 4902 74696567 4990

Volume(205_3) 71800095 7235 70428267 7790

Z-1 89693420 1812 71892044 1694

Z-36 85515601 4699 78888625 4398

Z-108 87481405 9702 83643996 8847

Z-144 6604651 6602 6241319 6529

Z1-MSL 11283646 4960 8167285 3712

Z5-MSL 7518302 7155 7025565 7116

(11)

the percentage match between the image ground obtained by expert and the segmented image by the approaches used in this research and stat-of-arts. The measurement of accuracy is Minkowski Score (MS) represented in Eq. (13).

The ground truth image is represented as T and the segmented image is assigned as S. The n11 is represented, when segment image S and ground truth image T have the elements in the same cluster. The n01 indicates pixel elements in segment image S with the same cluster. Here, the n10 indicates pixel elements in ground truth image T with the same cluster, where T and S represent the partitioning matrix for the images of ground truth and segmented image by AUTO-FCM-FMA, respectively. Meanwhile, the optimal value of MS is minimum value.

(13) MS(T, S) =

√

n₀₁+n₁₀ n₁₁+n₁₀

6.3.1 A full automatic normal simulated brain images segmentation

In this part, a group of seven images comprising MRI images of normal brain, was obtained. Table 3 accordingly illustrates a group of images in plane (Z) and lists their actual number of clusters which was generated by expert (marked as # AC). Also, Table 3 shows the number of clusters obtained by AUTO-FCM-FMA algorithm (marked as # OC) and the clusters’ corresponding segmentation accuracy rate (MS).

Table 3 shows the ability of AUTO-FCM-FMA in finding the optimum number of clusters for the majority group of normal MRI images plan (z1, z2, z3, z36, z72 and z144).

The rest of the MRI images plan (z108) is able to determine the near optimal number of clusters. Also, the MS accuracy rate proves that AUTO-FCM-FMA is able to perform effec- tive segmentation. Accordingly, Figs. 6a, 7a and 8a illustrate the original MRI images of normal brain obtained from Brain Web as T1WIs in z36, z72, and z144 planes, respectively. Further, Figs. 6b, 7b and 8b show the ground truth MRI of normal brain T1WIs in z36, z72, and z144 planes, respectively. On the other hand, Figs. 6c, 7c and 8c show

Table 3 Normal simulated MRI

images result Z plane #AC AUTO-FCM-FMA FCM-FA DCHS FVGAPS

#OC MS #OC MS #OC MS #OC MS

1 6 6 0.36 6 0.47 6 0.45 9 0.69

2 6 6 0.37 8 0.53 6 0.47 9 0.62

3 6 6 0.33 8 0.47 6 0.47 8 0.59

36 9 9 0.54 8 0.72 9 0.83 8 0.84

72 10 10 0.55 9 0.78 9 0.75 8 0.59

108 9 8 0.57 8 0.73 8 0.74 9 0.52

144 9 9 0.40 6 0.81 9 0.79 6 0.33

Fig. 6 a Original normal MRI T1 image in z1, b ground truth normal MRI T1 image in z1, c normal MRI T1 image in z1 segmented by DCFA, d normal MRI T1 image in z1 segmented by AUTO-FCM-FMA

(12)

the corresponding segmented images generated using FCM- FA. As for Figs. 6d, 7d and 8d, they show the outcomes of segmented images by the approach of AUTO-FCM-FMA.

Also, the performance of AUTO-FCM-FMA was compared against that of other related works. Table 3 accordingly illustrates the outcome obtained by DCHS (Alia et al.

2011) and FVGAPS clustering based algorithms (Saha and Bandyopadhyay 2007a, 2009) which were reported in Saha and Bandyopadhyay (2007a, 2009). Specifically, DCHS and FVGAPS are fully automatic segmentation algorithms (dynamic clustering) which are able to find the number of clusters with their corresponding center locations.

The dynamic clustering based harmony search (DHCS) and genetic based FVGAPS algorithms are coupled with index of point symmetry. Consequently, Table 3 details the number of clusters identified by AUTO-FCM-FMA, FCM-FA, DCHS and FVGAPS along with the associated

MS rates. Here, a closer look at the outcomes validates the fact that AUTO-FCM-FMA is far superior at conducting the clusters number. In fact, AUTO-FCM-FMA algorithm was able to obtain the optimal cluster number in images group of planes z1, z2, z3, z36, z144. On the other hand, FCM-FA algorithm was only able to determine the optimal number of clusters of a single image, z1. The accuracy rates (MS) for images (z1, z2, z3, z36, z72) are much better for AUTO- FCM-FMA as compared to those of DCHS, FVGAPS and FCM-FA algorithms.

For the rest of the images z108 and z144, the MS rates are comparable. This owes to the similar intensity level of the brain tissue images. However, this causes problem to all automated image segmentation approaches which solely rely upon intensity variations. In contrast, manual segmentation which is performed by medical experts (ground truth images) has more than one source

(13)

of information (e.g. human knowledge) rather than the intensity level. Therefore, this affects the matching test (MS) rate for the algorithms.

The values of objective function during iterations time are very important as they demonstrate the behavior of proposed algorithm outcomes and other related works.

Figure 9 shows objective function values during segmentation process of image z144 using AUTO-FCM-FMA and FCM-FA algorithms.

Also, the following graph depicts the obtained results in Table 4, and as we can see the proposed algorithm presents the lowest matching rate, which shows how much the

proposed algorithm is better than other algorithms, including DCHS, FCM_FA, and FVGAPS (Fig. 10).

To evaluate the overall performance of the proposed algorithm, a Non-parametric test, called Friedman test was applied at α = 5% significance level. This test was used to evaluate the overall performance of the proposed algorithm in contrast to other algorithms.

To establish comparative assessment, we conducted Friedman statistical test was conducted based on the mean results of Tables 3, 5, 7 and 9. The results presented in Tables 4, 7, 10 and 13 confirm that the new proposed algorithm outperforms all other algorithms because it provides the highest ranking. The results in Friedman test reflect the overall performance of each algorithm, whereby and the smaller the result the higher the rank. Based on that proposed algorithm has the highest rank as shown in Tables 4, 6, 8 and 10.

6.3.2 Fully‑automatic abnormal simulated brain images segmentation

In this part, a collection of ten images derived from MSLs affected brain tissue was utilized. Accordingly, the full segmentation approaches were used to segment the abnormal simulated brain images.

Table 5 illustrates the group of images and also lists their cluster numbers from ground truth image (marked as

# AC). In addition, the table illustrates the number of clusters which was obtained automatically using the algorithms of AUTO-FCM-FMA (marked as # OC) and their corresponding matching rate (MS). The table demonstrates the ability of AUTO-FCM-FMA in determining the number of clusters for all the groups of images studied. Also, the MS accuracy rate proves that AUTO-FCM-FMA is able to out- perform the segmentation outcomes. Figures 11a, 12a and 13a illustrate the MRI MSLs brain T1WIs in planes z1, z5 and z10, respectively. Meanwhile, the Figs. 11b, 12b and 13b show ground truth MSLs MRI T1 images in z1, z5 and z10 planes, respectively, the segmented images determined

Fig. 9 Objective function values of auto-FCM-FMA and FCM-FA clustering algorithms with image Z plane 144

Table 4 Friedman test results for normal simulated MRI images result

Algorithms Mean rank AUTO-FCM-FM 1.2857

FCM-FA 3.0714

DCHS 2.7857

FVGAPS 2.8571

Fig. 10 Box–Whisker graph to present the results from Table 3

(14)

Table 5 MSLs simulated

normal MRI images results Z plane #AC AUTO-FCM-

FMA DCHS FVGA FVGAPS FCM-FA

#OC MS #OC MS #OC MS #OC MS #OC MS

1 6 6 0.39 6 0.39 2 1.21 10 0.58 6 0.52

2 6 6 0.39 6 0.5 2 1.2 10 0.58 6 0.52

3 6 6 0.49 6 0.47 2 1.19 7 0.71 7 0.78

4 6 6 0.31 6 0.47 5 0.69 5 0.67 5 0.48

5 6 6 0.34 6 0.47 2 1.18 8 0.62 5 0.49

6 6 6 0.32 6 0.47 2 1.18 3 0.71 5 0.50

7 6 6 0.33 6 0.48 2 1.17 8 0.7 7 0.57

8 6 6 0.35 6 0.49 2 1.16 9 0.71 9 0.58

9 6 6 0.40 6 0.49 2 1.16 9 0.68 9 0.59

10 9 9 0.47 9 0.74 2 1.17 9 0.65 9 0.63

Fig. 11 a Original MSLs MRI T1 image in z1; b ground truth MSLs MRI T1 image in z1; c MSLs MRI T1 image in z1 segmented by FCM-FA;

d MSLs MRI T1 image in z1 segmented by AUTO-FCM-FMA

Fig. 12 a Original MSLs MRI T1 image in z5; b ground truth MSLs MRI T1 image in z; c MSLs MRI T1 image in z5 segmented by FCM-FA;

d MSLs MRI T1 image in z1 segmented by AUTO-FCM-FMA

(15)

by the FCM-FA show in Figs. 11c, 12c and 13c. Whereas the images segmented by AUTO-FCM-FMA are show in Figs. 10d, 11d and 12d.

In order to compare the performances of other state- of-the-art algorithms with that of AUTO-FCM-FMA and FCM-FA, the same experiments using DCHS, FVGAPS and FVGA were used. Table 5 lists the number of clusters identified by AUTO-FCM-FMA, FCM-FA, DCHS and FVGAPS along with their corresponding MS rates. Also, the number of clusters determined by FVGA for the same set of images and their corresponding MS rates are shown in the same table. From these results, it can be seen clearly that AUTO-FCM-FMA is far superior in determining the number of clusters than other approaches. To be more specific, FVGAPS was unsuccessful in finding the optimal number of clusters for all the images, except for the image z10. The poorest performance was demonstrated by FVGA as it was unable to find the optimal number for even a single image.

Meanwhile, the corresponding MS rates demonstrate better performance of the newly developed AUTO-FCM-FMA algorithm.

Figure 14 shows objective function values during segmentation process of image z plan 5 using AUTO-FCM- FMA and FCM-FA algorithms.

The following graph presents the obtained results in Table 6. It also shows how that the proposed algorithm pro- vided the best results against the other algorithms, including DCHS, FVGA, FVGAPS, and FCM-FA.

6.4 Synthetic images

This section details how the performance of AUTO-FCM- FMA and FCM-FA algorithms was tested for segmentation of synthetic images as illustrated in Fig. 16. The synthetic images were produced by using an automatic image

generating tool known as SIGT. This tool serves as a bench- marking method and has been tailored for the purpose of verifying and comparing the performances of different unsupervised image classification algorithms. The tool is very versatile and can easily be programmed to generate different synthetic images with different characteristics like variable

Fig. 13 a Original MSLs MRI T1 image in z10; b ground truth MSLs MRI T1 image in z10; c MSLs MRI T1 image in z10 segmented by FCM- FA; d MSLs MRI T1 image in z1 segmented by AUTO-FCM-FMA

Fig. 14 Objective function values of auto-FCM-FMA and FCM-FA clustering algorithms with image Z plane5

Table 6 Friedman test results for MSLs simulated normal MRI images results

Algorithms Mean rank AUTO-FCM-FM 1.1500

DCHS 2.0500

FVGA 5.0000

FVGAPS 3.8000

FCM-FA 3.0000

(16)

image size, number of regions, different image depths, and varied histogram characteristics. This experiment uses a set of 15 synthetic images with differing characteristics generated using this tool.

The 15 images were carefully generated and selected in order to demonstrate the robustness of AUTO-FCM-FMA in two ways. Firstly, these images had different associated number of known clusters with different degrees of complexities (i.e., they consisted of well-separated clusters, overlapping clusters, or a combination of both types). These varying complexities are evident from their histograms as shown in Fig. 15. Therefore, successful differentiation of these images into the correct number of clusters as well as

clustering them is a very difficult task. Secondly, the performance of AUTO-FCM-FMA was compared with that of other established clustering algorithms of DCPSO, SOM and UFA as in Omran et al. (2006).

The segmentation experiments with the 15 developed images were performed using AUTO-FCM-FMA and FCM-FA in the same way as detailed by Omran et al.

(2006). Accordingly, Table 7 gives evidence that AUTO- FCM-FMA outperforms FCM-FA and DCPSO in 8 datasets (synthetic images 3, 4, 5, 6, 7, 10, 12, and 14). In 7 other datasets (synthetic images: 1, 2, 8, 9, 11, 13, and 15), AUTO-FCM-FMA performs at the same level as DCPSO. On the other hand, AUTO-FCM-FMA is superior

Fig. 15 Box–Whisker graph to present the results from Table 5

Table 7 Comparison of AUTO- FCM-FMA and FCM-FA with DCPSO, SOM and UFA of over synthetic images

Image #AC Optimal no. of clusters Mean ± SD

AUTO-FCM-FMA FCM-FA DCPOS SOM UFA

1 2 2 ± 0 2 ± 0 2 ± 0 2 20

2 3 3 ± 0 3 ± 0 3 ± 0 3 20

3 3 3 ± 0.90 2 ± 0 2 ± 0 6 20

4 3 2.93 ± 0.42 6.2 ± 0.8 5.15 ± 0.357 10 20

5 4 4 ± 0.980 6 ± 0.2 5 ± 0 7 20

6 10 10 ± 0 7 ± 0.9 7.2 ± 0.872 9 20

7 6 6 ± 0 8 ± 0.45 7.9 ± 0.995 9 20

8 4 4.86 ± 0.19 5 ± 0.3 5 ± 0 4 20

9 7 5.5 ± 1 5 ± 0 5 ± 0 13 20

10 4 4.16 ± 0.50 7 ± 0.5 7 ± 0 9 20

11 10 10 ± 0 10 ± 0 10 ± 0 10 20

12 5 5.5 ± 0.86 6 ± 0.9 7.2 ± 0.4 6 20

13 5 5 ± 0. 2 5 ± 0.8 5 ± 0 5 20

14 7 7 ± 0.63 5 ± 0.53 5 ± 0 7 20

15 5 5 ± 0 5 ± 0 5 ± 0 5 20

(17)

in performance to SOM algorithm in 7 cases (synthetic images numbered 3, 4, 5, 6, 7, 10, and 12), while AUTO- FCM-FMA works just as well as SOM in the rest. Table 7 also shows one of the limitations of UFA, i.e., it over

fits the data as reported in Omran et al. (2006). This is because the algorithm is considerably influenced by certain parameters setting. From this table, therefore, it can be

Fig. 16 Synthetic images with their corresponding histograms. a Image #1 (2 clusters) with its histogram, b Image #2 (3 clusters) with its histogram, c Image #3 (3 clusters) with its histogram