THEORETICAL ADVANCES

Improved clustering criterion for image clustering with artificialbee colony algorithm

Celal Ozturk • Emrah Hancer • Dervis Karaboga

Received: 23 November 2012 / Accepted: 27 January 2014

� Springer-Verlag London 2014

Abstract In this paper, a new objective function is pro-

posed for image clustering and is applied with the artificial

bee colony (ABC) algorithm, the particle swarm optimi-

zation algorithm and the genetic algorithm. The perfor-

mance of the proposed objective function is tested on seven

benchmark images by comparing it with the three well-

known objective functions in the literature and the

K-means algorithm in terms of separateness and com-

pactness which are the main criterions of the clustering

problem. Moreover, the Davies–Bouldin Index and the XB

Index are also employed to compare the quality of the

proposed objective function with the other objective

functions. The simulated results show that the ABC-based

image clustering method with the improved objective

function obtains well-distinguished clusters.

Keywords Image clustering � Artificial bee colony

algorithm � Genetic algorithms � K-means �Particle swarm optimization � Validity indexes

1 Introduction

It is known that a large-scale data set needs to be stored on

account of high development in data acquisition, in com-

puter performances, in technological devices and on

account of requirement for detail information. For instance,

highly improved scanners are used to minimize data loss in

order to accomplish successful raster to vector conversa-

tion of topographic maps [1, 2]. Therefore, the requirement

of storing data plays important role in many areas. How-

ever, some pre-processing implementations are needed to

distinguish significant information in this type of data sets

before processing and analysing. Data mining, one of the

young and interdisciplinary fields of computer science,

simply aims to extract knowledge from data to overcome

difficulties of analysing data. Data mining involves various

approaches like clustering, data summarization, classifica-

tion, finding dependency networks, regression, analysing

changes, detecting anomalies, etc. [3]. This paper concen-

trates on clustering, specifically on image clustering

approach, which is considered as an unsupervised learning

problem.

Clustering is the task of separating groups or clusters

based on some similar measures like distance, intervals

within multidimensional data [4]. Data members have

maximum similarities among all data members assigned to

one group or one cluster. A cluster is generally distin-

guished by its cluster centre, namely centroid [5]. Due to its

crucial role, clustering is used in many applications,

including machine learning, image analysis, pattern rec-

ognition and statistical data analysis.

Most clustering algorithms are based on two popular

techniques: hierarchical and partitional clustering. In hier-

archical clustering algorithms, the output is ‘‘a tree show-

ing a sequence of clustering, each of which is a partition of

the data set [6]’’. Hierarchical clustering algorithms can be

categorized into two groups: divisive or agglomerative.

Agglomerative hierarchical clustering algorithms (bottom

up approaches) start with each data member as a separate

cluster and the most similar clusters are merged into larger

C. Ozturk � E. Hancer (&) � D. Karaboga

Engineering Faculty, Computer Engineering Department,

Erciyes University, Kayseri, Turkey

e-mail: emrahhancer@erciyes.edu.tr

C. Ozturk

e-mail: celal@erciyes.edu.tr

D. Karaboga

e-mail: karaboga@erciyes.edu.tr

123

Pattern Anal Applic

DOI 10.1007/s10044-014-0365-y

clusters. The most known agglomerative algorithms are

single link and complete link. Single link algorithms are

based on merging two clusters which have the smallest

maximum pairwise distance. On the other hand, complete

link algorithms are based on merging two clusters which

have the smallest minimum pairwise distance [7]. Hierar-

chical algorithms have the following advantages: the

number of clusters does not need to be specified and they

are not very dependent to initial conditions [8]. However,

they are not dynamic, i.e. after a data point assigned to a

cluster, its cluster cannot be modified, and the lack of

information about global size and shape can cause over-

lapping clusters [9]. The other disadvantage of hierarchical

algorithms is the time complexity (O(Np2 log Np)), i.e. time

computation is expensive.

Partitional clustering attempts to decompose the data set

into a set of disjoint clusters by generating various partitions.

The algorithms evaluate the partitions by some criterions

which are tried to be minimized by assigning clusters to

peaks in the probability density function or the global

structure [10]. Typically the global criteria involve mini-

mizing a measure of dissimilarity in the samples within each

cluster while maximizing the dissimilarity of different

clusters. Since a criterion (e.g. square error function) is tried

to be minimized, the partitional clustering algorithms can be

regarded as optimization problems [11]. In the usage of the

partitional clustering algorithms, the disadvantages of the

hierarchical algorithms are the advantages of the partitional

clustering algorithms, and vice versa [11, 12].

The most widely used partitional clustering algorithm

developed by MacQueen is known as K-means [13]. In

K-means, K centroids are first initialized with the help of

random number generator or data information. After that,

each pattern is assigned to its corresponding closest centroid

based on similarity measure, Euclidean distance. Finally, the

positions of centroids are recalculated. These steps are

repeated until convergence. K-means algorithm is simple to

implement and is convenient for large data sets on account of

its time complexity (O(Np)). However, the algorithm is data

dependent and depends on the initial conditions, which cause

the algorithm to converge to the nearest local optimum from

the starting position of the search [14, 15]. The fuzzy version

of K-means algorithm, developed by Dunn [16] and

improved by Bezdek [17], is the Fuzzy C-means (FCM)

algorithm. FCM aims to minimize fuzzy version of the least-

square error criterion and is less influenced by existence of

uncertainty in the data set [18]. In this way, FCM performs

better than K-means. However, it might also converge to

local minima and it is highly sensitive to noise and imaging

artefacts as in K-means [19]. Researchers have developed

new versions of FCM by modifying equations based on

objective functions such as possibilistic C-means (PCM)

[20], robust fuzzy C-means (RFCM) [21], generalized FCM

(GFCM) [22], etc. K-means also has some other versions:

K-harmonic means (KHM) [23], spatial kernel-based KHM

(SKKHM) [24], etc.

Sorting out local minima problem of clustering process

is another demanded subject for researchers. To overcome

local minima problem, evolutionary algorithms, particu-

larly swarm-based algorithms, were employed in recent

years. Particle swarm optimization (PSO), proposed by

Kennedy and Eberhart [25, 26], is a population-based

approach inspired by social behaviours of bird flicks and

fish swarms [27]. It has been widely applied to image

clustering. The first PSO-based image clustering algorithm

is proposed by Omran et al. [28]. The other PSO-based

image clustering studies of Omran et al. can be seen in [29–

31]. Ant colony optimization (ACO) [32] is based on the

behaviours of ants searching a path between their colony

and a source of food, also applied to image clustering [3,

33]. The other applied algorithm to the clustering problems

is the differential evolution (DE) algorithm with the mod-

ifications [34–37]. Artificial bee colony (ABC) algorithm,

introduced by Karaboga [38], is known to be a new swarm

intelligence-based evolutionary algorithm. The model of

the ABC algorithm is inspired by the intelligent behaviours

of honeybee swarm and it has been applied to many

problems such as optimization of numerical problems [39],

data clustering [14], neural networks training for pattern

recognition [40], image analysis [41], and wireless sensor

network deployment [42] and routing [43]. The ABC

algorithm was also applied to the image clustering problem

in Refs. [44, 45]. However, in these works [44, 45] the

objective fitness function was only based on quantization

error which was not sufficient to satisfy the simple prop-

erties of the clustering process: compactness and sepa-

rateness. For instance, the maximum distance between the

centroids is generally very closest in Ref. [44] which

cannot be acceptable for the separateness property. On the

other hand, Hancer et al. [46] applied the ABC and PSO

algorithms to the clustering problem with the widely used

objective fitness function, which was also used in the

studies of Omran et al. [29–31]. Yet, the ABC and PSO

algorithms with this objective function could not obtain

well-minimized quantization error. Therefore, a new

objective function is proposed to obtain well-separated and

well-compacted clusters in this study. The effectiveness of

the proposed objective function is verified on seven

benchmark images (airplane, house, Lena, Morro Bay,

MRI, 42049 and 48025).

The rest of the paper organized as follows: Sect. 2

describes the ABC algorithm briefly; Sect. 3 describes the

clustering problem, objective functions and validity

indexes; Sect. 4 summarizes the results and comparisons of

the ABC, K-means, PSO and GA algorithms; finally, Sect.

5 concludes the paper.

Pattern Anal Applic

123

2 Artificial bee colony algorithm

As in the colony of real honey bees, the colony of real

artificial bees consists of three kinds of bees: employed

bees, onlooker bees and scout bees. A bee associated with a

food resource is called employed bee, a bee waiting on

dance area to get information concerning food sources is

called onlooker bee, and a bee performing random search is

called scout. The area of food sources denotes the possible

solutions of optimization problem and the nectar amount of

a food source represents the quality of relevant solution. It is

assumed that only one employed bee is associated with one

food source. Therefore, the number of employed bees is

equal to the number of food sources which are equal to half

of the colony size. After randomly generation of food

sources, each employed bee searches a new position in the

neighborhood of its associated food source. The employed

bee keeps its new food source position when the nec-

tar amount of new food source is more abundant than the

position of old one (greedy selection). After all employed

bees complete their search process, they share the infor-

mation belonging to nectar amount, direction and distance

of food sources with onlooker bees by waggle dance in

dancing area. Each onlooker bee selects a food source

probabilistically and searches for a new food source and

evaluates its quality. Then, greedy selection is applied same

as in the implementation of employed bees. After that, the

position of each food source is checked whether it is

improved or not. If a position of food source cannot be

improved for a number of predetermined trials (limit), that

food source is assigned as abandoned and a scout randomly

produces a new food source position for the abandoned one.

After the food production, scout bee becomes an employed

bee. The flow chart of ABC algorithm can be seen in Fig. 1

and the main steps of ABC algorithm [47] can be defined as

1. The production of initial food sources (x0is) and

evaluation of population by Eq. (1)

xij ¼ xminj þ rand 0; 1ð Þ xmax

j � xminj

� �ð1Þ

where i ¼ 1; . . .; SN and SN is the number of food sources;

j ¼ 1; . . .;D; and D is the number of parameters; xjmin is the

minimum and xjmax is the maximum values of parameter j.

2. Produce new food source in the neighbourhood of xi by

Eq. (2)

mij ¼ xij þ /ij xij � xkj

� �ð2Þ

where j and k are randomly chosen parameters and

neighbourhoods, respectively, and /ij is a random number

within [-1,1].

3. Compute the fitness value of mi, if mi is better than xi,

employed bee memorizes mi and removes xi from its

memory.Fig. 1 Flow chart of ABC algorithm

Pattern Anal Applic

123

4. The probability of pi is calculated by Eqs. (3) and (4).

pi ¼fitnessiPSNj¼1 fitnessi

ð3Þ

fitnessi ¼1

1þ fiti

; fiti� 0

1þ abs fitið Þ; fiti\0

8<: ð4Þ

where fiti is the fitness value of source xi.

5. Select food sources probabilistically depending on pi

values. After that, produce new sources by Eq. (2) and

update the position of food sources using the greedy

selection approach as in step (3).

6. Check whether there exists an abandoned source or

not; if so, it is replaced with a new randomly generated

food source by the scout using Eq. (1).

7. Repeat stages (2)–(6) until the maximum number of

cycles.

3 The clustering problem using the ABC algorithm

Patterns are the physical definition of an object used by

clustering and they are distinguished from one another

through their attributes known as features. Given a data set

Z ¼ z1; z2; . . .; zp; . . .; zN

� �; zp is a pattern with D-dimen-

sional feature space, and N is the number of patterns in Z. To

partition Z into K number of clusters as C ¼C1;C2; . . .;Ckf g via a crisp clustering process, the following

properties are need to be considered:

• Each cluster should have at least one pattern,

Ci 6¼ ;8i ¼ 1; 2; . . .;K

• Each pattern should be assigned to a cluster,

[Kk¼1

Ck ¼ Z

• Each pattern should be assigned to only one cluster,

Ci \ Cj ¼ ; where i 6¼ j

Having these properties, clustering is done based on

some similarity measures. Distance measurement is

generally used for evaluating similarities between

patterns, which constructs fundamental part of an

objective function (fit) in a clustering problem. In this

paper, four objective functions are used in experiments. The

first two of them (fit1 [30] and fit2 [48]) are very popular in

the literature, the third one (Pfit) is the proposed objective

function in this study by improving fit1 and fit2 together, and

the last one (CSfit [49]) is employed as an objective

function, recently used in automatic clustering problem

[34]. The objective functions can be described as follows:

1. The first applied objective function: fit1 [30] applies

the three main criterions, comprising error measure,

intra-cluster distance and inter-cluster separation in a

weighted manner;

fit1 xi; Zð Þ ¼ w1dmax Z; xið Þ þ w2 zmax � dmin Z; xið Þð Þ þ w3Je

ð5Þ

where Z is the image data set; xi is the ith food source and

defined by xi ¼ ðmi;1;mi;2; . . .;mi;k. . .;mi;KÞ, mi,k is the

centre of kth cluster of the ith food source; zmax is the

maximum value of data set (for a s-bit image, zmax = 2s - 1);

and w1, w2 and w3 are user-defined parameters having

constant values. In addition, dmax(Z, xi), known as intra-

cluster distance, is the maximum average distance of a

cluster calculated by distances of all pixels to their

respective centre. To satisfy well-collected pixel groups,

intra-cluster distance [29] should be minimized, defined by

Eq. (6);

dmax Z; xið Þ ¼ maxX8zp2Ci;k

d zp;mi;k

� �ni;k

8<:

9=; ð6Þ

where ni;k is the number of pixels in cluster, Ci,k and

d(zp, mi,k) is the Euclidean distance between zp and mi,k.

The second component of fit1 is the inter-cluster separation

which is known as the minimum average Euclidean

distance (dmin) [50] between any pairs of clusters. The

dmin criterion, shown as follows, should be maximized to

obtain well-separated clusters;

dmin Z; xið Þ ¼ min d mi;j;mi;k

� �� �; j 6¼ k: ð7Þ

And the last component of fit1, known as the quantization

error (Je) and defined by Eq. (8), is used to express the

general quality of a clustering algorithm;

Je ¼PK

k¼1

P8zp2Ck

dðzp;mkÞ=nk

K: ð8Þ

2. The second applied objective function: The parameters

of fit1 are needed to be selected empirically for every

image, which is not flexible on account of high time

consumption. To overcome this problem, fit2 [48] was

improved;

fit2 xi; Zð Þ ¼ dmax Z; xið Þ þ Je;i

dmin Z; xið Þ : ð9Þ

3. The proposed objective function: Although fit2requires no user-defined parameters, the performance

of fit2 is almost similar to fit1. To tackle the drawbacks

of the above-mentioned objective functions, a new

fitness function (Pfit) is improved. Contrary to the

others, Pfit considers the mean square error (MSE) and

the quantization error in an efficient way. Moreover,

Pattern Anal Applic

123

Pfit aims to keep the error rate in minor rates while

maximizing the dmax and minimizing the dmin

criterions at the same time. The equation of Pfit is as

follows:

Pfit xi; Zð Þ ¼ Je �dmax Z; xið Þdmin Z; xið Þ

� dmax Z; xið Þ þ zmax � dmin Z; xið Þ þMSEð Þð10Þ

where MSE is defined by Eq. (11);

MSE ¼ 1

N

XK

k¼1

X8zp2Ck

dðzp;mkÞ2 ð11Þ

where N is the total number of patterns in a data set.

4. The last applied objective function: The CS measurement

[49] looks for clusters having minimum within cluster-

scatter and maximum between-cluster separation;

CS ¼

PKk¼1

1nk

Pzp2Ck

maxzq2Ck

dðzp; zqÞPK

k¼1 minj2K;j 6¼k

dðmk;mjÞð12Þ

CSfit xi; Zð Þ ¼ CS: ð13Þ

In addition to the Je, dmax and dmin criterions, two

clustering validity indexes such as the Davies–Bouldin

index (DBI) [51] and the XB index (XBI) [52] are chosen

to measure and compare the clustering qualities of the

above-mentioned objective functions:

1. The Davies–Bouldin Index: The Davies–Bouldin

index (DBI) [51], one of the most well-known

internal indexes, is a function of the ratio of the

sum of within cluster-scatter to between-cluster

separation. The scatter within the ith cluster (Si) and

between the ith and jth cluster (Rij) are defined by

Eqs. (14) and (15);

Si ¼1

ni

Xxj2Ci

d xj;mi

� �2 ð14Þ

Rij ¼Si þ Sj

dðmi;mjÞ2; i 6¼ j: ð15Þ

Last, DBI can be described as

DBI ¼ 1

K

XK

k¼1

Rk ð16Þ

where Rk ¼ maxj¼1;2;...;KRij and i ¼ 1; 2; . . .;K:2. The XB Index: The sum squares within (SSW) clusters

is a measurement of compactness and the sum squares

between (SSB) clusters is a measurement criterion of

separateness. The equations of SSW and SSB are given

as follows:

SSW ¼XK

k¼1

X8zp2Ck

dðzp;mkÞ2 ð17Þ

SSB ¼XK

k¼1

nkdðmk;MÞ2: ð18Þ

According to Zhao et al. [52], the SSW in the ratio of

SSW/SSB plays more important role in clustering and the

indexes such as Calinski–Harabasz [53], Ball–Hubert [54],

Hartigan [55], etc. based on the SSW and SSB criterions

have such drawback that they can be monotonously

increased or decreased. Therefore, they proposed XB

index (XBI) [52];

XBI ¼ K � SSW

SSB: ð19Þ

The fast implementation of image clustering algorithm

has two stages: histogram process of an image and creation

of data structure including grey levels, related clusters and

total pixel numbers. When an empty cluster is created for a

xi solution, this solution is generated again until receiving a

non-empty cluster set. Figure 2 briefly explains the basic

stages of the ABC-based clustering algorithm.

4 Experimental results

Experimental results can be considered in six stages: (1)

comparing the ABC, PSO and GA algorithms according to

the obtained fitness values, (2) analysing the objective

functions such as fit1, fit2, Pfit and CSfit within the ABC,

PSO and GA algorithms in terms of the separateness and

compactness properties (Je, dmax and dmin) as in [29–31], (3)

Fig. 2 The main steps of ABC-based image clustering algorithm

Pattern Anal Applic

123

measuring the qualities of the objective functions within the

ABC, PSO and GA algorithms according to their DBI and

XBI values, (4) determining whether there exist significant

differences between the results of the ABC, PSO and GA

algorithms over Pfit, (5) comparing the results obtained by

ABC, PSO and GA algorithms with the results of K-means

algorithm, and (6) judging whether there exist significance

differences between the employed fitness functions and the

proposed fitness function over the above-mentioned crite-

rions. The experimental results are summarized through the

Fig. 3 a 1. Airplane Image; 2. Clustered by ABC with Pfit, b 1.

House Image; 2. Clustered by ABC with Pfit, c 1. Lena Image; 2.

Clustered by ABC with Pfit, d 1. Morro Bay Image; 2. Clustered by

ABC with Pfit, e 1. MRI Image; 2. Clustered by ABC with pfit, f 1.

42049 Image; 2. Clustered by ABC with Pfit, g 1. 48025 Image; 2.

Clustered by ABC with Pfit

Pattern Anal Applic

123

following images: the five of them are airplane, house,

Lena, Morro Bay and MRI shown in Fig. 3a.1, b.1, c.1, d.1

and e.1, respectively; and the last two images are 42049

(3f.1) and 48025 (3g.1) taken from Berkeley segmentation

data set and benchmark images [56].

To evaluate the results in a statistically significant way,

Friedman test [57], which is a non-parametric statistical

test and based on ranks between the independent samples

(blocks or rows), is employed;

FR ¼12

rcðcþ 1ÞXc

j¼1

R2j � 3r cþ 1ð Þ ð20Þ

where Rj is the square of the sum of ranks for each group j,

r is the number of blocks (rows) and c is the number of

groups. Through the FR value, P value is obtained. When

the obtained P value is smaller than 0.05, the difference

between the independent samples is significant. Otherwise,

there exists no such a significant difference between the

compared results.

In the experiments, the following parameters are used

for the ABC, PSO, GA and K-means algorithms: the colony

size is taken 30 and the value of limit is 75 for ABC; and the

number of particles (chromosomes) is taken 30 for PSO and

GA. The maximum number of iterations (cycles) for ABC,

PSO and GA is set to 250, i.e. the number of evaluations is

7,500. Therefore, the number of iterations of the K-means

algorithm is selected as 7,500. The number of clusters is

taken 5 in all cases. The parameters of the PSO algorithm are

chosen as used in [30] (c1 = 1.49, c2 = 1.49, wstart = 0.72,

wend = 0.4 and Vmax = 255) and the parameters of GA such

as crossover rate and mutation rate are chosen as 0.8 and 0.2,

respectively. The weights of fit1 are chosen as w1 = 0.4,

w2 = 0.2, w3 = 0.4 for each image.

The simulation runs are studied on the computer with

29CPU 2.4 Ghz Dual (6 cores), 4 9 8 GB Ram, and it is

observed that the simulation times of the algorithms are

very close to each other. The results of the mean values and

standard deviations of the considered criterions are pre-

sented over the 30 simulation runs in the tables. Further-

more, the best mean values for the compared algorithms

and the statistically significant results are written in bold,

and standard deviation values of produced results are

reported in parenthesis in the tables. The obtained fitness

values of the ABC, PSO and GA algorithms are summa-

rized for each objective function in the Table 1. In addi-

tion, the separateness and compactness properties (Je, dmax,

dmin, DBI and XBI) of each objective function are dem-

onstrated in the Table 2 for the images: airplane (Fig. 3a),

house (Fig. 3b), Lena (Fig. 3c), Morro Bay (Fig. 3d), MRI

(Fig. 3e), 42049 (Fig. 3f) and 48025 (Fig. 3g) in terms of

ABC,PSO and GA algorithms, respectively. The obtained

results of the K-means algorithm for each criterion with the

results of the algorithms such as ABC, PSO and GA based Ta

ble

1T

he

resu

lts

of

fitn

ess

val

ues

for

the

ob

ject

ive

fun

ctio

ns

(Fit

CS

,F

it1,

Fit

2an

dP

Fit

)

Imag

esF

itC

SF

it1

Fit

2P

Fit

AB

CP

SO

GA

AB

CP

SO

GA

AB

CP

SO

GA

AB

CP

SO

GA

Air

pla

ne

0.4

99

(0.0

01)

0.5

01

(0.0

03

)

0.5

20

(0.0

21

)

46

.15

9(0

.07

9)

46

.17

5

(0.2

73

)

47

.32

8

(1.7

41

)

0.4

98

(0.0

05)

0.5

00

(0.0

13

)

0.5

70

(0.0

59

)

79

0.6

18

(8.7

89

)

77

7.7

67

(8.6

76)

92

4.5

96

(13

3.1

09

)

Ho

use

0.6

05

(0.0

01)

0.6

06

(0.0

03

)

0.6

19

(0.0

20

)

50

.79

5(0

.15

5)

51

.11

6

(0.8

41

)

51

.73

2

(0.9

93

)

0.4

90

(0.0

08)

0.5

05

(0.0

47

)

0.5

45

(0.0

53

)

1,0

70

.08

(11.8

40

)

1,0

86

.34

(13

7.1

40

)

1,1

92

.32

4

(21

2.8

51

)

Len

a0

.57

8

(0.0

01

)

0.5

77

(0.0

01)

0.5

96

(0.0

20

)

50

.06

5(0

.08

8)

50

.08

7

(0.2

36

)

50

.66

6

(0.6

16

)

0.5

03

(0.0

06)

0.5

03

(0.0

11

)

0.5

44

(0.0

44

)

91

9.3

07

(13

.98

7)

89

9.3

65

(11

.46

4)

1,0

25

.06

5

(10

0.0

02

)

Mo

rro

Bay

0.6

04

(0.0

01)

0.6

05

(0.0

03

)

0.6

19

(0.0

20

)

50

.09

7(0

.10

4)

50

.16

8

(0.1

86

)

50

.74

1

(0.5

22

)

0.4

54

(0.0

06)

0.4

58

(0.0

08

)

0.4

83

(0.0

23

)

88

6.0

52

(9. 8

25)

89

8.8

66

(64

.64

8)

99

8.2

67

(13

0.4

35

)

MR

I0

.49

0(0

.00

1)

0.4

96

(0.0

09

)

0.5

25

(0.0

41

)

40

.49

0

(0.0

73)

40.4

48

(0.0

96

)

40

.92

9

(0.4

66

)

0.4

46

(0.0

03)

0.4

48

(0.0

05

)

0.4

90

(0.0

55

)

52

3.1

45

(7.2

82)

52

3.7

51

(16

.78

6)

59

5.7

81

(61

.69

5)

42

04

90

.52

5

(0.0

01

)

0.5

23

(0.0

01)

0.5

96

(0.0

11

)

50

.78

9

(0.0

58

)

50.7

68

(0.1

70

)

52

.08

9

(1.0

65

)

0.4

87

(0.0

05)

0.4

88

(0.0

11

)

0.6

03

(0.0

92

)

67

3.3

70

(7.7

85

)

66

7.9

83

(9.1

10

)

71

4.8

39

(45

.81

8)

48

02

50

.58

6

(0.0

01

)

0.5

85

(0.0

01

)

0. 5

61

(0.0

04

)

51

.06

4(0

.10

4)

51

.07

4

(0.2

77

)

51

.67

5

(0.5

62

)

0.5

05

(0.0

07)

0.5

06

(0.0

16

)

0.5

31

(0.0

29

)

1,1

92

.57

(16.6

02

)

1,2

02

.15

(24

.53

6)

1,3

32

.13

(20

2.3

9)

Pattern Anal Applic

123

Ta

ble

2T

he

resu

lts

of

com

par

iso

ncr

iter

ion

s(J

e,

dm

ax,

dm

in,

DB

Ian

dX

BI)

Fit

s.A

BC

PS

OG

A

J ed

max

dm

inD

BI

XB

IJ e

dm

ax

dm

inD

BI

XB

IJ e

dm

ax

dm

inD

BI

XB

I

Air

pla

ne

Fit

CS

19.7

01

(0.1

54)

36.9

82

(0.9

12)

40.6

02

(1.4

87)

0.2

401

(0.0

117)

0.4

951

(0.0

223)

19.7

60

(0.2

96)

34.7

68

(1.9

03)

51.8

83

(3.1

99)

0.2

276

(0.0

069)

0.4

699

(0.0

503)

21.1

09

(1.6

87)

36.6

44

(6.0

05)

49.1

80

(9.0

33)

0.3

254

(0.1

452)

0.8

844

(0.4

591)

Fit

19.8

89

(0.2

04)

11.0

16

(0.4

38)

42.0

13

(1.2

79)

0.1

590

(0.0

021)

0.2

826

(0.0

340)

9.9

13

(0.3

56)

11.2

12

(0.4

93)

42.3

73

(1.2

86)

0.1

592

(0.0

045)

0.2

792

(0.0

377)

10.4

84

(1.0

17)

12.0

51

(1.3

63)

40.6

89

(2.7

39)

0.1

739

(0.0

222)

0.3

101

(0.0

624)

Fit

29.9

34

(0.3

62)

11.0

77

(0.5

72)

42.1

05

(1.6

51)

0.1

586

(0.0

023)

0.2

828

(0.0

349)

10.0

81

(0.3

38)

11.5

13

(0.4

71)

43.1

91

(1.3

49)

0.1

597

(0.0

036)

0.2

770

(0.0

512)

11.3

36

(1.4

46)

12.9

48

(2.0

81)

42.5

46

(3.4

00)

0.1

799

(0.0

227)

0.3

287

(0.0

627)

PF

it9

.710

(0. 0

46

)

10.7

34

(0.3

22

)

40.5

31

(1.2

34)

0.1

569

(0.0

008)

0.2

533

(0.0

111

)

9.7

23

(0.1

69)

10.8

35

(0.3

61)

41.3

29

(1.2

02)

0.1

571

(0.0

012)

0.2

566

(0.0

153)

10.0

83

(0.5

02)

11.4

35

(0.6

59)

39.6

97

(2.3

02)

0.1

66

(0.0

126)

0.2

535

(0.0

236)

House

Fit

CS

18.2

88

(0.1

46)

26.4

47

(1.1

73)

56.0

87

(2.2

14)

0.1

973

(0.0

047)

0.4

808

(0.0

118)

18.0

94

(0.3

77)

27.4

71

(2.0

92)

54.3

77

(4.9

98)

0.1

943

(0.0

048)

0.5

464

(0.0

966)

18.0

81

(0.9

79)

28.0

84

(4.5

47)

49.5

75

(7.9

69)

0.2

174

(0.0

389)

0.5

912

(0.2

031)

Fit

111.3

47

(0.2

91)

11.9

42

(0.4

16)

47.6

01

(1.3

53)

0.1

470

(0.0

033)

0.2

998

(0.0

086)

11.6

04

(0.7

76)

12.4

36

(0.9

76)

47.4

98

(2.7

09)

0.1

636

(0.0

298)

0.3

179

(0.0

429)

11.6

36

(0.7

83)

13.0

24

(0.8

87)

45.6

61

(4.3

51)

0.1

78

(0.0

039)

0.3

322

(0.0

553)

Fit

211.3

19

(0.3

06)

11.9

63

(0.3

97)

47.4

94

(1.3

77)

0.1

468

(0.0

029)

0.2

992

(0.0

089)

11.8

42

(0.9

05)

12.7

14

(1.3

91)

48.6

59

(1.8

29)

0.1

610

(0.0

260)

0.3

226

(0.0

514)

12.4

10

(1.1

40)

13.9

19

(1.7

22)

48.3

49

(3.2

31)

0.1

71

(0.0

028)

0.3

520

(0.0

478)

PF

it10.8

18

(0.1

09)

11.4

27

(0.0

99

)

44.2

63

(1.6

11)

0.1

452

(0.0

027

)

0.2

759

(0.0

083)

10.8

21

(0.1

48)

11.4

83

(0.4

68)

44.6

80

(1.9

86)

0.1

506

(0.0

229)

0.2

831

(0.0

279)

11.0

21

(0.3

65)

11.9

50

(0.7

38)

43.6

47

(3.1

36)

0.1

57

(0.0

272)

0.3

161

(0.0

625)

Len

a

Fit

CS

18.2

51

(0.1

18)

26.5

51

(0.7

97)

59.8

92

(1.2

23)

0.2

059

(0.0

034)

0.5

023

(0.0

046)

18.2

21

(0.1

06)

26.4

71

(0.7

71)

60.4

29

(1.1

83)

0.2

046

(0.0

034)

0.5

003

(0.0

046)

18.8

79

(1.5

79)

30.3

69

(10.3

1)

52.3

87

(9.3

32)

0.2

831

(0.2

452)

0.6

689

(0.2

935)

Fit

110.2

09

(0.3

84)

11.1

20

(0.4

72)

42.3

33

(1.5

98)

0.1

584

(0.0

042)

0.2

474

(0.0

112)

10.6

08

(0.4

62)

11.4

96

(0.6

09)

43.7

73

(1.1

35)

0.1

628

(0.0

049)

0.2

589

(0.0

192)

10.8

09

(0.9

73)

11.9

15

(1.3

41)

42.1

17

(2.3

72)

0.1

66

(0.0

121)

0.2

852

(0.0

386)

Fit

210.1

39

(0.3

03)

11.0

96

(0.3

62)

41.9

87

(1.1

99)

0.1

586

(0.0

058)

0.2

416

(0.0

083)

10.5

37

(0.5

33)

11.6

05

(0.7

07)

44.0

03

(1.5

99)

0.1

625

(0.0

057)

0.2

621

(0.0

174)

11.2

41

(0.9

16)

12.3

92

(1.2

85)

43.8

93

(3.1

24)

0.1

69

(0.0

094)

0.3

062

(0.0

490)

PF

it9.7

36

(0.1

08)

10.6

76

(0.1

33

)

39.9

55

(0.9

67)

0.1

548

(0.0

044

)

0.2

297

(0.0

047)

9.7

93

(0.1

98)

10.6

91

(0.2

24)

40.5

64

(0.5

03)

0.1

554

(0.0

045)

0.2

361

(0.0

083)

10.2

11

(0.4

89)

11.0

46

(0.5

83)

40.2

23

(1.9

45)

0.1

629

(0.0

073)

0.2

585

(0.0

287)

Morr

o

Bay

Fit

CS

17.6

11

(0.0

63)

31.9

67

(0.0

75)

57.4

13

(0.7

24)

0.1

797

(0.0

016)

0.3

268

(0.0

013)

17.6

59

(0.0

31)

31.9

81

(0.0

01)

58.3

32

(0.3

47

)

0.1

804

(0.0

009)

0.3

264

(0.0

007)

17.2

87

(0.9

42)

30.2

56

(3.2

96)

50.0

74

(7.9

47)

0.2

071

(0.0

508)

0.3

478

(0.0

535)

Fit

110,1

92

(0.4

79)

12,2

85

(0.7

66)

49,4

63

(1,8

28)

0.1

37

(0.0

042)

0.1

626

(0.0

112)

10,4

79

(0.4

55)

13,1

96

(0.9

15)

51,5

09

(2,2

82)

0.1

422

(0.0

055)

0.1

839

(0.0

128)

10,9

49

(1.0

61)

13,5

51

(1.0

07)

50,2

94

(2,9

74)

0.1

468

(0.0

098)

0.1

853

(0.0

171)

Fit

210,0

10

(0.3

19)

12,0

84

(0.7

54)

48,6

88

(1.7

18)

0.1

361

(0.0

039)

0.1

566

(0.0

057)

10,6

23

(0.6

53)

13,2

98

(0.9

29)

52,1

96

(2,5

12)

0.1

423

(0.0

057)

0.1

759

(0.0

150)

11,2

14

(1.0

10)

13,8

24

(0.9

83)

51,8

08

(2,5

66)

0.1

480

(0.0

088)

0.1

822

(0.0

193)

PF

it9.6

96

(0.1

37)

11.7

10

(0.1

67

)

46,9

55

(0.4

66)

0.1

352

(0.0

013

)

0.1

536

(0.0

019)

9,7

63

(0.2

48)

12.0

9

(0.8

73)

47,9

72

(0.9

99)

0.1

360

(0.0

047)

0.1

559

(0.0

093)

10,0

04

(0.2

85)

12,4

51

(1.1

73)

47,1

32

(1.7

33)

0.1

412

(0.0

100)

0.1

673

(0.0

207)

MR

I

Fit

CS

18.0

94

(0.0

29)

47.9

93

(0.0

38)

52.1

74

(0.3

78)

0.1

98

(0.0

01)

0.3

12

(0.0

02)

18.1

63

(0.4

60)

47.7

74

(2.0

35)

50.3

44

(2.9

80)

0.2

009

(0.0

087)

0.3

178

(0.0

126)

18.4

64

(1.5

72)

48.7

53

(9.2

38)

46.2

21

(10.0

7)

0.2

513

(0.1

187)

0.3

912

(0.1

324)

Fit

18.4

69

(0.2

55)

11.4

76

(0.3

95)

44.4

38

(1.3

36)

0.1

322

(0.0

019)

0.2

653

(0.0

069)

8.6

45

(0.3

61)

11.6

51

(0.4

69)

45.3

52

(1.5

96)

0.1

316

(0.0

029)

0.2

668

(0.0

065)

8.7

99

(0.4

73)

11.6

21

(0.7

04)

43.1

96

(2.7

53)

0.1

410

(0.0

101)

0.2

712

(0.0

139)

Fit

28.3

31

(0.3

03)

11.2

43

(0.5

11)

43.5

09

(1.8

46)

0.1

321

(0.0

018)

0.2

654

(0.0

060)

8.6

07

(0.3

17)

11.6

44

(0.4

92)

45.1

65

(1.4

30)

0.1

329

(0.0

037)

0.2

667

(0.0

058)

9.0

26

(0.9

03)

12.1

74

(1.7

81)

43.3

16

(2.5

49)

0.1

465

(0.0

175)

0.2

715

(0.0

382)

Pattern Anal Applic

123

Ta

ble

2co

nti

nu

ed

Fit

s.A

BC

PS

OG

A

J ed

max

dm

inD

BI

XB

IJ e

dm

ax

dm

inD

BI

XB

IJ e

dm

ax

dm

inD

BI

XB

I

PF

it8

.049

(0.1

02

)

10.6

01

(0.3

73)

41.4

83

(1.1

94)

0.1

338

(0.0

024)

0.2

648

(0.0

051

)

8.1

06

(0.1

75)

10.6

93

(0.2

82)

41.5

78

(0.9

22)

0.1

327

(0.0

014)

0.2

661

(0.0

096)

8.4

84

(0.3

27)

11.3

23

(0.6

19)

40.3

69

(2.7

63)

0.1

442

(0.0

096)

0.2

685

(0.0

386)

42049

Fit

CS

18.7

52

(0.0

99)

27.1

19

(0.4

09)

62.9

44

(0.6

26)

0.2

213

(0.0

010)

0.3

351

(0.0

103)

18.6

85

(0.0

38)

27.1

05

(0.2

28)

63.6

12

(0.4

43

)

0.2

199

(0.0

004)

0.3

363

(0.0

036)

17.7

63

(0.6

51)

27.9

17

(5.1

61)

51.6

88

(6.5

63)

0.2

133

(0.0

379)

0.5

284

(0.1

378)

Fit

18.9

44

(0.2

82)

10.3

25

(0.2

63)

39.5

96

(0.9

60)

0.1

554

(0.0

026)

0.1

349

(0.0

136)

9.3

75

(0.5

20)

10.5

55

(0.4

36)

41.0

18

(1.3

11)

0.1

574

(0.0

057)

0.1

514

(0.0

277)

10.9

20

(0.8

78)

12.3

71

(1.6

07)

41.1

34

(2.2

40)

0.1

799

(0.0

191)

0.2

074

(0.0

352)

Fit

28.8

92

(0.2

48)

10.2

01

(0.3

27)

39.1

42

(1.1

58)

0.1

569

(0.0

036)

0.1

327

(0.0

059)

9.4

88

(0.6

31)

10.6

62

(0.5

63)

41.2

51

(1.7

98)

0.1

582

(0.0

055)

0.1

517

(0.0

240)

11.6

67

(1.7

76)

13.8

11

(3.0

85)

42.2

31

(3.8

90)

0.1

912

(0.0

310)

0.2

425

(0.0

765)

PF

it8

.699

(0.0

53

)

9.9

00

(0.1

58

)

38.0

24

(0.7

66)

0.1

558

(0.0

025)

0.1

281

(0.0

025

)

8.7

44

(0.1

39)

10.0

26

(0.2

54)

38.9

30

(0.8

63)

0.1

558

(0.0

022)

0.1

286

(0.0

036)

9.1

58

(0.7

18)

10.5

21

(0.9

90)

37.5

43

(2.8

78)

0.1

655

(0.0

173)

0.1

486

(0.0

314)

48025

Fit

CS

17.7

81

(0.2

27)

31.3

01

(2.2

18)

51.3

20

(3.7

47)

0.1

987

(0.0

101)

0.5

423

(0.0

400)

17.9

82

(0.3

71)

29.6

26

(3.6

68)

54.6

18

(6.0

64)

0.2

011

(0.0

102)

0.5

318

(0.0

384)

17.7

31

(0.7

95)

28.0

26

(5.2

14)

50.5

84

(6.9

59)

0.2

180

(0.0

511)

0.5

299

(0.1

454)

Fit

111.1

49

(0.2

45)

13.3

62

(0.6

60)

48.6

99

(1.9

88)

0.1

537

(0.0

041)

0.2

935

(0.0

101)

11.5

15

(0.3

99)

13.9

31

(0.5

55)

50.6

72

(1.9

99)

0.1

575

(0.0

038)

0.3

024

(0.0

129)

11.6

78

(0.9

50)

13.2

51

(1.0

07)

46.4

87

(4.3

18)

0.1

612

(0.0

142)

0.2

991

(0.0

311)

Fit

211.2

01

(0.3

89)

13.4

49

(0.9

61)

48.8

36

(2.6

77)

0.1

542

(0.0

055)

0.2

971

(0.0

138)

11.7

78

(0.8

64)

14.1

17

(0.9

24)

51.2

47

(2.5

15)

0.1

578

(0.0

059)

0.3

089

(0.0

183)

12.1

75

(1.0

93)

14.1

04

(1.4

87)

49.5

00

(4.4

06)

0.1

639

(0.0

118)

0.3

180

(0.0

332)

PF

it10.8

95

(0.1

03)

12.2

76

(0.3

02)

44.7

16

(1.9

14)

0.1

504

(0.0

023)

0.2

781

(0.0

100)

10.8

72

(0.0

98

)

12.0

79

(0.2

72

)

41.9

17

(2.7

50)

0.1

552

(0.0

052)

0.2

699

(0.0

141)

11.2

77

(0.7

01)

13.0

82

(1.2

74)

45.7

98

(4.5

53)

0.1

594

(0.0

089)

0.2

871

(0.0

294)

Pattern Anal Applic

123

Table 3 The comparison

criterions values of ABC, PSO,

GA and K-means algorithms

ABC (Pfit) PSO (Pfit) GA (Pfit) K-means

Airplane Je 9.710 (0.046) 9.723 (0.169) 10.083 (0.502) 9.854 (0.121)

dmax 10.734 (0.322) 10.835 (0.361) 11.435 (0.659) 15.434 (0.577)

dmin 40.531 (1.234) 41.329 (1.202) 39.697 (2.302) 21.167 (3.573)

DBI 0.1569 (0.0008) 0.1571 (0.0012) 0.166 (0.0126) 0.233 (0.001)

XBI 0.2533 (0.0111) 0.2566 (0.0153) 0.2535 (0.0236) 0.2048 (0.0003)

House Je 10.818 (0.109) 10.821 (0.148) 11.021 (0.365) 10.330 (0.061)

dmax 11.427 (0.099) 11.483 (0.468) 11.950 (0.738) 12.545 (0.783)

dmin 44.263 (1.611) 44.680 (1.986) 43.647 (3.136) 31.833 (1.053)

DBI 0.1452 (0.0027) 0.1506 (0.0229) 0.157 (0.0272) 0.168 (0.001)

XBI 0.2759 (0.0083) 0.2831 (0.0279) 0.3161 (0.0625) 0.2454 (0.0001)

Lena Je 9.736 (0.108) 9.793 (0.198) 10.211 (0.489) 9.430 (0.036)

dmax 10.676 (0.133) 10.691 (0.224) 11.046 (0.583) 11.712 (0.261)

dmin 39.955 (0.967) 40.564 (0.503) 40.223 (1.945) 34.833 (0.461)

DBI 0.1548 (0.0044) 0.1554 (0.0045) 0.1629 (0.0073) 0.154 (0.001)

XBI 0.2297 (0.0047) 0.2361 (0.0083) 0.2585 (0.0287) 0.2266 (5.9E-5)

MorroBay Je 9.696 (0.137) 9,763 (0.248) 10,004 (0.285) 9,998 (0.076)

dmax 11.710 (0.167) 12,09 (0.873) 12,451 (1.173) 14,032 (1.263)

dmin 46,955 (0.466) 47,972 (0.999) 47,132 (1.733) 41,565 (3,863)

DBI 0.1352 (0.0013) 0.1360 (0.0047) 0.1412 (0.0100) 0.148 (0.005)

XBI 0.1536 (0.0019) 0.1559 (0.0093) 0.1673 (0.0207) 0.1666 (0.0071)

MRI Je 8.049 (0.102) 8.106 (0.175) 8.484 (0.327) 8,142 (0.056)

dmax 10.601 (0.373) 10.693 (0.282) 11.323 (0.619) 17,198 (0.280)

dmin 41.483 (1.194) 41.578 (0.922) 40.369 (2.763) 21,698 (0.492)

DBI 0.1338 (0.0024) 0.1327 (0.0014) 0.1442 (0.0096) 0.154 (0.001)

XBI 0.2648 (0.0051) 0.2661 (0.0096) 0.2685 (0.0386) 0.1704 (0.0001)

42049 Je 8.699 (0.053) 8.744 (0.139) 9.158 (0.718) 8.989 (0.244)

dmax 9.900 (0.158) 10.026 (0.254) 10.521 (0.990) 12.759 (1.470)

dmin 38.024 (0.766) 38.930 (0.863) 37.543 (2.878) 19.838 (7.271)

DBI 0.1558 (0.0025) 0.1558 (0.0022) 0.1655 (0.0173) 0.2088 (0.2088)

XBI 0.1281 (0.0025) 0.1286 (0.0036) 0.1486 (0.0314) 0.1298 (0.0089)

48025 Je 10.895 (0.103) 10.872 (0.098) 11.277 (0.701) 10.526 (0.021)

dmax 12.276 (0.302) 12.079 (0.272) 13.082 (1.274) 12.512 (0.232)

dmin 44.716 (1.914) 41.917 (2.750) 45.798 (4.553) 31.350 (0.249)

DBI 0.1504 (0.0023) 0.1552 (0.0052) 0.1594 (0.0089) 0.1683 (0.0012)

XBI 0.2781 (0.0100) 0.2699 (0.0141) 0.2871 (0.0294) 0.2440 (0.0002)

Table 4 The Friedman test

results of ABC, PSO and GA

over Pfit objective function

Images Je dmax DBI XBI

v2 p val v2 p val v2 p val v2 p val

Airplane 19.04 6.1e2005 23.15 9.4e2006 29.87 3.2e2007 1.67 0.4346

House 6.07 0.0482 40.78 1.3e2009 11.04 0.0033 5.60 0.0608

Lena 24.47 4.8e2006 3.58 0.1668 24.27 5.3e2006 29.60 3.7e2007

Morro Bay 21.07 2.6e2005 7.2 0.0273 2.07 0.3558 18.47 9.7e2005

MRI 27.80 9.1e2007 19.66 5.3e2005 5.34 0.0692 4.27 0.1184

42049 24.87 3.9e2006 11.37 0.0034 13.07 0.0015 20.60 3.3e2005

48025 5.4 0.0672 19.76 5.1e2005 16.47 0.0003 8.87 0.0119

Pattern Anal Applic

123

on Pfit are demonstrated in the Table 3. In addition, the

statistical significance of the proposed objective function

results over the algorithms: ABC, PSO and GA; summa-

rized in Table 4. Last, the statistical significance of the

employed and proposed objective functions on the evalu-

ation criterions through the ABC algorithm is presented in

Table 5.

The most important evaluation criterion for comparing

evolutionary algorithms is to analyse obtained fitness

values. As clearly seen from the Table 1, the ABC algo-

rithm is almost superior to the PSO and GA algorithms in

terms of minimizing the objective functions. Table 1 also

reveals that the ABC algorithm gets the minimum stan-

dard deviation values in most cases. Indeed, ABC out-

performs the others in minimizing the Je and dmax

criterions which can be indicated from the Table 2.

Although the PSO and GA algorithms perform better than

the ABC algorithm in maximizing the dmin criterion, the

Table 2 demonstrates that the ABC algorithm almost

outperforms the others in terms of minimizing the DB and

XBI values which are accepted as the well-known crite-

rions of measuring the quality of clustering process. That

makes the ABC algorithm is the most convenient algo-

rithm for clustering process. Therefore, this disadvantage

of the ABC algorithm in maximizing the dmin criterion can

be ignored. Moreover, the GA algorithm is always

arranged in the last position in minimizing the Je, dmax

criterions, objective functions, DBI and XBI values

against the ABC and PSO algorithms. It can be apparently

inferred from this situation that the GA algorithm is not

very suitable to the clustering process against the ABC

and PSO algorithms. In addition, the statistically signifi-

cant difference between the ABC and other algorithms

over Pfit in terms of the criterions except dmin can be

observed in Table 4, where the results clearly show that

almost in all cases ABC obtains much better performances

against the others. As for the K-means algorithm, the

performance of the K-means in minimizing XBI is suc-

cessful for almost all cases and in minimizing Je is suc-

cessful in some situations against the ABC and PSO

algorithms. However, the K-means algorithm actually

obtains the most awful performance in minimizing the

dmax criterion and in maximizing the dmin criterion. In

addition, the performance of the K-means algorithm in

minimizing the DBI values is also not sufficient when

compared with the other algorithms.

Considering the comparison of the objective functions,

it can be observed from the Table 2 that the proposed

objective function Pfit is almost superior to the other

objective functions in minimizing the Je and dmax criterions

and obtaining the best standard deviation values. Also, the

implementations of the ABC, PSO and GA algorithms with

Pfit almost get better results than the implementations of

the algorithms with the other objective functions. More-

over, it should be notified that the best performance using

Pfit is obtained by the ABC algorithm. In this way, the

ABC algorithm with the help of Pfit also gets advantages

versus the K-means algorithm in terms of the Je criterion.

Moreover, Pfit also performs better than fit1, fit2 and CSfit

in terms of obtaining the best DBI values except some

situations and Pfit gets the best XBI values in all cases

which can be extracted from the Table 2. As for CSfit, it

exhibits the best dmin values, yet the performance of this

objective function is not very sufficient for the other cri-

terions. The superiority of Pfit over the other criterions

through the ABC algorithm can be also inferred from

Table 5. It can therefore be concluded that Pfit is the most

convenient objective function to get well-separated and

well-compacted clusters despite its disadvantage in mini-

mizing the dmin criterion.

5 Conclusion

Clustering is a very comprehensive research area and has

been used in many applications. In this paper, the ABC-

based clustering algorithm with the improved function is

proposed, in which the mean square error (MSE) is effi-

ciently used. The algorithms such as ABC, PSO, GA and

K-means with the handled objective functions are com-

pared in terms of many criterions on seven benchmark

images. The results indicate that ABC-based clustering

Table 5 The Friedman test

results of objective functions

(FitCS, Fit1, Fit2 and Pfit)

through ABC

Images Je dmax DBI XBI

v2 p val v2 p val v2 p val v2 p val

Airplane 66.16 2.8e2014 55.92 4.4e2012 58.48 1.2e2012 66.04 3.0e2014

House 81 1.8e2017 75.96 2.2e2016 57.72 1.8e2012 81.04 1.8e2017

Lena 66.16 2.8e2014 69.96 4.3e2015 59.16 8.8e2012 75.28 3.1e2016

Morro Bay 66.64 2.3e2014 64.08 7.8e2014 54.00 1.1e2011 66.96 1.9e2014

MRI 71.56 1.9e2015 69.96 4.3e2015 56.92 2.6e2012 54.12 1.1e2011

42049 70.68 3.1e2015 69.96 4.3e2015 54.52 8.7e2012 58.28 1.2e2012

48025 66.64 2.2e2014 75.96 2.2e2016 59.16 8.9e2012 74.32 5.1e2016

Pattern Anal Applic

123

algorithm with the proposed objective function outper-

forms the others. Moreover, it is also indicated from the

results that all the criterions satisfying separateness and

compactness are needed to be handled together to obtain

well-separated clusters. In other words, one criterion is not

sufficient to satisfy well-separated and well-compacted

clusters. In conclusion, it can be suggested that the ABC-

based clustering algorithm with the proposed fitness func-

tion can be efficiently employed for clustering problems.

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