Modified seeker optimization algorithm for image segmentation by multilevel thresholding

Abstract

Image segmentation is among most often used techniques for image analysis and one standard way to do it is multilevel tresholding. The selection of optimum thresholds has remained a challenge in image segmentation. High computational cost and inefficiency of the conventional multilevel thresholding methods lead to the use of global search heuristics to find the optimal thresholds. The optimal thresholds are often determined by either minimizing or maximizing some criterion functions defined on images. In this paper, a new swarm intelligence algorithm, modified seeker optimization (MSO) algorithm, is presented for image segmentation by multilevel thresholding. This algorithm is used to maximize the Kapur's and Tsallis' objective functions. Experiments have been performed on four test images using various numbers of thresholds. The experimental results demonstrate that the proposed MSO algorithm can find multiple thresholds that are very close to the optimal ones determined by the exhaustive search method. Compared to the particle swarm optimization (PSO) algorithm, the MSO algorithm performs satisfactory in terms of solution quality, robustness and convergence.

Full text


Abstract— Image segmentation is among most often used
techniques for image analysis and one standard way to do it is
multilevel tresholding. The selection of optimum thresholds has
remained a challenge in image segmentation. High computational
cost and inefficiency of the conventional multilevel thresholding
methods lead to the use of global search heuristics to find the optimal
thresholds. The optimal thresholds are often determined by either
minimizing or maximizing some criterion functions defined on
images. In this paper, a new swarm intelligence algorithm, modified
seeker optimization (MSO) algorithm, is presented for image
segmentation by multilevel thresholding. This algorithm is used to
maximize the Kapur's and Tsallis' objective functions. Experiments
have been performed on four test images using various numbers of
thresholds. The experimental results demonstrate that the proposed
MSO algorithm can find multiple thresholds that are very close to the
optimal ones determined by the exhaustive search method. Compared
to the particle swarm optimization (PSO) algorithm, the MSO
algorithm performs satisfactory in terms of solution quality,
robustness and convergence.
Keywords—Maximum entropy thresholding, Image thresholding,
Multilevel tresholding, Seeker optimization algorithm, Swarm
intelligence
I. INTRODUCTION
MAGE segmentation is one of the most important operations
in image analysis and computer vision [1], [2], [3], [4], [5].
Thresholding is one of the simplest techniques for
performing image segmentation and it is very useful in
separating objects from background image, or discriminating
objects from objects that have distinct gray-levels.
Thresholding involves bi-level thresholding and multilevel
thresholding. For bi-level thresholding the main objective is to
determine one threshold which separates the pixels into two
groups, one including those pixels with gray levels above
certain threshold, the other including the rest. For multilevel
thresholding the main objective is to determine multiple
thresholds which divide pixels into several groups. The pixels
which belong to the same class have gray levels within a
specific range defined by several thresholds. The global
thresholding methods [6], belonging to parametric and
Manuscript received October 12, 2012. Revised April 29, 2013.
The work was supported by the Ministry of Science, Republic of Serbia,
Grant No. III-44006
M. Tuba is with the Faculty of Computer Science, Megatrend University,
Belgrade, Serbia (e-mail: tuba@ieee.org)
I. Brajevic is with the Faculty of Mathematics, University of Belgrade,
Serbia (e-mail: ivona.brajevic@googlemail.com)
nonparametric approaches, select thresholds by optimizing
(maximizing or minimizing) some criterion functions defined
from images.
The maximum entropy thresholding has been widely used in
determining the optimal thresholding in image segmentation
[7]. Of particular interest is an information theoretic approach
that is based on the concept of entropy introduced by Shannon.
The principle of entropy is to use uncertainty as a measure to
describe the information contained in a source. The maximum
entropy criterion for image thresholding was first proposed by
Pun, and later it was corrected and improved by Kapur [6].
Basically, the entropy thresholding considers an image
histogram as a probability distribution, and then selects as an
optimal threshold value that yields the maximum entropy.
More precisely, a best entropy thresholded image is the one
that preserves as much information as possible that is
contained in the original unthresholded image in terms of
Shannon's entropy. Recent developments of statistical
mechanics based on a concept of nonextensive entropy, also
called Tsallis entropy, have intensified the interest of
investigating a possible extension of Shannon’s entropy to
information theory [8], [9]. This interest appears mainly due to
similarities between Shannon’s and Boltzmann/Gibbs entropy
functions. The Tsallis entropy is a new proposal in order to
generalize the Boltzmann/Gibbs traditional entropy to
nonextensive physical systems.
Optimization techniques inspired by swarm intelligence
have become very popular during the last decade. Three of the
most successful examples of optimization techniques inspired
by swarm intelligence are ant colony optimization [10],
particle swarm optimization [11] and artificial bee colony
optimization [12], [13]. Basic versions of these algorithms are
later enhanced to improve performance in general or for some
class of the problems [13], [15], [16], [17], [18], [19], [20],
[21], [22], [23], [24], [25], [26]. Also, swarm intelligence
algorithms are successfully used for wide range of practical
problems. Many of these metaheuristics were adopted to
search for the multilevel thresholds [27], [28], [29]. Yin [30]
presented a new method that adopts the particle swarm
optimization to select the thresholds based on the minimum
cross-entropy. Horng applied the honey bee mating
optimization (HBMO), the artificial bee colony (ABC)
algorithm [31] and the firefly algorithm [32] to search for the
thresholds using the maximum entropy criterion. In [33] the
adaptation and comparison of six meta-heuristic algorithms:
genetic algorithm, particle swarm optimization, differential
evolution, ant colony, simulated annealing and tabu search
were presented. The experimental results have shown that the
Modified seeker optimization algorithm for
image segmentation by multilevel thresholding
Milan Tuba, and Ivona Brajevic
I
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370

genetic algorithm, the particle swarm optimization and the
differential evolution outperformed the other algorithms.
Seeker optimization algorithm (SOA) is a novel swarm
intelligence algorithm based on simulating the act of human
searching, which has been shown to be a promising candidate
of search algorithms for unconstrained function optimization
[34]. The SOA results for multimodal test functions were not
very satisfactory and in order to enhance its performance, the
modified seeker optimization algorithm named MSO was
proposed [35]. This paper proposes the Kapur and Tsallis
based MSO algorithm for solving multilevel thresholding
problem. The PSO algorithm is implemented for purposes of
comparison. Also, the exhaustive search method is conducted
for deriving the optimal solutions for comparison with the
results generated from PSO and MSO algorithms.
The rest of the paper is organized as follows. In the Section
2 the problem of the multilevel thresholding is formulated and
the objective functions treated are presented. In the Section 3
the SOA algorithm is briefly described. Section 4 presents our
proposed MSO algorithm for multilevel thresholding problem.
Section 5 gives comparative results of the implemented MSO
and PSO algorithms.
II. MULTILEVEL THRESHOLDING PROBLEM FORMULATION
Let there be L gray levels in a given image I having M pixels
and these gray levels are in the range
}1,...1,0{ L
. The
multilevel thresholding problem can be configured as a k-
dimensional optimization problem, for determination of k
optimal thresholds
],...,,[ 21 k
ttt
which optimizes an objective
function.
Several objective functions devoted to the thresholding have
been proposed in the literature [6]. Generally, these functions
are determined from the histogram of the image, denoted by
the SOA algorithm
)(ih
,
1...1,0  Li
, where
)(ih
represents
the number of pixels having the gray level i. The normalized
probability at level i is defined by the ratio
MihPi/)(
. One
of the most popular objective function is defined by Kapur.
The aim is to maximize the objective function:
kk HHHtttf  ...]),...,,([ 1021
(1)
where
0
1
00
0ln
1
w
P
w
P
Hi
t
i
i




,




1
0
0
1
t
i
i
Pw
,
1
1
1
1ln
2
1w
P
w
P
Hi
t
ti
i




,




1
1
2
1
t
ti
i
Pw
,
2
1
02
2ln
3
w
P
w
P
Hi
t
i
i




,




1
2
3
2
t
ti
i
Pw
,...
k
i
L
ti k
i
kw
P
w
P
H
k
ln
1




,



1L
ti
iK
k
Pw
The method similar to the maximum entropy sum method
of Kapur is Tsallis non-extensive entropy concept. The aim is
to maximize the objective function:
k
k
A
q
A
q
A
q
A
q
A
q
A
qk
SSSq
SSStttf


...}1(
...]),...,,([
10
10
21
(2)
where
1
)(1 1
0
1
0
0





q
p
p
S
t
i
q
A
i
A
q
,




1
0
1
0
t
i
i
App
1
)(1 1
2
1
1
1






q
p
p
S
t
ti
q
A
i
A
q
,




1
2
1
1
t
ti
i
App
1
)(1 1
3
2
2
2






q
p
p
S
t
ti
q
A
i
A
q
,




1
3
2
2
t
ti
i
App
,...
1
)(1 1






q
p
p
S
L
ti
q
A
i
A
qk
k
k
,



1L
ti
i
A
k
kpp
III. SEEKER OPTIMIZATION ALGORITHM
Seeker optimization algorithm (SOA) mimics the behaviour of
human search population based on their memory, experience,
uncertainty reasoning and communication with each other.
SOA is a population-based heuristic algorithm. The algorithm
operates on a set of solutions called search population. Each
individual of the population is called a seeker or agent. The
total population is equally categorized into K subpopulations
according to the indexes of the seekers (all the subpopulations
have the same size, shown as Fig. 1). All the seekers in the
same subpopulation constitute a neighborhood which
represents the social component for the social sharing of
information.
The main characteristic features of this algorithm are the
following:
1. The algorithm uses search direction and step length to
update the positions of seekers
2. The calculation of the search direction is based on a
compromise among egotistic behaviour, altruistic
behaviour and pro-activeness behaviour
3. Fuzzy reasoning is used to generate the step length because
the uncertain reasoning of human searching could be the
best described by natural linguistic variables and a simple
if-else control rule: If {objective function value is small}
(i.e., condition part), then {step length is short} (i.e., action
part)
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A search direction
)(tdij
and a step length
)(t
ij

are separa-
tely computed for each individual i on each dimension j at
each iteration iter, where
0)iter( 
ij

and
}1,0,1{)iter( 
ij
d
. At each iteration the position of each seek-
er is updated by:
)iter()iter()iter()1iter( ijijijij dxx 

(3)
where
DjSNi,...2,1;,...2,1 
(SN is the number of seekers).
Also, at each iteration, the current positions of the worst two
individuals of each subpopulation are exchanged for both of
the best one in each of the other two subpopulations, which is
called inter-subpopulation learning.
Fig.1: Relationship chart of population and subpopulation
Short pseudo–code of the SOA algorithm is given below:
1. Generating s positions uniformly and randomly in search
space;
2. cycle = 0;
3. Repeat
4. For i = 1 to s do
Computing
)iter(
i
d
;
Computing
)iter(
i

;
Updating each seeker’s position using Eq. (3);
5. End of For
6. Evaluating all the seekers and saving the historical best
position;
7. Implementing the inter-subpopulation learning operation;
8. cycle = cycle+1;
Until the termination criterion is satisfied
Since its invention, SOA has been applied to solve the other
kinds of problems beside numerical function optimization. In
[36], the application of the SOA to tuning the structures and
parameters of artificial neural networks is presented, while in
[37] SOA-based evolutionary method is proposed for digital
IIR filter design. Also, a new optimized model of proton
exchange membrane fuel cell (PEMFC) was proposed by using
SOA [38].
IV. MODIFIED SEEKER OPTIMIZATION ALGORITHM FOR
MULTILEVEL THRESHOLDING PROBLEM
SOA was analyzed with a challenging set of benchmark
problems for function optimization. The simulation results
showed that the proposed algorithm is a promising candidate
of swarm algorithms for numerical function optimization. For
multimodal test functions the results were not very satisfactory
because it was noticed that for this type of problems SOA may
be stuck at a local optimum. In order to enhance the
performance of SOA, the modified seeker optimization (MSO)
algorithm is developed [35]. MSO algorithm uses two search
equations for producing new population: search equation of
artificial bee colony (ABC) algorithm [18] and the search
equation of seeker optimization algorithm. Also, MSO
algorithm implements the modified inter-subpopulation
learning using the binomial crossover operator.
The proposed MSO algorithm based on maximum entropy
criterion tries to obtain this optimum K-dimensional vector
[
k
ttt ,...,,21
] which can maximize Eq.(1) in the case of
Kapur's method Eq.(2) in the case of Tsallis' method. The
details of the developed approach are introduced as follows.
Step 1. Initialize population
MSO algorithm generates a randomly distributed initial
population of SN solutions or seekers
i
t
(i = 1, 2, ..., SN ) with
K dimensions denoted by matrix T.
],...,[ 21 SN
tttT 
and
),...,( ,2,1, Kiiii tttt 
(4)
where
ij
t
is the jth component value that is restricted into
[0,…,L-1] and the
1
ijij tt
for all j. Each seeker
i
t
(i = 1, 2,
..., SN ) is generated by:
)()1,0( minmaxmin, ttrandtt ji 
(5)
where
min
t
and
max
t
are the minimum and the maximum gray
values in the image, the rand(0, 1) is a random number
between 0 and 1. In MSO algorithm, as in SOA, the total
population is categorized into N subpopulations according to
the indexes of the seekers. Each seeker
i
t
, beside of its
current position and its objective function value, has the
following attributes: the personal best position
ibest
p
so far
and the neighborhood best position
best
g
so far.
Step 2. Evaluate population
For each seeker
i
t
(i = 1, 2 ..., SN) evaluate the objective
function values by Eq.(1) or Eq.(2).
Step 3. Record the best solution
In this step, the best solution vector, i.e. the solution vector
with the highest objective function value is recorded.
Step 4. is repeated a fixed number of iterations. It consists of
three parts. The details of each part are described as follows.
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Part 1. Calculate new population
Perform an update process for each solution in the search
population using a randomly selected search equation. The
MSO included a new control parameter which is called
behaviour rate (BR) in order to select the search equation in
the following way: If a uniformly distributed real random
number between [0,1) is less then BR, the SOA search
equation is used, otherwise the search equation of ABC
algorithm is performed.
The variant of ABC search equation for producing a new
solution
i
v
, i

{ 1, 2, ..., SN} which is used in MSO
algorithm is:
otherwise
Rif
t
ttt
vj
ji
jkjiiji
ji
5.0
,
,
)(
,
,,,
,








(6)
where k is a randomly chosen index of a solution from the
subpopulation to which the ith seeker belongs, k has to be
different from i,
Kj ,...,2,1
,
i

is a uniformly distributed
real random number between [-1, 1) and
j
R
is a uniformly
random real number within [0, 1).
The SOA search solution equation uses search direction
ji
d,
and step length
ji,

for producing a new solution
i
v
, i

{ 1, 2, ..., SN} . It can be described by:
jijijiji dtv ,,,, 

,
Kj ,...2,1
(7)
A search direction
ji
d,
and a step length
ji,

are separately
computed for each individual i on each dimension j at each
iteration. The calculation of the search direction is based on a
compromise among egotistic behavior, altruistic behavior and
pro-activeness behavior. The egotistic behavior of each seeker
i
t
may be modeled by vector called egotistic direction
iego
d
by:
jijibestjiego tpd ,,, 
,
Kj ,...2,1
(8)
The altruistic behavior of each seeker
i
t
may be modeled by
vector called altruistic direction
ialt
d
by:
jijbestjialt tgd ,,, 
,
Kj ,...2,1
(9)
where
best
g
represents the neighbourhood best position so far.
The pro-active behavior of each seeker
i
t
may be modeled by
vector called pro-activeness direction
ipro
d
by:
)()( 2,1,, itertitertd jijijipro 
,
Kj ,...2,1
(10)
where
}2,1,{, 21  iteriteriteriteriter
,
)( 1
iterti
and
)( 2
iterti
are the best and the worst positions in the set
)}(),1(),2({ itertitertitert iii 
respectively. Here, iter
denotes the current iteration, while
1iter
and
2iter
denote the previous two iterations.
The expression of search direction for the ith seeker is set to
the stochastic combination of egotistic direction, altruistic
direction and pro-activeness direction by:
)( ,2,1, jialtji egojiproij dddwsignd 

(11)
where
Kj ,...2,1
, the function sign (·) is a signum function
on each dimension of the input vector,
w
is the inertia weight
and
1

and
2

are real numbers chosen uniformly and
randomly in the range [0,1]. Inertia weight is used to gradually
reduce the local search effect of pro-activeness direction
ipro
d
and provide a balance between global and local exploration
and exploitation. Inertia weight is linearly decreased from 0.9
to 0.1 during a run.
Fuzzy reasoning is used to generate the step length because
the uncertain reasoning of human searching. From the view
point of human searching behavior, it may be assumed that
better points are likely to be found in the neighborhood of
families of good points. For calculating the step length of ith
seeker we need to calculate vector
i

by:
)(
1minmaxmax




 S
IS i
i
(12)
where S denotes the size of the subpopulation to which the
seekers belong,
i
I
is the sequence number of
i
t
after sorting
the objective function values in ascending order,
max

is the
maximum membership degree value which is equal to or a
little less than 1.0,
min

is set to 0.0111. Beside of vector
i

,
we need to calculate vector
i

by :
)( minmax ttabsw
i

(13)
In Eq.(11), the absolute value of the input vector as the
corresponding output vector is represented by the symbol abs
(⋅),
max
t
and
min
t
are the positions of the best and the worst
seeker in the subpopulation to which the ith seeker belongs,
respectively. In order to introduce the randomness in each
variable and to improve the local search capability, the
following equation is introduced to convert
i

into a vector
with elements as given by:
)1,( iij rand


,
Kj ,...2,1
(14)
The equation used for generating the step length
ji,

for ith
seeker is :
)ln( ,,, jijiji


,
Kj ,...2,1
(15)
For each seeker
i
t
(i = 1, 2, ..., SN) evaluate the objective
function values.
Part 2. Evaluating all the seekers and saving the historical
best position.
Part 3. Apply the modified inter-subpopulation learning
operation
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The modified inter-subpopulation learning is implemented as
follows: The positions of seekers with the lowest objective
function values of each subpopulation l are combined with the
positions of seekers with the highest objective function values
of (l+z) mod N subpopulations respectively, where z=1,2,..
NSC. NSC denotes the number of the worst seekers of each
population which are combined with the best seekers. The
appropriate seekers are combined using the following binomial
crossover operator as expressed in:
otherwise
Rif
t
t
tj
worstjl
besti
worstjl
n
j
n
5.0
,
,
,
,
,







(16)
where
j
R
is a uniformly random real number within [0, 1),
worstjln
t,
is denoted as the jth variable of the nth worst position
in the lth subpopulation,
bestij
t
is the jth variable of the best
position in the ith subpopulation. Additionally, we included a
new parameter which we named inter-subpopulation learning
increase period (ILIP). After ILIP iterations the number of the
worst seekers of each subpopulation which are combined with
the best seekers is increased to
NSC2
.
Step 5. Output best recorded solution
After a predefined number of iterations the positions of the
best recorded solution are the optimal threshold values.
V. EXPERIMENTAL RESULTS AND DISCUSSION
The MSO and PSO algorithms have been implemented in Java
programming language. Four well-known images, namely
Lena, Peppers, Cameraman and Boats with 256 gray levels are
taken as the test images. All the images are of size (512 x 512).
These original images with their histograms are shown in Fig
2. Tests were done on a PC with Intel® Core™ i3-2310M
processor @2,10 GHz with 2GB of RAM and Windows 7 x64
Professional operating system.
In all experiments for both algorithms the same size of
population (SP) of 40 is used and the same size of maximum
number of iterations (MCN) of 100 is taken. In proposed MSO
algorithm the number of subpopulations (N) is 5, the behavior
rate (BR) is 0.4, the number of seekers of each subpopulation
for combination (NSC) is 1 and the inter-subpopulation
learning increase period (ILIP) is
MCN4.0
. Parameters of
PSO algorithm are: inertia weight (w) is 0.5, minimum velocity
(
min
v
) is -5, maximum velocity (
max
v
) is 5 ,
min

is 0 and
max

is 2. Since MSO and PSO algorithms are of stochastic
type and therefore the results of experiments are not absolutely
the same in each run of algorithm, each experiment was
repeated 50 times.
Table 1 shows the optimal thresholds, the optimal objective
function values and the processing time provided by the
exhaustive search method for Kapur’s and Tsallis' method.
Table 2 and Table 3 present the mean values, standard
deviations and average processing time over 50 runs provided
by both algorithms for each image with a threshold numbers
from 1 to 5 for Kapur's and Tsallis' method respectively.
Lena
Peppers
Cameraman
Boats
Fig 2: Test images and their histograms
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374

TABLE I
THRESHOLDS, OBJECTIVE FUNCTION VALUES AND TIME PROCESSING PROVIDED BY THE EXHAUSTIVE SEARCH
FOR KAPUR’S AND TSALLIS’ METHOD
TABLE II
MEAN VALUES, STANDARD DEVIATIONS AND AVERAGE PROCESSING TIME OVER 50 RUNS FOR KAPUR’S METHOD
Image
k
PSO
MSO
Mean value
St. Dev.
Time(s)
Mean value
St. Dev.
Time(s)
Lena
1
8.941944
1.24E-14
0.0631
8.941944
1.24E-14
0.1781
2
12.347015
5.33E-15
0.0635
12.347015
5.33E-15
0.1769
3
15.318053
1.24E-14
0.0656
15.318053
1.24E-14
0.1829
4
18.000658
1.02E-02
0.0618
18.008059
6.98E-03
0.1826
5
20.610531
1.40E-04
0.0682
20.610500
1.98E-04
0.1856
Pepper
1
9.118984
7.11E-15
0.0551
9.118984
7.11E-15
0.1923
2
12.557434
1.07E-14
0.0568
12.557434
1.07E-14
0.1934
3
15.621959
1.42E-14
0.0668
15.621959
1.42E-14
0.1926
4
18.392850
2.42E-02
0.0719
18.400501
1.24E-04
0.1920
5
21.037588
2.17E-02
0.0686
21.052916
1.92E-03
0.2013
Cameraman
1
8.786829
8.88E-15
0.0606
8.786829
8.88E-15
0.1954
2
12.286490
5.33E-15
0.0657
12.286490
5.33E-15
0.2078
3
15.392280
1.39E-02
0.0661
15.394271
1.07E-14
0.2102
4
18.556636
1.35E-04
0.0691
18.556655
2.13E-14
0.2132
5
21.294869
3.68E-02
0.0656
21.310750
2.67E-02
0.2149
Boats
1
8.964219
5.33E-15
0.0642
8.964219
5.33E-15
0.2010
2
12.574798
1.42E-14
0.0648
12.574798
1.42E-14
0.2058
3
15.820754
7.29E-04
0.0679
15.820828
5.21E-04
0.2091
4
18.624686
3.77E-02
0.0706
18.652739
1.38E-02
0.2110
5
21.384345
5.88E-02
0.0752
21.400989
1.74E-03
0.2104
Image
k
Kapur
Tsallis
Threshold values
Objective
function
Time (s)
Threshold values
Objective
function
Time (s)
Lena
1
123
8.941944
0.015
164
0.333333
0.031
2
97, 164
12.347015
0.827
104, 164
0. 888885
2.291
3
82, 126, 175
15.318053
30.576
84, 126, 173
1.296279
203.582
4
64, 97, 138, 179
18.012432
1873.643
NA
NA
NA
Pepper
1
97
9.118984
0.015
94
0.333333
0.035
2
74, 149
12.557434
0.773
72, 153
0.888885
5.266
3
69, 119, 167
15.621959
28.461
66, 120, 166
1.296281
223.095
4
55, 94, 134, 177
18.400522
1881.104
NA
NA
NA
Cameraman
1
196
8.786829
0.022
201
0.333333
0.042
2
125, 196
12.286490
0.847
145, 201
0.888877
6.523
3
44, 102, 196
15.394271
27.311
124, 155, 203
1.296252
250.41
4
42, 96, 145, 198
18.556655
1900.083
NA
NA
NA
Boats
1
115
8.964219
0.024
119
0.333333
0.045
2
107, 176
12.574798
0.846
64, 119
0.888882
6.446
3
64, 119, 176
15.820903
30.687
64, 119, 186
1.296281
246.327
4
48, 88, 128, 181
18.655734
1856.990
NA
NA
NA
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TABLE III
MEAN VALUES, STANDARD DEVIATIONS AND AVERAGE PROCESSING TIME OVER 50 RUNS FOR TSALLIS’ METHOD
Image
k
PSO
MSO
Mean value
St. Dev.
Time(s)
Mean value
St. Dev.
Time(s)
Lena
1
0.333333
4.44E-16
0.3035
0.333333
3.02E-15
1.2168
2
0.888885
3.33E-16
0.3101
0.888885
3.33E-16
1.2287
3
1.296279
1.65E-08
0.3111
1.296279
1.37E-09
1.2253
4
1.654273
1.28E-06
0.3163
1.654273
3.16E-08
1.2494
5
1.995795
1.67E-06
0.3229
1.995796
8.32E-07
1.2944
Pepper
1
0.333333
5.55E-17
0.3405
0.333333
5.55E-17
1.3112
2
0.888885
6.66E-16
0.3413
0.888885
6.66E-16
1.3120
3
1.296281
1.33E-15
0.3432
1.296281
1.33E-15
1.3750
4
1.654282
1.03E-06
0.3491
1.654283
7.97E-10
1.3802
5
1.995809
7.31E-06
0.3531
1.995811
6.89E-08
1.3942
Cameraman
1
0.333333
5.55E-17
0.3593
0.333333
5.55E-17
1.3897
2
0.888877
6.66E-16
0.3616
0.888877
6.66E-16
1.3095
3
1.296252
1.55E-15
0.3646
1.296252
1.55E-15
1.3241
4
1.654207
1.16E-05
0.3846
1.654210
1.11E-05
1.3386
5
1.995700
1.02E-05
0.3890
1.995701
1.00E-05
1.3674
Boats
1
0.333333
2.22E-16
0.3563
0.333333
2.22E-16
1.3470
2
0.888882
7.97E-08
0.3617
0.888882
6.35E-08
1.3610
3
1.296280
1.76E-08
0.3647
1.296281
6.66E-16
1.3717
4
1.654283
2.66E-06
0.3708
1.654283
5.94E-08
1.3738
5
1.995800
2.72E-06
0.3709
1.995802
9.69E-07
1.3909
The mean values and standard deviations obtained by MSO
and PSO algorithms can be compared to the optimal
objective function values derived by the exhaustive search
method. From Table 1 we found that the computation times
of exhaustive search method is exponential. For Kapur
function in the case k

5 and for Tsallis function in the case
k

4 and k

5, the optimal thresholds and objective
function values aren’t counted because the time needed to
find these values was unacceptable.
From Table 2 it can be seen that both algorithms give
good results both in terms of accuracy (mean fitness) and
robustness (similar results of repeated runs or small
standard deviation), for the threshold numbers from 1 to 2.
For each image, for the threshold numbers from 1 to 2,
MSO and PSO algorithms converged consistently to the
same solution which is equal to the optimal solution. In this
case, the standard deviations provided by both algorithms
are very low. In the case when the number of thresholds is
higher or equal to 3, the MSO algorithm performs better
than PSO algorithm for each image, except for the image
Lena (k=5). We can see that for the threshold numbers from
3 to 5, the mean values of MSO are closer to the optimal
ones than the same of PSO. Also, in that case, the standard
deviations obtained by MSO are lower than the standard
deviations obtained by PSO, which is specially noticeable
for the image Cameraman. It can be concluded that MSO
algorithm is superior to PSO in terms of precision and
robustness of the results for the Kapur’s method.
From Table 3 it is observed that for the threshold
numbers from 1 to 5, the MSO algorithm perform well as
compared with the PSO algorithm. The mean results show
that MSO algorithm performs slightly better than PSO
algorithm for each image. Also, MSO algorithm gives
smaller standard deviations than the same of PSO. It can be
concluded that MSO algorithm is more stable than PSO
algorithm for the Tsallis' method.
The reported results from these tables show that as for
the exhaustive search, for both algorithms, the number of
iterations and the run time increase with the threshold
number, but not in the same manner. The convergence times
of the MSO and PSO are faster than those of the exhaustive
search, except for k

1 for both methods. From Table 2 and
Table 3, for the threshold numbers from 1 to 5, for each
image, we can see that PSO is more efficient in terms of
time execution than MSO. It is also observed that the
computation time of Tsallis-based PSO is higher than the
Kapur-based MSO.
VI. CONCLUSION
In this paper, the modified seeker optimization (MSO)
algorithm based on simulating the act of human searching is
proposed for multilevel thresholds selection. In order to
verify the effectiveness of the proposed MSO approach,
INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES
Issue 4, Volume 7, 2013
376

four standard test images are investigated. Particle swarm
optimization (PSO) algorithm is also implemented for
comparison. The experimental results show that MSO
algorithm performs better than PSO algorithm with respect
to precision and robustness, while in term of execution time
the PSO is more efficient than MSO. Even though the
Tsallis-based MSO gives lower standard deviation values,
compared with all the cases, the Kapur-based MSO
converges faster than the Tsallis-based MSO and the
Tsallis-based PSO. Therefore, the proposed Kapur-based
MSO method is a promising approach for image
segmentation due to quality of its segmentation results and
computational efficiency.
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Milan Tuba received B.S. in mathematics, M.S.
in mathematics, M.S. in computer Science, M.Ph.
in computer science, Ph.D. in computer science
from University of Belgrade and New York
University.
From 1983 to 1994 he was in the U.S.A. first as
a graduate student and teaching and research
assistant at Vanderbilt University in Nashville
and Courant Institute of Mathematical Sciences,
New York University and later as an assistant
professor of electrical engineering at Cooper
Union Graduate School of Engineering, New York. During that time he
was the founder and director of Microprocessor Lab and VLSI Lab, leader
of scientific projects and supervisor of many theses. From 1994 he was
associate professor of computer science and Director of Computer Center
at University of Belgrade, Faculty of Mathematics, and from 2004 also a
Professor of Computer Science and Dean of the College of Computer
Science, Megatrend University Belgrade. He was teaching more than 20
graduate and undergraduate courses, from VLSI design and Computer
architecture to Computer networks, Operating systems, Image processing,
Calculus and Queuing theory. His research interest includes mathematical,
queuing theory and heuristic optimizations applied to computer networks,
image processing and combinatorial problems. He is the author of more
than 100 scientific papers and a monograph. He is coeditor or member of
the editorial board or scientific committee of number of scientific journals
and conferences.
Prof. Tuba is member of the ACM since 1983, IEEE 1984, New York
Academy of Sciences 1987, AMS 1995, SIAM 2009, IFNA. He
participated in many WSEAS Conferences with plenary lectures and
articles in Proceedings and Transactions.
Ivona Brajevic received B.S. in mathematics in 2006 and M.S. in
mathematics in 2008 from University of Belgrade,
Faculty of Mathematics.
She is currently Ph.D. student at Faculty of
Mathematics, Computer science department,
University of Belgrade and works as teaching
assistant at College of Business, Economy and
Entrepreneurship in Belgrade. She is the coauthor
of six research papers. Her current research
interest includes nature inspired metaheuristics,
particularly swarm intelligence.
Ms. Brajevic participated in WSEAS conferences.
INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES
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378