# A Global Multilevel Thresholding Using Differential Evolution Approach

Authors: Kanjana Charansiriphaisan; Sirapat Chiewchanwattana; Khamron Sunat
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

Otsu’s function measures the properness of threshold values in multilevel image thresholding. Optimal threshold values are necessary for some applications and a global search algorithm is required. Differential evolution (DE) is an algorithm that has been used successfully for solving this problem. Because the difficulty of a problem grows exponentially when the number of thresholds increases, the ordinary DE fails when the number of thresholds is greater than 12. An improved DE, using a new mutation strategy, is proposed to overcome this problem. Experiments were conducted on 20 real images and the number of thresholds varied from 2 to 16. Existing global optimization algorithms were compared with the proposed algorithms, that is, DE, rank-DE, artificial bee colony (ABC), particle swarm optimization (PSO), DPSO, and FODPSO. The experimental results show that the proposed algorithm not only achieves a more successful rate but also yields a lower threshold value distortion than its competitors in the search for optimal threshold values, especially when the number of thresholds is large.


---

## Body

## 1. Introduction

Thresholding is the simplest and most commonly used method of image segmentation. It can be bilevel or multilevel [1]. Both of these types can be classified into parametric and nonparametric approaches [1]. Surveys of thresholding techniques for image segmentation can be found in [2–7]. The surveys revealed that Otsu’s method is a commonly used technique [4, 8]. This method finds the optimal thresholds by maximizing the weighted sum of between-class variances (BCV) [9]. The BCV function is also called Otsu's function. However, the solution finding process is an exhaustive search and it is a very time-consuming process because the complexity grows exponentially with the number of thresholds.Multilevel image thresholding based on Otsu’s function has been used as a benchmark for comparing the capability of evolutionary algorithms (EA). The EA is a nongradient based optimization algorithm. Several algorithms have been widely applied to solve multilevel thresholding. A group of successful works were based on a combination of Otsu’s function with some state-of-the-art algorithms: PSO [10], DE [11], ABC [12], and FOSPSO [13]. Kulkarni and Venayagamoorthy [14] showed that PSO was faster than Otsu’s method in searching the optimal thresholds of multilevel image thresholding. Akay [15] presented a comprehensive comparative study of the ABC and PSO algorithms. The results showed that the ABC algorithm with both the between-class variance and the entropy criterion can be efficiently used in multilevel thresholding. Hammouche et al. [16] focused on solving the image thresholding problem by combining Otsu’s function with metaheuristic techniques, that is, genetic algorithm (GA), PSO, DE, ant colony, simulated annealing, and Tabu search. Their results revealed that DE was the most efficient with respect to the quality of solution. Osuna-Enciso et al. [17] presented an empirical comparative study of the ABC, PSO, and DE algorithms to perform image thresholding using a mixture of Gaussian functions. The results showed that the DE algorithm was superior in performance in minimizing the Hellinger distance and used less evaluations of the Hellinger distance. Ghamisia et al. [18] showed that a global optimal search for optimal threshold values of Otsu’s function was essential for the multilevel segmentation of multispectral and hyperspectral images.The DE algorithm was selected for multilevel image thresholding. It is simple to implement and produces good results. However, based on our experiments, DE could not reach an optimal solution when it was applied to a very difficult problem. Therefore, a better DE algorithm is required. We noticed that the mechanism of vector selection and the size of the higher ranked population are an important criterion for success.The contribution of this paper is as follows.DE with the onlooker and ranking-based mutation operation, namedO
(
β
)
R
-DE, is proposed to overcome the drawback of the DE algorithm for multilevel image thresholding, especially when the number of thresholds is large. The proposed algorithm homogenizes the onlooker phase of the ABC algorithm and the ranking-based mutation operator of the rank-DE [19]. The main advantage of the proposed algorithm is that a user can adjust the balancing of the exploitation and exploration capabilities of the algorithm.To verify the capabilities of the proposedO
(
β
)
R
-DE algorithm, experiments to find the optimal solutions in the multilevel image thresholding, when the number of thresholds ranged from two to 16, were set up. It was found that the optimal solutions could be effectively reached using the proposed O
(
β
)
R
-DE algorithm.The remainder of the paper is organized as follows. Section2 describes the multilevel thresholding problem. Section 3 presents a brief review of the differential evolution algorithm (DE). In Section 4, the proposed new version of the DE algorithm with the onlooker and ranking-based mutation operator algorithm, O
(
β
)
R
-DE, is described in detail. Section 5 shows the experimental results of applying the proposed method to multilevel segmentation in different images. Finally, the conclusion of the paper is discussed in Section 6.


## 2. Multilevel Thresholding Problem Formulation

Otsu’s method [9] is based on the maximization of the between-class variance. Consider a digital image having the size H
×
W, where W is the width and H is the height. The pixels of a given picture are represented in L gray levels and they are in {
0
,
1
,
2
,
.
…
,
L
-
1
}. The number of pixels at level i is denoted by n
i and the total number of pixels by N
=
n
1
+
n
2
+
⋯
+
n
L. The gray-level histogram is normalized and regarded as a probability distribution and is written as follows:
(1)
p
i
=
n
i
N
,
p
i
≥
0
,
∑
i
=
1
L
p
i
=
1
.
The total mean of the image can be defined as
(2)
μ
T
=
∑
i
=
1
L
i
×
p
i
.
The multilevel thresholding with respect to the given n
-
1 threshold values t
j, j
=
1
,
…
,
n
-
1 can be performed as follows:
(3)
F
(
x
,
y
)
=
{
0
,
f
(
x
,
y
)
≤
t
1
,
1
2
(
t
1
+
t
2
)
,
t
1
<
f
(
x
,
y
)
≤
t
2
,
⋮
1
2
(
t
n
-
2
+
t
n
-
1
)
,
t
n
-
2
<
f
(
x
,
y
)
≤
t
n
-
1
,
L
,
f
(
x
,
y
)
>
t
n
-
1
,
where (
x
,
y
) is the coordinate of a pixel and f
(
x
,
y
) denotes the intensity level of a pixel. The pixels of a given image will be divided into n classes D
1
,
…
,
D
n in this regard.The optimal threshold can be determined by maximizing the between-class variance function (BCV),σ
B
2, which can be defined by
(4)
σ
B
2
=
∑
j
=
1
n
w
j
(
μ
j
-
μ
T
)
2
,
where j represents a specific class in such a way that w
j and μ
j are the probability of occurrence and the mean of class j, respectively. Equation (4) is also called Otsu’s function. The probabilities of occurrence w
j of classes D
1
,
…
,
D
n are defined by
(5)
w
j
=
{
∑
i
=
1
t
j
p
i
,
j
=
1
,
∑
i
=
t
j
-
1
+
1
t
j
p
i
,
1
<
j
<
n
,
∑
i
=
t
j
-
1
+
1
L
p
i
,
j
=
n
.
The mean of each class μ
j can be given by
(6)
μ
j
=
{
∑
i
=
1
t
j
i
×
p
i
w
j
,
j
=
1
,
∑
i
=
t
j
-
1
+
1
t
j
i
×
p
i
w
j
,
1
<
j
<
n
,
∑
i
=
t
j
-
1
+
1
L
i
×
p
i
w
j
,
j
=
n
.
Thus, the n-level thresholding problem is transformed to an optimization problem. The process is to search for n
-
1 thresholds t
j that maximize the value φ, which is generally defined as
(7)
φ
=
max
⁡
1
<
t
1
<
⋯
<
t
n
-
1
<
L
σ
B
2
(
t
j
)
.


## 3. Differential Evolution Algorithm

The DE algorithm is an evolutionary optimization technique proposed by Storn and Price [11]. The main procedures of DE are briefly described as follows.


### 3.1. Initialization

The DE algorithm starts with a population of initial solutions, each of dimensionD, X
i
,
g
=
(
x
i
,
1
,
x
i
,
2
,
…
,
x
i
,
D
)
,
i
=
1
,
…
,
NP, where the index i denotes the i
th solution, or vector, of the population, g is the generation, and NP is the population size. The initial population (at g = 0) is randomly generated to be within the search space constrained by the minimum and maximum bounds, X
min
⁡
=
{
x
1
,
min
⁡
,
x
2
,
min
⁡
,
…
,
x
D
,
min
⁡
} and X
max
⁡
=
{
x
1
,
max
⁡
,
x
2
,
max
⁡
,
…
,
x
D
,
max
⁡
}. The i
th vector x
i is initialized as follows:
(8)
x
j
,
i
,
0
=
x
j
,
min
⁡
+
rndreal
i
,
j
[
0,1
)
·
(
x
j
,
max
⁡
-
x
j
,
min
⁡
)
,
where rndreal
i
,
j
[
0,1
) is a uniformly distributed random real number between 0 and 1, (
0
≤
rndreal
i
,
j
[
0,1
)
<
1
).


### 3.2. Mutation Operators

The differential mutation operator is one of the three operators of DE. The mutation operator is applied to generate the mutant vectorv
i for each target vector x
i in the current population. A mutant vector is generated according to
(9)
v
i
,
g
+
1
=
x
r
1
,
g
+
F
·
(
x
r
2
,
g
-
x
r
3
,
g
)
,
where the randomly chosen indexes, random indexes, r
1
,
r
2
,
r
3
∈
{
1,2
,
…
,
NP
} are mutually different random integer indices and they are also different from the running index i. Further, i
,
r
1
,
r
2, and r
3 are different so that NP
≥
4. F is a real and constant factor, F
∈
[
0
,
2
], which controls the amplification of the differential variation; x
r
1
,
g is called the base vector, x
r
2
,
g is called the terminal vector, x
r
3
,
g is called the other vector, and (
x
r
2
,
g
-
x
r
3
,
g
) is called the difference vector.There have been many proposed mutation strategies for DE [20, 21]. Each different strategy has different characteristics and is suitable for a set of problems. However, the choice of the best mutation operators for DE is difficult for a specific problem [22–24]. The “DE/rand/1/bin” strategy has been widely used in DE literature [25–28]. It is more reliable than the strategies based on the best-so-far solution such as “DE/best/1” and “DE/current-to-best/1”. However, “DE/rand/1/bin” has slower convergence. Simply put, it has high exploration but low exploitation abilities.


### 3.3. Crossover

DE utilizes the crossover operation to generate new solutions by shuffling competing vectors and to increase the diversity of the population. The classical version of the DE (DE/rand/1/bin) uses the binary crossover. It defines the following trial vector:(10)
u
i
,
g
+
1
=
(
u
1
i
,
g
+
1
,
u
2
i
,
g
+
1
,
…
,
u
D
i
,
g
+
1
)
,
where j
=
1
,
…
,
D (D = problem dimension) and
(11)
u
j
i
,
g
+
1
=
{
v
j
i
,
g
+
1
if
(
randb
(
j
)
≤
CR
)
and
j
=
rnbr
(
i
)
x
j
i
,
g
if
(
randb
(
j
)
>
CR
)
and
j
≠
rnbr
(
i
)
.CR is the crossover rate∈
[
0
,
1
], randb
(
j
) is the jth evaluation of a uniform random number generator with outcome ∈
[
0,1
], and rnbr
(
i
) is a randomly chosen index ∈
1
,
2
,
…
,
D that ensures u
i
,
g
+
1 will get at least one parameter from v
i
,
g
+
1.


### 3.4. Selection

Selection determines whether the target or the trial vector survives to the next generation. The selection operation is described as(12)
x
i
,
g
+
1
=
{
u
i
,
g
,
if
f
(
u
i
,
g
)
≤
f
(
x
i
,
g
)
x
i
,
g
,
if
f
(
u
i
,
g
)
>
f
(
x
i
,
g
)
,
where f
(
x
) is the objective function to be minimized. Therefore, if the objective of the new trial vector, f
(
u
i
,
g
), is equal to or less than the objective of the old trial vector, f
(
x
i
,
g
), then x
i
,
g
+
1 is set to u
i
,
g; otherwise, the old value x
i
,
g is retained.The pseudocode of basic DE with “DE/rand/1/bin” strategy is shown in Algorithm1.Algorithm 1:The DE algorithm with “DE/rand/1/bin” strategy.
(1)  Generate the initial population randomly
(2)  Evaluate the fitness for each individual in the population
(3)  while the maximum generationG is not reached do
(4)for i
=
1 to NP do
(5)  Select uniform randomly   r
1
≠
r
2
≠
r
3
≠
i
(6)  j
rand
=
rndint
[
1
,
D
]
(7)  for j
=
1 to D do
(8)     if rndreal
j
[
0,1
)
≤
CR or j is equal to j
rand then
(9)     u
i
,
j
=
x
r
1
,
j
+
F
·
(
x
r
2
,
j
-
x
r
3
,
j
)
(10)    else
(11)    u
i
,
j
=
x
i
,
j
(12)    end if
(13)  end for
(14)end for
(15)for i
=
1 to NP do
(16)    Evaluate the offspring u
i
(17)     if f
(
u
i
) is better than or equal to f
(
x
i
) then
(18)    Replace x
i with u
i
(19)    end if
(20)    end for
(21) end whileThe functionrndint
[
1
,
D
] returns a uniformly distributed random integer number between 1 andD. rndreal
j
[
0,1
) is a uniformly distributed random real value of [
0,1
). The word “better” in line 17 means “less than” if the problem requires minimization, see (12) and its explanation, and it means “greater than,” if the problem requires maximization. The best X
i
,
G, where G is the maximum number of generations, is the solution of the algorithm. The word “best” also depends on the type of problem.


## 3.1. Initialization

The DE algorithm starts with a population of initial solutions, each of dimensionD, X
i
,
g
=
(
x
i
,
1
,
x
i
,
2
,
…
,
x
i
,
D
)
,
i
=
1
,
…
,
NP, where the index i denotes the i
th solution, or vector, of the population, g is the generation, and NP is the population size. The initial population (at g = 0) is randomly generated to be within the search space constrained by the minimum and maximum bounds, X
min
⁡
=
{
x
1
,
min
⁡
,
x
2
,
min
⁡
,
…
,
x
D
,
min
⁡
} and X
max
⁡
=
{
x
1
,
max
⁡
,
x
2
,
max
⁡
,
…
,
x
D
,
max
⁡
}. The i
th vector x
i is initialized as follows:
(8)
x
j
,
i
,
0
=
x
j
,
min
⁡
+
rndreal
i
,
j
[
0,1
)
·
(
x
j
,
max
⁡
-
x
j
,
min
⁡
)
,
where rndreal
i
,
j
[
0,1
) is a uniformly distributed random real number between 0 and 1, (
0
≤
rndreal
i
,
j
[
0,1
)
<
1
).


## 3.2. Mutation Operators

The differential mutation operator is one of the three operators of DE. The mutation operator is applied to generate the mutant vectorv
i for each target vector x
i in the current population. A mutant vector is generated according to
(9)
v
i
,
g
+
1
=
x
r
1
,
g
+
F
·
(
x
r
2
,
g
-
x
r
3
,
g
)
,
where the randomly chosen indexes, random indexes, r
1
,
r
2
,
r
3
∈
{
1,2
,
…
,
NP
} are mutually different random integer indices and they are also different from the running index i. Further, i
,
r
1
,
r
2, and r
3 are different so that NP
≥
4. F is a real and constant factor, F
∈
[
0
,
2
], which controls the amplification of the differential variation; x
r
1
,
g is called the base vector, x
r
2
,
g is called the terminal vector, x
r
3
,
g is called the other vector, and (
x
r
2
,
g
-
x
r
3
,
g
) is called the difference vector.There have been many proposed mutation strategies for DE [20, 21]. Each different strategy has different characteristics and is suitable for a set of problems. However, the choice of the best mutation operators for DE is difficult for a specific problem [22–24]. The “DE/rand/1/bin” strategy has been widely used in DE literature [25–28]. It is more reliable than the strategies based on the best-so-far solution such as “DE/best/1” and “DE/current-to-best/1”. However, “DE/rand/1/bin” has slower convergence. Simply put, it has high exploration but low exploitation abilities.


## 3.3. Crossover

DE utilizes the crossover operation to generate new solutions by shuffling competing vectors and to increase the diversity of the population. The classical version of the DE (DE/rand/1/bin) uses the binary crossover. It defines the following trial vector:(10)
u
i
,
g
+
1
=
(
u
1
i
,
g
+
1
,
u
2
i
,
g
+
1
,
…
,
u
D
i
,
g
+
1
)
,
where j
=
1
,
…
,
D (D = problem dimension) and
(11)
u
j
i
,
g
+
1
=
{
v
j
i
,
g
+
1
if
(
randb
(
j
)
≤
CR
)
and
j
=
rnbr
(
i
)
x
j
i
,
g
if
(
randb
(
j
)
>
CR
)
and
j
≠
rnbr
(
i
)
.CR is the crossover rate∈
[
0
,
1
], randb
(
j
) is the jth evaluation of a uniform random number generator with outcome ∈
[
0,1
], and rnbr
(
i
) is a randomly chosen index ∈
1
,
2
,
…
,
D that ensures u
i
,
g
+
1 will get at least one parameter from v
i
,
g
+
1.


## 3.4. Selection

Selection determines whether the target or the trial vector survives to the next generation. The selection operation is described as(12)
x
i
,
g
+
1
=
{
u
i
,
g
,
if
f
(
u
i
,
g
)
≤
f
(
x
i
,
g
)
x
i
,
g
,
if
f
(
u
i
,
g
)
>
f
(
x
i
,
g
)
,
where f
(
x
) is the objective function to be minimized. Therefore, if the objective of the new trial vector, f
(
u
i
,
g
), is equal to or less than the objective of the old trial vector, f
(
x
i
,
g
), then x
i
,
g
+
1 is set to u
i
,
g; otherwise, the old value x
i
,
g is retained.The pseudocode of basic DE with “DE/rand/1/bin” strategy is shown in Algorithm1.Algorithm 1:The DE algorithm with “DE/rand/1/bin” strategy.
(1)  Generate the initial population randomly
(2)  Evaluate the fitness for each individual in the population
(3)  while the maximum generationG is not reached do
(4)for i
=
1 to NP do
(5)  Select uniform randomly   r
1
≠
r
2
≠
r
3
≠
i
(6)  j
rand
=
rndint
[
1
,
D
]
(7)  for j
=
1 to D do
(8)     if rndreal
j
[
0,1
)
≤
CR or j is equal to j
rand then
(9)     u
i
,
j
=
x
r
1
,
j
+
F
·
(
x
r
2
,
j
-
x
r
3
,
j
)
(10)    else
(11)    u
i
,
j
=
x
i
,
j
(12)    end if
(13)  end for
(14)end for
(15)for i
=
1 to NP do
(16)    Evaluate the offspring u
i
(17)     if f
(
u
i
) is better than or equal to f
(
x
i
) then
(18)    Replace x
i with u
i
(19)    end if
(20)    end for
(21) end whileThe functionrndint
[
1
,
D
] returns a uniformly distributed random integer number between 1 andD. rndreal
j
[
0,1
) is a uniformly distributed random real value of [
0,1
). The word “better” in line 17 means “less than” if the problem requires minimization, see (12) and its explanation, and it means “greater than,” if the problem requires maximization. The best X
i
,
G, where G is the maximum number of generations, is the solution of the algorithm. The word “best” also depends on the type of problem.


## 4. The Proposed DE with Onlooker Ranking-Based Mutation Operator

In 2013 Gong and Cai [19] proposed a rank-DE algorithm. They claimed that probabilistically selecting the vectors x
r
1 and x
r
2 in the mutation operator from the better population can improve the exploitation ability of basic DE. To the best of the authors’ knowledge, rank-DE may, however, also lead to premature convergence (this will be shown in the experiments). That means that the rank-DE has too much exploitation ability. Furthermore, it cannot balance between the exploration and the exploitation abilities. In order to balance between the two abilities, we propose DE with the onlooker and ranking-based mutation operator, named O
(
β
)
R
-DE. The proposed algorithm is an improvement of the rank-DE by homogenizing the rank-DE with the onlooker phase of ABC algorithm. The detail of the O
(
β
)
R
-DE algorithm is described as follows.


### 4.1. Ranking Assignment

To perform the maximization, the fitness of each vector is sorted in ascending order (i.e., from worst to best). Then, the rank of theith vector, R
i, is assigned based on its sorted ordering as follows:
(13)
R
order
=
order
,
order
=
1
,
2
,
…
,
NP
.
As a result, the best vector in the current population will obtain the highest ranking, that is, NP.


### 4.2. Probabilistic Selection

After assigning the ranking for each vector, the selection probabilityp
i of the ith vector x
i is calculated as
(14)
p
i
=
R
i
NP
,
i
=
1
,
2
,
…
,
NP
.


### 4.3. A New Strategy for Base Vector, Terminal Point, and the Other Vector Selections

Definition 1 (a worse population and a better population).
Letζ be a real value and 0
≤
ζ
<
1. A population having probability less than ζ is called a worse population and a population having probability greater than or equal to ζ is called a better population.In therank-DE, the base vector x
r
1 and the terminal point x
r
2 were based on their selection probabilities. The other vector in the mutation operator, x
r
3, is selected randomly as in the original DE algorithm. The vectors with higher rankings (higher selection probabilities) are more likely to be chosen as the base vector or the terminal point in the mutation operator.Our investigation revealed that if bothx
r
1 and x
r
2 vectors of rank-DE were chosen from better vectors, then the distribution of the target vector may collapse quickly and possibly lead to premature convergence. Accordingly, when the rank-DE was applied to a very difficult problem, it could not reach the optimal solution.If the steps of the DE algorithm are compared with the ABC algorithm, the population in the current generation can be considered as the employed bees and the population in the next generation can be considered as the onlooker bees. To follow the concept of ABC, a new vector,x
r
1, which is called the base vector, chooses a food source with respect to the probability that is computed from the fitness values of the current population. The probability value, p
t, of which x
t is chosen by a base vector x
r
1 can be calculated by using the expression given in (14). After a base source x
r
1 for a new vector is probabilistically chosen, both x
r
2 and x
r
3 are also chosen in the same manner as the terminal point and the other vector selections in the rank-DE. The target vector is created by a mutation formula of DE. The mutant vector u
i is created after the target vector is crossed with a randomly selected vector, and then the fitness value is computed. As in the ordinary DE, a greedy selection is applied between u
i and x
i. Hence, the new population contains better sources and positive feedback behavior appears. This idea can be expressed as pseudocode, as in Algorithm 2. Since the selection of x
r
1 is the onlooker selection and the selections of x
r
2 and x
r
3 are brought from the rank-DE, then the algorithm is called onlooker and ranking-based vector selection.Algorithm 2:Onlooker and ranking-based vector selection for DE.
(1)  Input: The target vector indexi, the last index of onlooker r
1, and β
⇐
(2)  Output: The selected vector indexesr
1
,
r
2
,
r
3
(3)r
1
=
r
1
+
1; if r
1
>
NP then r
1
=
1; end if              ⇐
(4)  whileminprop
(
β
)
>
p
r
1//onlooker-like selection        ⇐
(5) r
1
=
r
1
+
1; if r
1
>
NP then r
1
=
1; end if            ⇐
(6)  end while
(7)  Randomly selectr
2
∈
{
1
,
NP
}
/
/terminal vector index
(8)  whilerndreal
[
0,1
)
>
p
r
2 or r
2
=
=
i or r
2
=
=
r
1 do
(9) Randomly select r
2
∈
{
1
,
NP
}
(10) end while
(11) Randomly selectr
3
∈
{
1
,
NP
}
/
/the other vector index
(12) whiler
3
=
=
r
2 or r
3
=
=
r
1 or r
3
=
=
i do
(13) Randomly select r
3
∈
{
1
,
NP
}
(14) end whileThe pseudocode of onlooker and ranking-based vector selection is shown in Algorithm2. The differences between the original ranking-based and onlooker and ranking based selection are highlighted by “⇐”.The functionminprop
(
β
) is added to generalize the algorithm. Its output depends on the parameter β. The outcome can be either a constant value of [
0,1
) or a value of the uniform random function rndreal
[
0,1
). The balance of the exploration and exploitation ability can be set by the parameter β. And the function is defined by
(15)
minprop
(
β
)
=
{
β
,
if
β
is
a
constant
and
0
≤
β
<
1
rndreal
[
0,1
)
,
otherwise
.


### 4.4. The DE with Onlooker-Ranking-Based Mutation Operator

The procedures in Sections4.1, 4.2, and 4.3 are combined together to create a better DE algorithm. The parameter 0
≤
β
<
1 determines the fraction of the worse population to be eliminated. When β
=
0 there is no worse population; each single vector in the current population will act as the base vector. If 0
<
β
<
1, then each single vector having a probability less than β is a worse vector and will not be selected as the base vector. If β is not a constant or is outside [
0
,
1
), each single base vector is an onlooker bee. Accordingly, the name of the algorithm is Onlooker
(
β
) Ranking-Base Differential Evolution (
O
(
β
)
R
-DE
). To achieve the global solution, a user can set a proper value for β to control the balance of the exploration and exploitation abilities of the algorithm. The pseudocode of O
(
β
)
R
-DE is shown in Algorithm 3 and the differences between the rank-DE and O
(
β
)
R
-DE are highlighted by “⇐”.Algorithm 3:DE with onlooker and ranking-based mutation.
(1)  Randomly generate the initial population
(2)  Evaluate the fitness for each individual in the population
(3)  while the maximum generationG is not reached do
(4)     Sort and rank the fitness values of population according to (13)
(5)     Calculate the selection probability for each individual according to (14)
(6) r
1
=
0                                   ⇐
(7)for i
=
1 to NP do
(8)       Selectr
1, r
2, r
3 as shown in Algorithm
2 based on the current r
1 and β
⇐
(9)j
rand
=
rndint
[
1
,
D
]
(10)      forj
=
1 to D do
(11)          ifrndreal
j
[
0,1
)
≤
CR or j is equal to j
rand then
(12) u
i
,
j
=
x
r
1
,
j
+
F
·
(
x
r
2
,
j
-
x
r
3
,
j
)
(13)          else
(14) u
i
,
j
=
x
i
,
j
(15)          end if
(16)      end for
(17)    end for
(18)    fori
=
1 to NP do
(19)          Evaluate the offspringu
i
(20)          iff
(
u
i
) is better than or equal to f
(
x
i
) then
(21) Replace x
i with u
i
(22)          end if
(23)    end for
(24) end while


## 4.1. Ranking Assignment

To perform the maximization, the fitness of each vector is sorted in ascending order (i.e., from worst to best). Then, the rank of theith vector, R
i, is assigned based on its sorted ordering as follows:
(13)
R
order
=
order
,
order
=
1
,
2
,
…
,
NP
.
As a result, the best vector in the current population will obtain the highest ranking, that is, NP.


## 4.2. Probabilistic Selection

After assigning the ranking for each vector, the selection probabilityp
i of the ith vector x
i is calculated as
(14)
p
i
=
R
i
NP
,
i
=
1
,
2
,
…
,
NP
.


## 4.3. A New Strategy for Base Vector, Terminal Point, and the Other Vector Selections

Definition 1 (a worse population and a better population).
Letζ be a real value and 0
≤
ζ
<
1. A population having probability less than ζ is called a worse population and a population having probability greater than or equal to ζ is called a better population.In therank-DE, the base vector x
r
1 and the terminal point x
r
2 were based on their selection probabilities. The other vector in the mutation operator, x
r
3, is selected randomly as in the original DE algorithm. The vectors with higher rankings (higher selection probabilities) are more likely to be chosen as the base vector or the terminal point in the mutation operator.Our investigation revealed that if bothx
r
1 and x
r
2 vectors of rank-DE were chosen from better vectors, then the distribution of the target vector may collapse quickly and possibly lead to premature convergence. Accordingly, when the rank-DE was applied to a very difficult problem, it could not reach the optimal solution.If the steps of the DE algorithm are compared with the ABC algorithm, the population in the current generation can be considered as the employed bees and the population in the next generation can be considered as the onlooker bees. To follow the concept of ABC, a new vector,x
r
1, which is called the base vector, chooses a food source with respect to the probability that is computed from the fitness values of the current population. The probability value, p
t, of which x
t is chosen by a base vector x
r
1 can be calculated by using the expression given in (14). After a base source x
r
1 for a new vector is probabilistically chosen, both x
r
2 and x
r
3 are also chosen in the same manner as the terminal point and the other vector selections in the rank-DE. The target vector is created by a mutation formula of DE. The mutant vector u
i is created after the target vector is crossed with a randomly selected vector, and then the fitness value is computed. As in the ordinary DE, a greedy selection is applied between u
i and x
i. Hence, the new population contains better sources and positive feedback behavior appears. This idea can be expressed as pseudocode, as in Algorithm 2. Since the selection of x
r
1 is the onlooker selection and the selections of x
r
2 and x
r
3 are brought from the rank-DE, then the algorithm is called onlooker and ranking-based vector selection.Algorithm 2:Onlooker and ranking-based vector selection for DE.
(1)  Input: The target vector indexi, the last index of onlooker r
1, and β
⇐
(2)  Output: The selected vector indexesr
1
,
r
2
,
r
3
(3)r
1
=
r
1
+
1; if r
1
>
NP then r
1
=
1; end if              ⇐
(4)  whileminprop
(
β
)
>
p
r
1//onlooker-like selection        ⇐
(5) r
1
=
r
1
+
1; if r
1
>
NP then r
1
=
1; end if            ⇐
(6)  end while
(7)  Randomly selectr
2
∈
{
1
,
NP
}
/
/terminal vector index
(8)  whilerndreal
[
0,1
)
>
p
r
2 or r
2
=
=
i or r
2
=
=
r
1 do
(9) Randomly select r
2
∈
{
1
,
NP
}
(10) end while
(11) Randomly selectr
3
∈
{
1
,
NP
}
/
/the other vector index
(12) whiler
3
=
=
r
2 or r
3
=
=
r
1 or r
3
=
=
i do
(13) Randomly select r
3
∈
{
1
,
NP
}
(14) end whileThe pseudocode of onlooker and ranking-based vector selection is shown in Algorithm2. The differences between the original ranking-based and onlooker and ranking based selection are highlighted by “⇐”.The functionminprop
(
β
) is added to generalize the algorithm. Its output depends on the parameter β. The outcome can be either a constant value of [
0,1
) or a value of the uniform random function rndreal
[
0,1
). The balance of the exploration and exploitation ability can be set by the parameter β. And the function is defined by
(15)
minprop
(
β
)
=
{
β
,
if
β
is
a
constant
and
0
≤
β
<
1
rndreal
[
0,1
)
,
otherwise
.


## 4.4. The DE with Onlooker-Ranking-Based Mutation Operator

The procedures in Sections4.1, 4.2, and 4.3 are combined together to create a better DE algorithm. The parameter 0
≤
β
<
1 determines the fraction of the worse population to be eliminated. When β
=
0 there is no worse population; each single vector in the current population will act as the base vector. If 0
<
β
<
1, then each single vector having a probability less than β is a worse vector and will not be selected as the base vector. If β is not a constant or is outside [
0
,
1
), each single base vector is an onlooker bee. Accordingly, the name of the algorithm is Onlooker
(
β
) Ranking-Base Differential Evolution (
O
(
β
)
R
-DE
). To achieve the global solution, a user can set a proper value for β to control the balance of the exploration and exploitation abilities of the algorithm. The pseudocode of O
(
β
)
R
-DE is shown in Algorithm 3 and the differences between the rank-DE and O
(
β
)
R
-DE are highlighted by “⇐”.Algorithm 3:DE with onlooker and ranking-based mutation.
(1)  Randomly generate the initial population
(2)  Evaluate the fitness for each individual in the population
(3)  while the maximum generationG is not reached do
(4)     Sort and rank the fitness values of population according to (13)
(5)     Calculate the selection probability for each individual according to (14)
(6) r
1
=
0                                   ⇐
(7)for i
=
1 to NP do
(8)       Selectr
1, r
2, r
3 as shown in Algorithm
2 based on the current r
1 and β
⇐
(9)j
rand
=
rndint
[
1
,
D
]
(10)      forj
=
1 to D do
(11)          ifrndreal
j
[
0,1
)
≤
CR or j is equal to j
rand then
(12) u
i
,
j
=
x
r
1
,
j
+
F
·
(
x
r
2
,
j
-
x
r
3
,
j
)
(13)          else
(14) u
i
,
j
=
x
i
,
j
(15)          end if
(16)      end for
(17)    end for
(18)    fori
=
1 to NP do
(19)          Evaluate the offspringu
i
(20)          iff
(
u
i
) is better than or equal to f
(
x
i
) then
(21) Replace x
i with u
i
(22)          end if
(23)    end for
(24) end while


## 5. Experiments and Results

### 5.1. Experimental Setup

The global multilevel thresholding problem deals with finding optimal thresholds within the range[
0
,
L
-
1
] that maximize the BCV function. The dimension of the optimization problem is the number of thresholds, n, and the search space is [
0
,
L
-
1
]
n. The parameter β of O
(
β
)
R
-DE is rndreal
[
0,1
) or is set to be one of 0.0, 0.1
,
…
,
0.9. The variation of the proposed O
(
β
)
R
-DE was implemented and compared with the existing metaheuristics that performed image thresholding, that is, PSO, DPSO, FODPSO, ABC, and several variations of DE algorithms. All the methods were programmed in Matlab R2013a and were run on a personal computer with a 3.4 GHz CPU, 8 GB RAM with Microsoft Windows 7 64-bit operating system. The experiments were conducted on 20 real images. The 19 images, namely, starfish, mountain, cactus, butterfly, circus, snow, palace, flower, wherry, waterfall, bird, police, ostrich, viaduct, fish, houses, mushroom, snow mountain, and snake, were taken from the Berkeley Segmentation Dataset and Benchmark [29]. The last image, namely, Riosanpablo, is a satellite image “New ISS Eyes see Rio San Pablo”, March 1, 2013 (http://visibleearth.nasa.gov/view.php?id=80561). Each image has a unique gray level histogram. These original images and their histograms are depicted in Figure 1. An experiment of an image with a specific number of thresholds is called a “subproblem.” The number of thresholds investigated in the experiments was 2,3
,
…
,
16. Thus, there are 20
×
15 subproblems per algorithm. Each subproblem was repeated 50 times and each time is called a run.The test images and corresponding histograms.
(a)
(b)To compare with PSO, ABC, and DEs algorithms, the objective function evaluation is computed forNP
×
N
i, where N
P is population size and N
i is the number of generations. A population of PSO and the DEs calls Otsu’s function one time per generation. The population size in the PSO and DEs algorithms was set to 50. A bee in the ABC calls Otsu’s function two times per generation; therefore their number of food sources were set to a half of the PSO’s size, that is, 25. The stopping criteria were set by the maximum amount of generations G. In this experiment, G was set to 50, 100, 150, 200, 300, 400, 600, 800, 1000, 1500, 2000, 3000, 4000, 5000, and 6000 when n was 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, and 16, respectively. For the PSO, DPSO, and FODPSO algorithms, the parameters were set as per the suggestion in [30] and is shown in Table 1. The other control parameter of the ABC algorithm,limit, was set to 50 [15]. The control parameters F and CR of the DE algorithms were set to 0.5 and 0.9, respectively [31, 32].Table 1
Essential parameters of the PSO, DPSO, and FODPSO taken from [30].
Parameter
PSO
DPSO
FODPSO
Population
50
50
50
ρ
1
1.5
1.5
1.5
ρ
2
1.5
1.5
1.5
W
1.2
1.2
1.2
V
max
⁡
2
2
2
V
min
⁡
−2
−2
−2
x
max
⁡
255
255
255
x
min
⁡
0
0
0
Min population
—
10
10
Max population
—
50
50
No. of swarms
—
4
4
Min swarms
—
2
2
Max swarms
—
6
6
Stagnancy
—
10
10
Fractional coefficient
—
—
0.75


### 5.2. Comparison Strategies and Metrics

To compare the performance of different algorithms, there are three metrics:(
1
) the convergence rate of algorithms was compared by the average of generations (NG
¯), a lower NG
¯ means a faster convergence rate; (
2
) the stability of algorithms was compared by the average of the success rate, (SRHM), a higher SRHM means higher stability; (
3
) the reliability was compared by the threshold value distortion measure (TVD), a lower TVD means higher reliability. The details of the three metrics are described as follows.When all 50 runs of an algorithm performing on an image with a specific number of thresholds are terminated, the outcomes will be analyzed. Runr’th is called a successful run if there is a generation of t
≤
G such that BCV
r
(
t
)
≥
VTR and the number of generations (NG) of the successful run is recorded. Thus, the number can be defined by
(16)
NG
r
=
ArgMin
t
(
BCV
r
(
t
)
≥
VTR
)
,
if
r
is
a
successful
run
and
otherwise
undefined
.
The average of N
G
r from those successful runs is represented by NG
¯ as follows:
(17)
NG
¯
=
1
number
of
successful
runs
∑
All
successful
runs
NG
r
.
The ratio of success rate (SR) for which the algorithm succeeds to reach the VTR for each subproblem is computed as
(18)
SR
=
number
of
successful
runs
total
number
of
runs
.
The experiments were conducted on 20 images. The arithmetic mean (AM) of NG
¯ (NG
¯
AM) over the entire set of images with a specific number of thresholds is calculated as
(19)
NG
¯
AM
=
1
N
∑
All
images
NG
¯
,
where N is the total number of images. NG
¯
AM is shown in Table 3. The worst-case scenario is that there is no successful run for a subproblem; this subproblem is called an “unsuccessful subproblem.” If an algorithm encounters this scenario, the subproblem will be grouped by its number of thresholds and the number of images in the group will be counted and assigned to x. These scenarios will be represented by NA
(
x
), as shown in Tables 3 and 4.The average of the success rate over the entire dataset with a specific number of thresholds (S
R
HM) is averaged by the Harmonic mean, HM, as follows:
(20)
S
R
HM
=
N
∑
All
images
(
1
/
SR
)
.TheS
R
HM is very important in measuring the stability of an algorithm and it means the ratio of runs that are achieving the target solution. Because the evolutionary methods are based on stochastic searching algorithms, the solutions are not the same in each run of the algorithm and depend on the search ability of the algorithm. Therefore, the S
R
HM is vital in evaluating the stability of the algorithms. The comparison of the stability gives us valuable information in terms of the ratio representing the success rates (S
R
HM). A higher S
R
HM means better stability of the algorithm.An algorithm producingS
R
HM
<
0.5 means that more than 50 percent of the independent runs of the algorithm cannot reach the global solution. Thus, the algorithm that yields S
R
HM
<
0.5 should not be selected to solve the problem. The experiments were conducted for the number of thresholds varying from 2 to 16. These experiments contained the maximum number of thresholds such that the algorithm yields S
R
HM
≥
0.5, which is represented by n
0.5 in Table 4. Furthermore, the experiments also contained the maximum number of thresholds that the algorithm can solve; and above this value there was the case such that all 50 runs of some subproblems missed the VTR. This number is represented by n
max
⁡. In this case the success rate was zero and the associated S
R
HM was zero too. And the definitions of the two values are presented in (21)
(21)
n
0.5
=
max
⁡
(
{
S
R
HM
n
∣
n
=
number
of
thresholds
that
has
S
R
HM
≥
0.5
S
R
HM
}
)
n
max
⁡
=
min
⁡
(
{
S
R
HM
n
∣
n
=
number
of
thresholds
that
has
S
R
HM
=
0
S
R
HM
}
)
-
1
.Letn be the number of thresholds. The reliability of a solution is measured by threshold value distortion measure (TVD) and is computed as
(22)
TVD
=
∑
All
images
∑
r
=
1
run
∑
i
=
1
n
|
T
r
i
*
-
T
r
i
m
|
1
+
∑
All
images
∑
r
=
1
run
∑
i
=
1
n
1
{
T
r
i
*
≠
T
r
i
m
}
×
(
1
-
SR
)
×
100
,
where T
* is the threshold value producing the VTR, T
m is the threshold value obtained from the algorithm, and 1
{
T
r
i
*
≠
T
r
i
m
} is the indicator function, which is equal to 1 when T
r
i
*
≠
T
r
i
m and is zero otherwise. TVD is zero if the algorithm can reach the VTR in every run. The lower the TVD the more reliable the algorithm is.


#### 5.2.1. The Value to Reach (VTR)

Following the completion of all of the experiments the best values of the between-class variance and the corresponding thresholds were collected and are shown in Table6. The results are shown image by image and the numbers of thresholds vary from 2 to 16. The between-class variance values in column 3 are used as the VTR values.


#### 5.2.2. Results Produced by Local Search Method

The multithresh function of the Matlab toolbox was conducted on the same images and number of thresholds as the other search methods. The capabilities of solving the optimal solution between a local search and a global search will be discussed here. This is the reason we focused on the global search, that is, the proposedO
(
0.0
)
R
-DE algorithm. Table 2 shows the between-class variances and threshold values of the “mountain” image. These values were the best outcomes of 50 runs produced by the multithresh function in the Matlab R2013a toolbox and by the proposed O
(
0.0
)
R
-DE algorithm. The terminated condition of the multithresh function was set by “MaxFunEvals” = 500000. That is the multithresh function performs more function calls than that of the O
(
0.0
)
R
-DE algorithm. It can be seen from columns 3 and 5 that all the BCVs produced by the O
(
0.0
)
R
-DE algorithm are better than the BCVs produced by the multithresh function; the difference of the BCVs is shown in column 7. The differences in the thresholds from the two algorithms, shown in column 8, tended to be large if the number of thresholds increased.Table 2
The best values of the between-class variance and thresholds of the “mountain” image produced by the multithresh function of the Matlab toolbox and by the proposedO(0.0)R-DE algorithm. The number of thresholds varies from 2 to 16.
Image(1)
n  (2)
Produced by multithresh
Produced by theO(0.0)R-DE algorithm
Objdiff
(
7
)
=
(
5
)
-
(
3
)
Threshdiff
(
8
)
=
∑
|
(
4
)
-
(
6
)
|
Between-class variance(3)
Thresholds(4)
Between-class variance(5)
Thresholds(6)
Mountain
2
2372.886
60, 127
2372.923
61, 128
0.037
2
3
2495.852
32, 76, 130
2496.113
33, 77, 131
0.261
3
4
2551.778
32, 72, 109, 145
2551.955
33, 73, 109, 147
0.177
4
5
2580.154
31, 68, 98, 124, 158
2580.336
32, 69, 99, 125, 159
0.182
5
6
2588.264
35, 75, 98, 118, 152, 175
2596.956
24, 46, 74, 101, 126, 160
8.692
122
7
2598.800
33, 70, 98, 118, 141, 166, 207
2608.807
24, 46, 73, 98, 119, 145, 175
10.007
153
8
2605.785
31, 68, 91, 105, 121, 140, 163, 192
2616.294
24, 46, 73, 97, 115, 135, 160, 191
10.509
70
9
2619.512
27, 52, 75, 95, 111, 129, 148, 168, 197
2622.314
20, 37, 54, 76, 98, 116, 136, 160, 191
2.802
114
10
2620.392
23, 46, 73, 97, 116, 135, 157, 175, 204, 232
2627.194
20, 36, 53, 74, 93, 106, 121, 140, 163, 194
6.802
258
11
2625.275
23, 45, 72, 93, 105, 119, 137, 156, 175, 204, 228
2630.496
20, 36, 53, 74, 93, 105, 118, 134, 152, 172, 201
5.221
199
12
2629.641
22, 39, 58, 78, 95, 109, 123, 139, 156, 177, 207, 247
2633.189
18, 31, 45, 59, 76, 93, 105, 118, 134, 152, 172, 201
3.548
246
13
2631.372
20, 38, 54, 74, 94, 107, 121, 139, 157, 173, 193, 224, 248
2635.290
18, 31, 45, 59, 76, 92, 103, 115, 129, 144, 161, 179, 207
3.918
283
14
2633.865
19, 36, 53, 74, 91, 103, 115, 129, 144, 160, 174, 190, 217, 245
2637.088
17, 29, 42, 56, 72, 86, 96, 106, 117, 130, 145, 161, 179, 207
3.223
307
15
2635.190
19, 33, 50, 69, 85, 96, 107, 121, 137, 152, 167, 178, 190, 212, 240
2638.449
17, 28, 39, 50, 61, 75, 88, 97, 107, 118, 131, 146, 162, 179, 207
3.259
351
16
2636.357
16, 30, 45, 59, 77, 93, 105, 118, 133, 148, 162, 176, 192, 209, 230, 244
2639.616
17, 27, 38, 49, 60, 74, 86, 95, 104, 113, 123, 135, 149, 164, 181, 209
3.259
415Table 3
The average of mean number of generation (NG
¯
AM) and ranks of the methods.
No.maxGen.
FODPSO
DPSO
PSO
ABC
Rank-DE
DE
O(rand)R-DE
O(0.9)R-DE
O(0.8)R-DE
O(0.7)R-DE
O(0.6)R-DE
O(0.5)R-DE
O(0.4)R-DE
O(0.3)R-DE
O(0.2)R-DE
O(0.1)R-DE
O(0.0)R-DE
n
=
2
G
=
50
NG
¯
AM
18.337
15.713
13.218
13.202
10.538
14.713
10.117
6.394
6.903
7.636
8.208
9.147
9.261
9.901
10.709
11.631
20.850
Rank
16
15
13
12
9
14
8
1
2
3
4
5
6
7
10
11
17
n
=
3
G
=
100
NG
¯
AM
35.200
36.210
27.479
33.210
19.488
29.049
18.958
10.492
12.296
13.587
14.807
16.969
17.107
18.100
19.546
21.749
44.192
Rank
15
16
12
14
9
13
8
1
2
3
4
5
6
7
10
11
17
n
=
4
G
=
150
NG
¯
AM
45.391
58.683
49.596
63.251
31.060
48.130
29.595
14.733
18.526
20.885
22.815
26.632
26.456
28.058
30.175
33.499
72.100
Rank
12
15
14
16
10
13
8
1
2
3
4
6
5
7
9
11
17
n
=
5
G
=
200
NG
¯
AM
55.664
81.877
87.441
108.131
44.157
69.088
41.283
19.508
24.524
28.097
30.963
36.935
36.748
38.647
41.455
45.631
100.693
Rank
12
14
15
17
10
13
8
1
2
3
4
6
5
7
9
11
16
n
=
6
G
=
300
NG
¯
AM
NA(2)
100.664
141.937
157.328
61.935
99.700
56.405
24.623
32.687
38.167
42.608
51.003
50.281
52.089
55.729
61.748
138.716
Rank
17
13
15
16
11
12
9
1
2
3
4
6
5
7
8
10
14
n
=
7
G
=
400
NG
¯
AM
NA(1)
NA(1)
192.267
236.928
86.938
142.096
75.340
29.763
41.691
49.408
56.193
67.619
67.063
69.696
74.389
83.326
184.371
Rank
17
16
14
15
11
12
9
1
2
3
4
6
5
7
8
10
13
n
=
8
G
=
600
NG
¯
AM
NA(7)
NA(6)
NA(2)
NA(1)
117.092
198.986
100.944
33.967
51.573
62.263
74.523
91.836
89.666
95.198
102.552
115.583
244.880
Rank
17
16
15
14
11
12
8
1
2
3
4
6
5
7
9
10
13
n
=
9
G
=
800
NG
¯
AM
NA(11)
NA(13)
NA(12)
NA(3)
151.004
264.253
125.299
41.228
63.742
79.432
90.201
113.530
110.416
114.397
122.555
137.538
299.492
Rank
17
16
15
14
11
12
9
1
2
3
4
6
5
7
8
10
13
n
=
10
G
=
1000
NG
¯
AM
NA(11)
NA(16)
NA(16)
NA(7)
NA(1)
321.373
147.770
NA(3)
73.564
91.121
105.020
132.968
128.999
132.691
141.825
157.997
356.546
Rank
17
16
15
14
12
10
8
13
1
2
3
6
4
5
7
9
11
n
=
11
G
=
1500
NG
¯
AM
NA(16)
NA(17)
NA(19)
NA(9)
NA(1)
405.090
185.020
NA(3)
NA(2)
NA(2)
127.135
162.822
157.184
160.123
175.152
198.522
434.877
Rank
17
16
15
14
12
8
6
13
10
10
1
4
2
3
5
7
9
n
=
12
G
=
2000
NG
¯
AM
NA(18)
NA(20)
NA(20)
NA(9)
278.743
535.542
229.965
NA(4)
NA(5)
135.266
160.047
205.266
197.386
203.814
222.094
251.535
574.104
Rank
17
16
15
14
12
8
6
13
11
10
1
4
2
3
5
7
9
n
=
13
G
=
3000
NG
¯
AM
NA(20)
NA(19)
NA(20)
NA(14)
NA(3)
NA(3)
275.860
NA(12)
NA(5)
NA(1)
NA(1)
NA(1)
NA(1)
261.522
278.082
316.468
726.521
Rank
17
16
15
14
12
9
2
13
11
10
6
6
6
1
3
4
5
n
=
14
G
=
4000
NG
¯
AM
NA(20)
NA(20)
NA(20)
NA(14)
NA(4)
NA(4)
330.194
NA(11)
NA(8)
NA(4)
NA(1)
NA(2)
296.414
NA(1)
331.626
385.951
889.711
Rank
17
16
15
14
12
9
1
13
11
10
7
8
6
5
2
3
4
n
=
15
G
=
5000
NG
¯
AM
NA(20)
NA(20)
NA(20)
NA(19)
NA(13)
NA(5)
399.228
NA(11)
NA(10)
NA(8)
NA(6)
363.506
362.665
383.788
417.958
481.830
1086.001
Rank
17
16
15
14
13
8
1
12
11
10
9
7
6
5
2
3
4
n
=
16
G
=
6000
NG
¯
AM
NA(20)
NA(20)
NA(20)
NA(18)
NA(10)
NA(8)
NA(4)
NA(18)
NA(11)
NA(9)
NA(5)
NA(1)
NA(2)
447.570
491.772
552.856
1374.438
Rank
17
16
15
14
12
9
4
13
11
10
8
6
7
5
1
2
3
Avg for
n
=
2 to 16
n
max
⁡,NA(x)
NG
¯
AM
5,NA(146), 38.648
6,NA(152), 58.629
7,NA(149), 85.323
7,NA(94), 102.008
9,NA(32), 88.995
12,NA(20), 193.456
15,NA(4), 144.713
9,NA(62), 22.589
10,NA(41), 36.167
10,NA(24), 52.586
12,NA(13), 66.593
12,NA(4), 106.519
12,NA(3), 119.204
13,NA(1), 143.971
16,NA(0), 167.708
16,NA(0), 190.391
16,NA(0), 436.499
Rank
17
16
15
14
12
9
4
13
11
10
8
7
6
5
1
2
3Table 4
The average success rate (SRHM) and ranks of the methods.
No
Multi-thresh
FODPSO
DPSO
PSO
ABC
Rank-DE
DE
O(rand)R-DE
O(0.9)R-DE
O(0.8)R-DE
O(0.7)R-DE
O(0.6)R-DE
O(0.5)R-DE
O(0.4)R-DE
O(0.3)R-DE
O(0.2)R-DE
O(0.1)R-DE
O(0.0)R-DE
2
SRHM
0
0.977
1
1
1
1
1
1
0.999
1
1
1
1
1
1
1
1
1
Rank
18
17
1
1
1
1
1
1
16
1
1
1
1
1
1
1
1
1
3
SRHM
0.000
0.635
0.972
0.991
0.979
0.998
1.000
0.997
0.986
0.993
0.999
0.997
0.998
0.999
1.000
1.000
1.000
1.000
Rank
18
17
16
13
15
8
1
10
14
12
6
10
8
6
1
1
1
1
4
SRHM
0.000
0.379
0.742
0.934
0.833
0.987
0.995
0.986
0.897
0.945
0.972
0.971
0.986
0.993
0.992
0.991
0.998
0.997
Rank
18
17
16
13
15
7
3
8
14
12
10
11
8
4
5
6
1
2
5
SRHM
0.000
0.185
0.391
0.767
0.706
0.936
0.977
0.960
0.782
0.878
0.874
0.944
0.950
0.959
0.968
0.960
0.985
1.000
Rank
18
17
16
14
15
10
3
5
13
11
12
9
8
7
4
5
2
1
6
SRHM
0.000
0.000
0.187
0.433
0.453
0.863
0.968
0.900
0.601
0.706
0.826
0.878
0.934
0.922
0.944
0.935
0.970
0.999
Rank
18
17
16
15
14
10
3
8
13
12
11
9
6
7
4
5
2
1
7
SRHM
0.000
0.000
0.000
0.153
0.164
0.696
0.884
0.835
0.366
0.561
0.675
0.741
0.840
0.851
0.883
0.914
0.949
0.998
Rank
18
17
16
15
14
10
4
8
13
12
11
9
7
6
5
3
2
1
8
SRHM
0.000
0.000
0.000
0.000
0.000
0.300
0.458
0.530
0.105
0.256
0.351
0.502
0.592
0.628
0.724
0.757
0.838
0.972
Rank
18
17
16
14
14
11
9
7
13
12
10
8
6
5
4
3
2
1
9
SRHM
0.000
0.000
0.000
0.000
0.000
0.495
0.695
0.681
0.083
0.274
0.372
0.548
0.721
0.725
0.769
0.815
0.852
0.999
Rank
18
17
16
14
14
10
7
8
13
12
11
9
6
5
4
3
2
1
10
SRHM
0.000
0.000
0.000
0.000
0.000
0.000
0.381
0.375
0.000
0.073
0.218
0.311
0.409
0.370
0.460
0.578
0.686
0.998
Rank
18
17
16
14
14
12
6
7
13
11
10
9
5
8
4
3
2
1
11
SRHM
0.000
0.000
0.000
0.000
0.000
0.000
0.271
0.267
0.000
0.000
0.000
0.144
0.407
0.316
0.424
0.584
0.704
0.999
Rank
18
17
16
14
14
12
7
8
13
10
10
9
5
6
4
3
2
1
12
SRHM
0.000
0.000
0.000
0.000
0.000
0.090
0.270
0.423
0.000
0.000
0.089
0.177
0.400
0.407
0.501
0.641
0.810
1.000
Rank
18
17
16
14
14
12
8
5
13
11
10
9
7
6
4
3
2
1
13
SRHM
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.151
0.000
0.000
0.000
0.000
0.000
0.000
0.254
0.267
0.381
0.974
Rank
18
17
16
14
14
12
6
5
13
10
10
6
6
6
4
3
2
1
14
SRHM
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.102
0.000
0.000
0.000
0.000
0.000
0.110
0.000
0.253
0.494
0.993
Rank
18
17
16
14
14
12
7
4
13
10
10
7
7
6
5
3
2
1
15
SRHM
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.090
0.000
0.000
0.000
0.000
0.095
0.117
0.149
0.288
0.466
1.000
Rank
18
17
16
14
14
12
8
4
13
10
10
8
7
6
5
3
2
1
16
SRHM
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.110
0.192
0.288
0.977
Rank
18
17
16
14
14
12
6
4
13
10
10
6
6
6
5
3
2
1
Avg. for
n
=
2 to 16
n
0.5,
n
max
⁡,  SRHM
<2,<2, 0.000
3,5, 0.376
4,6, 0.443
5,7, 0.454
5,7, 0.464
7,9, 0.665
9,12, 0.554
9,15, 0.304
6,9, 0.263
7,10, 0.313
7,10, 0.530
9,12, 0.420
9,12, 0.654
9,12, 0.625
9,13, 0.619
12,16, 0.498
12,16, 0.656
16,16, 0.994
Rank
18
17
16
15
14
12
8
4
13
11
10
9
6
7
5
3
2
1Figure2 shows the graph of the TVD of all the images and thresholds. These results are in the same pattern of the results of the “mountain” image in Table 2. That means the ability to search for the optimal solution of the proposed global search algorithm is higher than that of the multithresh function, especially when the number of thresholds is large. This goes to illustrate the difficulty of the problem. The problem with this kind is that it can be multimodal [33] or can be a nearly flat top surface [34]. The multithresh function solves the problem by performing the Nelder-Mead Simplex method [35], which is a local search method that cannot guarantee an optimal solution. Thus, its solutions are inferior to the solution produced by the algorithm using a global search.Figure 2
The threshold value distortion (TVD) of algorithms versus number of thresholds.


#### 5.2.3. Convergence Rate Comparison

The number of generations (NG) is a measure used for the convergence rate comparisons. If the target value, VTR, is achieved in a lesser number of generations (NG), it means a faster convergence rate for the algorithm. Table3 shows the average of NG
¯ (NG
¯
AM) for each specific number of thresholds. The results of each algorithm are represented in the corresponding column’s name. In each column, the cell containing NG
¯
AM starts from the row associated with n
=
2 until the row associated with n
=
n
max
⁡. The cells associated with n
=
n
max
⁡
+
1 to the row associated with n
=
16 are filled by NA
(
x
). The second last row of the table is filled by the triple:
(23)
(
(
NG
¯
AM
of
n
=
2
until
n
=
n
max
⁡
)
n
max
⁡
,
NA
(
number
of
unsuccessful
subproblem
)
,
AM
(
NG
¯
AM
of
n
=
2
until
n
=
n
max
⁡
)
)
.The algorithm with the highestn
max
⁡, that is, the lowest number of unsuccessful subproblems and the lowest average of generation is the winner. The ranking of the algorithms depends on the ordering of (a
1, b
1, c
1) and (a
2, b
2, c
2) as follows.First, rank ona
1 and a
2. Since both a
1 and a
2 are numeric, the higher value has the higher rank.
Second, rank onb
1 and b2. If a
1 is 16, then b
1 must be NA(0). If a
1 is less than 16, then b
1 must be NA (number of unsuccessful subproblems) and b
2 has the same characteristics as b
1. Perform the order of the two numeric values in reverse; the lower value has the higher rank.
Third, rank on c1 and c
2. They are numeric but the lower is better; perform the order of the two numeric values in reverse. The lower value has the higher rank.When the ordering is finished, assign the numeric value of “1” to the object having the highest rank, assign the numeric value of “2” to the first runner up, and so on. These values are represented in the last row of the table.If the row havingn
=
m must be ranked, it can be done in the same manner as above with some minor modifications. If m is less than n
max
⁡, the values of the triple pair will be m, NA
(
0
), NG
¯
AM. If m is greater than n
max
⁡, the values of the triple pair will be (n
max
⁡, NA
(
x
), ∞). Thus the ranking can now be performed.From the ranking results, see the second last row of Table3; the convergence rate can be ranked from best to worst in the following order: O
(
0.2
)
R
-DE, O
(
0.1
)
R
-DE, O
(
0.0
)
R
-DE, O
(
rand
)
R
-DE, O
(
0.3
)
R
-DE, O
(
0.4
)
R
-DE, O
(
0.5
)
R
-DE, O
(
0.6
)
R
-DE, DE, O
(
0.7
)
R
-DE, O
(
0.8
)
R
-DE, rank-DE, O
(
0.9
)
R
-DE, ABC, PSO, DPSO, FODPSO.As can be seen from Table3, the DE algorithm cannot complete the task when n
>
12 and the rank-DE algorithm cannot complete the task when n
>
9. Thus, rank-DE cannot compete with DE on searching for global multilevel thresholding.In order forO
(
β
)
R
-DE to outperform DE then β must be in the range of [
0.1
,
0.6
] or set to rndreal
[
0,1
).


#### 5.2.4. Stability Analysis

The harmonic mean of the success rate (S
R
HM) for each specific number of thresholds was computed and is presented in Table 4. The results of each algorithm are represented in the corresponding column’s name. The second row from the bottom shows the harmonic mean of the success rate of each algorithm for all threshold levels.In each column, the cells containingS
R
HM start from the row associated with n
=
2 to the row associated with n
=
n
0.5. The cells from the row associated with n
=
n
0.5
+
1 to the row associated with n
=
n
max
⁡ are filled by S
R
HM. The cells from the row associated with n
=
n
max
⁡
+
1 to the row associated with n
=
16 are the cells that have S
R
HM; these cells will be excluded from the comparison. The second last row of the table is filled with the triple:
(24)
(
n
0.5
,
n
max
⁡
,
HM
(
S
R
HM
of
n
=
2
until
n
=
n
max
⁡
)
)
.The algorithm with the highestn
0.5, the highest n
max
⁡, and the highest average success rate is the winner. The ranking of the algorithms depends on the ordering of (a1, b
1, c
1) and (a
2, b2, c
2) as follows.First, rank ona
1 and a2.
Second, rank onb
1 and b
2.
Third, rank onc
1 and c
2.Because they are numeric the higher value has the higher rank. When the ordering is finished, the numeric value of “1” is assigned to the object having the highest rank, the numeric value of “2” is assigned to the first runner up, and so on.If the row havingn
=
m must be ranked, it can be done in the same manner as above with some minor modifications. If m is less than or equal to n
0.5, the values of the triple pair will be (m, m, S
R
HM). If n
0.5
<
m
≤
n
max
⁡, then the values of the triple pair will be (n
0.5, m, S
R
HM). If m is greater than n
max
⁡, the values of the triple pair will be (n
0.5, n
max
⁡, S
R
HM). Thus, the ranking can now be performed.From the ranking results, see Table4, the success rate can be ranked from best to worst in the following order: O
(
0.0
)
R
-DE, O
(
0.1
)
R
-DE, O
(
0.2
)
R
-DE, O
(
rand
)
R
-DE, O
(
0.3
)
R
-DE, O
(
0.5
)
R
-DE, O
(
0.4
)
R
-DE, DE, O
(
0.6
)
R
-DE, O
(
0.7
)
R
-DE, O
(
0.8
)
R
-DE, rank-DE, O
(
0.9
)
R
-DE, ABC, PSO, DPSO, FODPSO, multithresh.As can be seen from Table4, the DE algorithm has an S
R
HM
≥
0.5 until n
=
9 and the rank-DE algorithm has an S
R
HM
≥
0.5 until n
=
7. This result confirms that rank-DE cannot compete with DE on searching for global multilevel thresholding. If the correct β is selected, the proposed algorithm can work very well. For β
≤
0.5, O
(
β
)
R
-DE has a higher rank than DE. The multithresh function cannot compete with any of the other algorithms. It can also be seen that the proposed O
(
0.0
)
R
-DE algorithm has the best stability because its S
R
HM is greater than 0.5 when n
=
2 to 16.


#### 5.2.5. Reliability Comparison

The threshold value distortion a.k.a.TVD for each specific threshold is computed, shown in Table 5 and depicted in Figure 2. The results of each algorithm are illustrated in the corresponding column’s name. The second last row of Table 5 shows the slope or the approximated growth rate, δ, of the TVD of each algorithm for all threshold levels. The δ is the slope of the robust linear regression computed by the Matlab function “robustfit.” The lower slope exhibits the better reliability. The δ of each algorithm is sorted in descending order. From the results, the reliability can be ranked from best to worst in the following order: O
(
0.0
)
R
-DE, O
(
0.1
)
R
-DE, O
(
0.2
)
R
-DE, O
(
0.3
)
R
-DE, O
(
0.4
)
R
-DE, O
(
0.5
)
R
-DE, O
(
rand
)
R
-DE, DE, O
(
0.6
)
R
-DE, ABC, rank-DE, O
(
0.7
)
R
-DE, O
(
0.8
)
R
-DE, O
(
0.9
)
R
-DE, DPSO, FODPSO, PSO, multithresh.Table 5
The average of threshold value distortion (TVD) and ranks of the methods.
No
Multi-thresh
FODPSO
DPSO
PSO
ABC
Rank-DE
DE
O(rand)R-DE
O(0.9)R-DE
O(0.8)R-DE
O(0.7)R-DE
O(0.6)R-DE
O(0.5)R-DE
O(0.4)R-DE
O(0.3)R-DE
O(0.2)R-DE
O(0.1)R-DE
O(0.0)R-DE
2
TVD
74.153
2.217
0.000
0.000
0.000
0.000
0.000
0.000
0.050
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
Rank
18
17
1
1
1
1
1
1
16
1
1
1
1
1
1
1
1
1
3
TVD
163.333
41.753
2.968
7.424
1.510
0.150
0.000
0.270
1.004
0.519
0.075
0.246
0.171
0.075
0.000
0.000
0.000
0.000
Rank
18
17
15
16
14
8
1
11
13
12
6
10
9
6
1
1
1
1
4
TVD
257.083
98.848
34.770
36.891
8.716
7.340
1.428
2.184
25.499
13.609
7.318
7.360
7.398
4.303
1.672
0.728
0.130
0.257
Rank
18
17
15
16
12
9
4
6
14
13
8
10
11
7
5
3
1
2
5
TVD
516.582
148.344
97.216
62.319
21.565
5.373
1.864
3.241
20.441
10.064
10.422
4.584
3.904
3.289
2.441
3.385
1.195
0.000
Rank
18
17
16
15
14
10
3
5
13
11
12
9
8
6
4
7
2
1
6
TVD
534.226
219.437
187.620
166.775
45.728
19.536
2.528
7.761
89.938
43.666
30.923
13.010
6.524
6.023
4.136
6.129
2.421
0.050
Rank
18
17
16
15
13
10
3
8
14
12
11
9
7
5
4
6
2
1
7
TVD
599.250
265.029
242.194
285.862
66.259
58.877
27.976
35.481
125.040
75.011
49.851
44.610
28.039
29.841
26.075
17.950
4.776
0.150
Rank
18
16
15
17
12
11
5
8
14
13
10
9
6
7
4
3
2
1
8
TVD
475.458
382.076
380.124
309.635
72.763
120.734
90.287
89.406
218.266
143.225
130.898
99.991
76.353
82.886
57.251
57.665
26.822
2.103
Rank
18
17
16
15
5
11
9
8
14
13
12
10
6
7
3
4
2
1
9
TVD
862.458
394.744
375.229
410.710
113.892
82.099
44.407
46.539
250.470
154.662
96.417
68.197
44.223
35.333
26.627
23.406
12.757
0.780
Rank
18
16
15
17
12
10
7
8
14
13
11
9
6
5
4
3
2
1
10
TVD
847.703
458.536
445.765
498.328
117.664
84.541
43.991
48.474
249.147
136.935
93.935
72.288
43.883
44.110
36.700
27.978
20.213
0.164
Rank
18
16
15
17
12
10
6
8
14
13
11
9
5
7
4
3
2
1
11
TVD
977.976
536.258
528.329
531.211
137.941
87.267
44.944
44.973
296.513
152.195
104.092
75.115
44.461
47.564
34.944
26.231
18.782
0.083
Rank
18
17
15
16
12
10
6
7
14
13
11
9
5
8
4
3
2
1
12
TVD
1070.300
592.750
589.931
605.280
157.169
174.580
109.138
101.282
388.491
258.312
196.098
150.932
101.112
82.365
79.966
51.281
23.429
0.000
Rank
18
16
15
17
10
11
8
7
14
13
12
9
6
5
4
3
2
1
13
TVD
1142.808
567.442
579.495
711.382
183.451
204.665
147.251
123.438
466.508
299.040
216.656
178.394
114.244
114.491
96.104
81.076
52.578
1.975
Rank
18
15
16
17
10
11
8
7
14
13
12
9
5
6
4
3
2
1
14
TVD
1148.641
632.482
660.908
770.319
205.714
218.408
141.431
116.716
524.783
325.936
241.997
177.121
113.426
104.073
88.593
65.908
39.647
0.564
Rank
18
15
16
17
10
11
8
7
14
13
12
9
6
5
4
3
2
1
15
TVD
1245.529
707.835
722.126
816.158
256.477
307.873
219.556
182.464
605.623
428.370
343.439
261.522
174.700
166.309
138.710
100.296
62.940
0.000
Rank
18
15
16
17
9
11
8
7
14
13
12
10
6
5
4
3
2
1
16
TVD
1182.989
768.714
835.747
828.004
249.042
279.753
211.614
195.676
605.790
416.711
311.314
251.408
174.118
167.316
133.922
113.776
76.661
2.539
Rank
18
15
17
16
9
11
8
7
14
13
12
10
6
5
4
3
2
1
Avg. for
n
=
2  to 16
TVD
739.899
387.764
378.828
402.687
109.193
110.080
72.428
66.527
257.838
163.884
122.229
93.652
62.170
59.199
48.476
38.387
22.823
0.578
Growth rate (δ)
86.86
55.25
61.13
67.98
19.25
21.76
15.61
13.48
48.43
32.25
23.91
18.93
12.52
11.86
9.99
7.69
4.94
0.09
Rank
18
15
16
17
10
11
8
7
14
13
12
9
6
5
4
3
2
1Table 6
The between-class variance criterion and the best thresholds for test images.
Image
n
Between-class variance (VTR)
Thresholds
Starfish
2
2546.885
85, 157
3
2779.925
68, 119, 177
4
2865.707
60, 101, 138, 187
5
2912.859
52, 86, 117, 150, 194
6
2941.728
47, 77, 105, 132, 162, 201
7
2960.158
44, 71, 95, 118, 142, 170, 206
8
2972.356
43, 68, 90, 110, 131, 153, 180, 212
9
2981.138
38, 58, 78, 97, 116, 136, 157, 183, 214
10
2988.206
37, 56, 75, 93, 110, 128, 146, 167, 192, 219
11
2993.348
35, 53, 71, 88, 103, 119, 135, 152, 172, 196, 221
12
2997.352
34, 51, 68, 84, 98, 112, 127, 142, 158, 177, 200, 223
13
3000.480
33, 48, 64, 79, 93, 106, 119, 133, 147, 162, 181, 203, 225
14
3003.076
32, 47, 62, 76, 89, 101, 114, 127, 140, 154, 169, 187, 207, 227
15
3005.235
30, 43, 56, 70, 83, 95, 107, 119, 131, 143, 156, 171, 189, 209, 228
16
3007.060
30, 42, 55, 68, 80, 92, 103, 114, 126, 138, 150, 163, 178, 195, 213, 230
Mountain
2
2372.923
61, 128
3
2496.113
33, 77, 131
4
2551.955
33, 73, 109, 147
5
2580.336
32, 69, 99, 125, 159
6
2596.956
24, 46, 74, 101, 126, 160
7
2608.807
24, 46, 73, 98, 119, 145, 175
8
2616.294
24, 46, 73, 97, 115, 135, 160, 191
9
2622.314
20, 37, 54, 76, 98, 116, 136, 160, 191
10
2627.194
20, 36, 53, 74, 93, 106, 121, 140, 163, 194
11
2630.496
20, 36, 53, 74, 93, 105, 118, 134, 152, 172, 201
12
2633.189
18, 31, 45, 59, 76, 93, 105, 118, 134, 152, 172, 201
13
2635.290
18, 31, 45, 59, 76, 92, 103, 115, 129, 144, 161, 179, 207
14
2637.088
17, 29, 42, 56, 72, 86, 96, 106, 117, 130, 145, 161, 179, 207
15
2638.449
17, 28, 39, 50, 61, 75, 88, 97, 107, 118, 131, 146, 162, 179, 207
16
2639.616
17, 27, 38, 49, 60, 74, 86, 95, 104, 113, 123, 135, 149, 164, 181, 209
Cactus
2
1816.448
73, 151
3
1970.112
64, 106, 173
4
2042.275
55, 87, 125, 187
5
2080.884
49, 76, 102, 138, 196
6
2104.147
46, 70, 93, 119, 157, 208
7
2119.670
44, 65, 85, 105, 131, 168, 215
8
2129.358
42, 60, 77, 94, 113, 138, 174, 218
9
2136.162
41, 58, 74, 89, 105, 125, 152, 186, 224
10
2141.579
40, 56, 71, 85, 99, 115, 135, 161, 193, 228
11
2145.397
39, 53, 66, 79, 91, 104, 119, 139, 165, 196, 229
12
2148.282
38, 51, 64, 76, 88, 100, 114, 131, 152, 176, 204, 233
13
2150.740
37, 49, 61, 72, 83, 94, 105, 118, 135, 155, 178, 205, 233
14
2152.626
36, 47, 58, 68, 78, 88, 98, 109, 122, 138, 158, 181, 207, 234
15
2154.162
36, 46, 56, 66, 76, 85, 94, 104, 115, 128, 144, 164, 187, 212, 237
16
2155.471
36, 46, 56, 66, 75, 84, 93, 102, 112, 124, 138, 155, 174, 195, 217, 239
Butterfly
2
3873.222
84, 155
3
3990.855
81, 144, 199
4
4051.357
77, 121, 167, 207
5
4092.929
60, 89, 129, 172, 209
6
4119.448
59, 87, 122, 161, 193, 221
7
4135.937
53, 74, 98, 128, 165, 195, 222
8
4148.556
53, 73, 96, 123, 156, 183, 203, 228
9
4156.528
52, 72, 94, 118, 144, 170, 190, 207, 231
10
4163.794
48, 63, 80, 99, 121, 147, 172, 191, 208, 231
11
4168.489
47, 62, 79, 98, 118, 141, 164, 183, 198, 213, 234
12
4172.517
46, 59, 73, 89, 104, 122, 144, 167, 185, 199, 214, 235
13
4175.487
46, 59, 72, 87, 102, 118, 137, 157, 175, 189, 202, 216, 236
14
4177.893
44, 55, 66, 79, 93, 106, 121, 141, 161, 178, 191, 203, 217, 237
15
4179.908
44, 55, 66, 78, 92, 105, 119, 137, 156, 173, 186, 197, 208, 221, 239
16
4181.559
42, 52, 61, 71, 83, 95, 107, 121, 139, 157, 173, 186, 197, 208, 221, 239
Circus
2
1651.257
118, 172
3
1760.512
105, 150, 187
4
1817.487
93, 132, 165, 195
5
1850.243
87, 122, 152, 177, 203
6
1870.083
82, 113, 141, 164, 185, 208
7
1883.966
77, 104, 129, 151, 171, 190, 212
8
1893.450
73, 98, 122, 143, 161, 178, 195, 216
9
1900.592
70, 92, 114, 134, 152, 168, 183, 199, 219
10
1905.992
68, 89, 109, 128, 145, 160, 174, 188, 203, 222
11
1909.887
66, 86, 105, 123, 139, 153, 166, 179, 192, 206, 225
12
1913.079
63, 81, 98, 114, 130, 145, 158, 170, 182, 194, 208, 226
13
1915.676
62, 79, 95, 111, 126, 140, 152, 164, 175, 186, 197, 210, 228
14
1917.756
61, 78, 94, 109, 123, 136, 148, 159, 170, 180, 190, 201, 214, 231
15
1919.524
59, 74, 88, 102, 115, 128, 140, 151, 161, 171, 181, 191, 202, 214, 231
16
1920.958
58, 73, 87, 100, 113, 126, 138, 149, 159, 169, 178, 187, 196, 206, 218, 234
Snow
2
5261.705
80, 169
3
5624.289
71, 139, 207
4
5729.116
50, 92, 144, 208
5
5785.138
49, 91, 140, 192, 231
6
5819.333
45, 81, 111, 148, 194, 232
7
5835.770
43, 76, 101, 127, 159, 196, 232
8
5850.190
30, 55, 83, 107, 133, 163, 197, 233
9
5862.437
30, 55, 82, 106, 129, 157, 185, 211, 237
10
5870.284
29, 52, 75, 94, 111, 132, 159, 186, 212, 237
11
5875.817
29, 52, 75, 94, 111, 132, 158, 183, 206, 227, 244
12
5880.335
29, 52, 74, 93, 109, 127, 149, 169, 188, 209, 228, 244
13
5884.513
23, 39, 57, 76, 94, 110, 128, 150, 170, 188, 209, 228, 244
14
5887.355
22, 37, 54, 70, 84, 98, 112, 129, 151, 170, 188, 209, 228, 244
15
5889.953
21, 36, 53, 69, 83, 97, 110, 124, 141, 159, 175, 192, 211, 229, 245
16
5891.998
21, 36, 53, 69, 83, 97, 110, 124, 141, 159, 174, 189, 207, 222, 236, 248
Palace
2
2623.440
99, 165
3
2791.488
84, 132, 186
4
2860.98
70, 103, 143, 191
5
2908.165
69, 101, 138, 177, 218
6
2934.330
64, 89, 117, 147, 181, 220
7
2953.745
54, 75, 99, 126, 153, 183, 220
8
2966.250
50, 69, 89, 111, 134, 158, 185, 221
9
2974.719
47, 65, 81, 100, 120, 141, 163, 188, 222
10
2981.875
47, 64, 80, 98, 117, 137, 157, 178, 199, 226
11
2986.898
45, 61, 75, 90, 106, 123, 141, 159, 179, 200, 227
12
2990.579
43, 58, 69, 82, 96, 111, 126, 143, 160, 180, 201, 227
13
2993.809
43, 58, 69, 81, 95, 110, 125, 141, 157, 173, 190, 208, 231
14
2996.112
43, 57, 68, 80, 94, 108, 123, 138, 153, 167, 182, 199, 218, 239
15
2998.213
42, 56, 66, 77, 88, 100, 113, 126, 140, 154, 167, 182, 199, 218, 239
16
2999.918
42, 55, 65, 75, 86, 98, 110, 123, 136, 149, 162, 176, 191, 205, 222, 241
Flower
2
1489.281
61, 130
3
1627.897
39, 77, 141
4
1685.956
36, 67, 105, 160
5
1715.220
28, 49, 75, 111, 164
6
1736.423
27, 47, 71, 102, 143, 192
7
1752.512
26, 43, 61, 83, 111, 151, 199
8
1761.968
25, 42, 58, 77, 99, 126, 161, 204
9
1768.354
24, 40, 54, 70, 88, 109, 135, 167, 207
10
1772.903
23, 37, 48, 60, 75, 92, 112, 138, 169, 208
11
1776.470
23, 36, 47, 58, 72, 87, 104, 124, 149, 177, 211
12
1779.106
22, 34, 44, 54, 65, 78, 92, 108, 128, 152, 179, 212
13
1781.251
20, 30, 39, 48, 58, 70, 83, 96, 112, 131, 154, 180, 213
14
1783.049
19, 29, 38, 47, 56, 66, 78, 90, 104, 120, 139, 161, 185, 215
15
1784.478
19, 29, 38, 46, 54, 63, 74, 85, 97, 111, 128, 147, 168, 190, 218
16
1785.641
19, 28, 37, 45, 53, 62, 72, 83, 94, 106, 120, 136, 155, 175, 196, 222
Wherry
2
3313.161
108, 189
3
3543.272
102, 161, 218
4
3599.924
83, 121, 163, 218
5
3634.708
81, 118, 152, 184, 224
6
3656.048
72, 103, 130, 156, 186, 225
7
3668.313
68, 94, 120, 139, 161, 189, 226
8
3678.216
60, 83, 110, 132, 152, 175, 198, 230
9
3686.001
56, 77, 100, 122, 138, 156, 179, 201, 231
10
3691.730
56, 76, 98, 120, 136, 153, 174, 194, 219, 243
11
3696.416
55, 74, 94, 114, 130, 142, 158, 178, 197, 222, 245
12
3699.866
54, 72, 91, 110, 126, 138, 151, 168, 185, 202, 225, 246
13
3702.619
52, 68, 83, 100, 117, 130, 140, 153, 169, 185, 202, 225, 246
14
3705.033
52, 68, 83, 100, 117, 130, 140, 152, 167, 182, 197, 215, 235, 249
15
3706.937
51, 66, 79, 94, 110, 123, 133, 142, 154, 169, 184, 199, 216, 235, 249
16
3708.484
49, 63, 75, 89, 104, 118, 129, 138, 147, 159, 173, 186, 200, 217, 235, 249
Waterfall
2
4512.801
88, 170
3
4646.137
72, 115, 182
4
4711.019
67, 103, 150, 204
5
4752.267
58, 87, 119, 164, 212
6
4777.063
53, 78, 104, 134, 176, 217
7
4793.510
48, 70, 92, 116, 147, 186, 221
8
4805.756
46, 66, 86, 107, 131, 163, 199, 226
9
4814.097
43, 61, 79, 97, 117, 141, 173, 205, 228
10
4820.724
40, 57, 73, 90, 108, 128, 152, 182, 210, 230
11
4825.739
38, 54, 69, 84, 100, 117, 137, 162, 191, 215, 232
12
4829.689
37, 53, 67, 81, 96, 111, 128, 148, 173, 198, 218, 233
13
4832.826
35, 50, 63, 76, 89, 103, 117, 133, 153, 177, 201, 220, 234
14
4835.371
32, 46, 58, 70, 82, 95, 108, 122, 138, 158, 182, 204, 221, 234
15
4837.537
32, 46, 57, 68, 80, 92, 104, 117, 131, 148, 169, 191, 210, 224, 236
16
4839.226
30, 43, 54, 64, 74, 85, 96, 108, 120, 134, 151, 172, 193, 211, 225, 236
Bird
2
901.450
71, 122
3
975.230
64, 111, 140
4
1027.509
61, 104, 131, 164
5
1051.482
54, 93, 119, 138, 169
6
1067.992
47, 82, 108, 127, 142, 172
7
1077.304
40, 70, 94, 113, 129, 143, 173
8
1086.005
39, 69, 93, 112, 128, 141, 159, 192
9
1091.341
37, 65, 88, 105, 119, 131, 142, 160, 193
10
1095.355
37, 64, 86, 103, 117, 129, 139, 149, 167, 200
11
1098.475
33, 56, 77, 93, 107, 119, 130, 140, 150, 169, 202
12
1100.749
33, 56, 77, 93, 107, 119, 129, 138, 146, 158, 178, 209
13
1102.639
31, 52, 71, 87, 100, 111, 121, 130, 139, 147, 159, 179, 210
14
1104.132
29, 48, 66, 82, 95, 107, 118, 127, 134, 141, 149, 161, 181, 211
15
1105.446
28, 45, 62, 77, 90, 101, 111, 120, 128, 135, 142, 150, 162, 182, 212
16
1106.500
28, 45, 62, 77, 90, 101, 111, 120, 128, 135, 141, 148, 157, 171, 191, 219
Police
2
3647.353
74, 150
3
3844.314
70, 135, 192
4
3966.225
63, 112, 158, 209
5
4013.875
61, 104, 140, 174, 214
6
4047.198
32, 67, 106, 141, 175, 214
7
4067.996
32, 67, 104, 133, 161, 186, 219
8
4084.933
29, 52, 78, 106, 134, 162, 187, 219
9
4094.705
29, 52, 78, 103, 125, 147, 169, 190, 220
10
4101.702
29, 52, 78, 102, 123, 143, 165, 184, 203, 228
11
4108.018
28, 49, 71, 90, 107, 126, 146, 166, 185, 204, 229
12
4112.304
28, 49, 71, 89, 105, 122, 139, 157, 174, 189, 207, 231
13
4115.165
28, 46, 62, 78, 92, 106, 123, 140, 158, 174, 189, 207, 231
14
4117.926
28, 46, 62, 78, 91, 105, 120, 135, 151, 166, 180, 193, 210, 232
15
4120.045
28, 46, 62, 78, 91, 103, 116, 129, 143, 158, 172, 185, 197, 214, 235
16
4121.861
28, 46, 62, 78, 91, 103, 116, 128, 142, 156, 170, 182, 193, 206, 223, 240
Ostrich
2
1073.452
75, 135
3
1139.260
69, 101, 149
4
1178.650
65, 92, 125, 176
5
1203.749
56, 78, 100, 131, 179
6
1218.643
47, 65, 85, 103, 133, 181
7
1228.925
47, 64, 83, 100, 122, 152, 192
8
1236.023
45, 59, 75, 90, 104, 125, 155, 194
9
1240.756
40, 52, 65, 80, 94, 107, 128, 157, 195
10
1244.909
40, 52, 64, 78, 91, 103, 119, 141, 168, 201
11
1247.879
40, 51, 62, 75, 87, 97, 108, 125, 148, 174, 205
12
1250.313
37, 48, 57, 68, 80, 91, 101, 112, 128, 150, 175, 206
13
1252.298
31, 44, 53, 63, 75, 86, 95, 105, 117, 134, 154, 178, 207
14
1253.956
29, 42, 50, 58, 68, 79, 89, 98, 108, 120, 137, 157, 181, 209
15
1255.366
29, 42, 50, 58, 67, 77, 86, 94, 102, 111, 124, 141, 160, 183, 210
16
1256.490
28, 41, 49, 57, 66, 76, 85, 93, 101, 110, 122, 137, 155, 174, 196, 219
Viaduct
2
7920.458
77, 180
3
8117.991
54, 109, 193
4
8203.807
42, 84, 131, 203
5
8246.806
35, 68, 103, 146, 210
6
8272.775
31, 59, 88, 120, 160, 216
7
8287.714
28, 53, 77, 103, 132, 169, 220
8
8298.322
27, 51, 75, 100, 128, 164, 212, 246
9
8308.240
24, 45, 66, 88, 112, 139, 172, 216, 247
10
8315.133
22, 41, 60, 79, 99, 121, 146, 178, 219, 247
11
8320.032
20, 37, 54, 71, 89, 108, 128, 152, 183, 221, 248
12
8323.799
20, 36, 52, 68, 84, 101, 119, 139, 162, 190, 224, 248
13
8326.793
18, 33, 48, 62, 77, 92, 108, 125, 144, 167, 195, 226, 248
14
8329.119
17, 31, 44, 57, 70, 84, 98, 113, 129, 148, 170, 197, 227, 248
15
8330.983
17, 31, 44, 57, 70, 84, 98, 113, 129, 147, 169, 195, 224, 243, 251
16
8332.744
16, 29, 42, 54, 66, 78, 91, 104, 118, 133, 151, 172, 196, 224, 243, 251
Fish
2
3593.389
64, 148
3
3870.456
44, 104, 177
4
3972.731
34, 81, 127, 188
5
4024.885
27, 63, 101, 139, 194
6
4054.836
24, 56, 90, 123, 156, 205
7
4075.236
22, 49, 78, 107, 135, 168, 213
8
4088.300
20, 44, 68, 93, 118, 142, 173, 216
9
4097.101
19, 40, 62, 85, 107, 128, 149, 178, 218
10
4103.194
18, 38, 58, 78, 98, 118, 137, 158, 185, 222
11
4107.954
16, 34, 51, 69, 88, 107, 125, 143, 163, 190, 224
12
4111.678
15, 31, 47, 64, 82, 100, 117, 133, 149, 169, 195, 227
13
4114.783
14, 28, 43, 58, 74, 90, 106, 121, 136, 152, 172, 198, 228
14
4116.954
14, 28, 43, 58, 73, 88, 103, 117, 131, 145, 160, 179, 203, 231
15
4118.931
13, 25, 38, 51, 64, 78, 92, 106, 120, 133, 147, 162, 181, 205, 232
16
4120.522
13, 25, 37, 49, 62, 75, 89, 102, 115, 127, 139, 152, 168, 188, 211, 235
Houses
2
2543.788
56, 116
3
2627.230
53, 105, 150
4
2663.698
42, 69, 110, 152
5
2694.902
39, 65, 96, 130, 158
6
2708.655
39, 65, 95, 127, 150, 169
7
2720.263
35, 55, 74, 98, 128, 151, 170
8
2726.676
35, 55, 74, 98, 127, 148, 164, 184
9
2732.853
32, 48, 65, 80, 101, 128, 148, 164, 184
10
2736.510
30, 45, 60, 74, 87, 106, 129, 149, 164, 184
11
2739.566
30, 45, 60, 73, 86, 105, 128, 145, 156, 168, 187
12
2742.148
29, 42, 55, 68, 79, 94, 112, 130, 145, 156, 168, 187
13
2744.071
28, 39, 51, 63, 74, 84, 98, 115, 131, 145, 156, 168, 187
14
2745.542
28, 39, 51, 63, 74, 84, 98, 115, 131, 145, 155, 165, 176, 193
15
2746.817
27, 37, 47, 58, 68, 77, 86, 100, 116, 131, 145, 155, 165, 176, 193
16
2747.991
27, 37, 47, 57, 67, 76, 85, 98, 113, 127, 139, 147, 156, 165, 176, 193
Mushroom
2
1988.328
76, 145
3
2153.037
65, 110, 174
4
2237.441
59, 93, 135, 193
5
2277.427
52, 78, 106, 145, 199
6
2301.629
48, 71, 94, 122, 157, 205
7
2317.538
46, 67, 87, 110, 138, 171, 213
8
2328.526
44, 62, 79, 97, 118, 145, 177, 216
9
2335.812
42, 58, 74, 90, 107, 127, 152, 181, 218
10
2340.981
41, 56, 70, 84, 99, 116, 136, 159, 186, 220
11
2345.160
39, 52, 65, 78, 92, 107, 124, 144, 166, 191, 223
12
2348.413
38, 51, 63, 75, 87, 100, 115, 132, 152, 174, 199, 227
13
2350.988
38, 50, 62, 74, 85, 97, 110, 125, 142, 160, 180, 203, 229
14
2353.119
37, 48, 59, 69, 79, 89, 100, 113, 127, 144, 162, 181, 204, 230
15
2354.748
36, 46, 56, 66, 75, 84, 94, 105, 117, 130, 146, 163, 182, 205, 230
16
2356.113
36, 46, 56, 65, 74, 83, 92, 102, 113, 125, 139, 154, 169, 187, 208, 232
Snow mountain
2
1912.613
85, 149
3
2135.274
79, 137, 197
4
2234.669
70, 119, 154, 204
5
2288.799
55, 96, 129, 160, 206
6
2317.224
52, 90, 118, 142, 167, 209
7
2333.078
50, 85, 111, 132, 153, 174, 212
8
2344.629
45, 76, 100, 120, 139, 158, 177, 214
9
2352.951
44, 74, 97, 116, 133, 150, 167, 187, 220
10
2359.175
41, 68, 90, 108, 124, 140, 156, 172, 193, 225
11
2364.304
38, 62, 84, 102, 117, 132, 146, 160, 175, 196, 227
12
2367.971
34, 56, 78, 96, 111, 125, 139, 152, 165, 179, 199, 228
13
2371.208
33, 53, 73, 90, 104, 117, 130, 143, 155, 167, 180, 200, 229
14
2373.567
32, 52, 72, 89, 103, 115, 127, 139, 150, 161, 172, 185, 205, 232
15
2375.586
31, 49, 68, 84, 98, 109, 120, 131, 142, 153, 164, 175, 188, 208, 233
16
2377.255
26, 42, 59, 75, 89, 101, 111, 121, 132, 143, 154, 165, 176, 189, 209, 234
Snake
2
1118.615
87, 134
3
1231.320
76, 114, 154
4
1286.555
69, 101, 129, 166
5
1317.027
63, 91, 115, 140, 175
6
1336.172
59, 84, 105, 126, 149, 182
7
1348.933
55, 78, 97, 115, 133, 155, 187
8
1357.665
52, 73, 91, 107, 123, 140, 161, 192
9
1364.159
50, 70, 87, 102, 116, 131, 148, 169, 198
10
1368.955
47, 65, 81, 95, 108, 121, 135, 151, 172, 200
11
1372.626
46, 63, 78, 91, 103, 115, 127, 140, 156, 176, 203
12
1375.513
45, 61, 75, 88, 99, 110, 121, 133, 146, 162, 182, 208
13
1377.847
43, 58, 72, 84, 95, 106, 116, 127, 138, 151, 166, 185, 211
14
1379.736
42, 56, 69, 81, 92, 102, 112, 122, 132, 143, 155, 170, 189, 214
15
1381.282
41, 54, 66, 77, 87, 97, 106, 115, 124, 134, 145, 157, 172, 191, 215
16
1382.603
40, 52, 63, 74, 84, 93, 102, 111, 120, 129, 139, 150, 162, 177, 195, 219
Riosanpablo
2
2667.020
95, 160
3
2818.660
75, 121, 177
4
2892.439
68, 102, 143, 189
5
2931.654
62, 89, 121, 158, 197
6
2957.018
58, 82, 107, 138, 171, 204
7
2973.269
53, 74, 95, 119, 147, 177, 207
8
2984.972
50, 69, 87, 108, 133, 159, 185, 212
9
2993.359
48, 66, 83, 101, 122, 145, 168, 191, 215
10
2999.615
46, 63, 78, 93, 110, 130, 151, 173, 195, 217
11
3004.374
45, 61, 75, 89, 104, 121, 140, 160, 180, 200, 220
12
3008.082
44, 59, 72, 84, 97, 112, 129, 147, 166, 185, 203, 222
13
3011.101
43, 57, 69, 81, 93, 107, 122, 138, 155, 172, 189, 206, 224
14
3013.465
41, 54, 66, 77, 88, 100, 113, 128, 144, 160, 176, 192, 208, 225
15
3015.423
40, 53, 64, 74, 84, 95, 107, 121, 135, 150, 165, 180, 195, 210, 226
16
3017.081
39, 51, 62, 72, 82, 92, 103, 115, 128, 142, 156, 170, 184, 198, 212, 227We can see from these results thatrank-DE has a higher approximated TVD growth rate than DE. O
(
β
)
R
-DE with β
≤
0.5 and O
(
rand
)
R
-DE are still better than DE. O
(
0.0
)
R
-DE produced the best result with a very flat slope and a very low y-intercept. The multithresh function yielded a higher growth rate of solution distortion and a higher y-intercept than the other algorithms. The higher growth rate of solution distortion means the quality of solution drops very fast if the number of thresholds increases. The higher y-intercept means that the solution distortion at the lowest number of thresholds is high. Thus, the global optimization algorithm is required for solving the multilevel thresholding.


## 5.1. Experimental Setup

The global multilevel thresholding problem deals with finding optimal thresholds within the range[
0
,
L
-
1
] that maximize the BCV function. The dimension of the optimization problem is the number of thresholds, n, and the search space is [
0
,
L
-
1
]
n. The parameter β of O
(
β
)
R
-DE is rndreal
[
0,1
) or is set to be one of 0.0, 0.1
,
…
,
0.9. The variation of the proposed O
(
β
)
R
-DE was implemented and compared with the existing metaheuristics that performed image thresholding, that is, PSO, DPSO, FODPSO, ABC, and several variations of DE algorithms. All the methods were programmed in Matlab R2013a and were run on a personal computer with a 3.4 GHz CPU, 8 GB RAM with Microsoft Windows 7 64-bit operating system. The experiments were conducted on 20 real images. The 19 images, namely, starfish, mountain, cactus, butterfly, circus, snow, palace, flower, wherry, waterfall, bird, police, ostrich, viaduct, fish, houses, mushroom, snow mountain, and snake, were taken from the Berkeley Segmentation Dataset and Benchmark [29]. The last image, namely, Riosanpablo, is a satellite image “New ISS Eyes see Rio San Pablo”, March 1, 2013 (http://visibleearth.nasa.gov/view.php?id=80561). Each image has a unique gray level histogram. These original images and their histograms are depicted in Figure 1. An experiment of an image with a specific number of thresholds is called a “subproblem.” The number of thresholds investigated in the experiments was 2,3
,
…
,
16. Thus, there are 20
×
15 subproblems per algorithm. Each subproblem was repeated 50 times and each time is called a run.The test images and corresponding histograms.
(a)
(b)To compare with PSO, ABC, and DEs algorithms, the objective function evaluation is computed forNP
×
N
i, where N
P is population size and N
i is the number of generations. A population of PSO and the DEs calls Otsu’s function one time per generation. The population size in the PSO and DEs algorithms was set to 50. A bee in the ABC calls Otsu’s function two times per generation; therefore their number of food sources were set to a half of the PSO’s size, that is, 25. The stopping criteria were set by the maximum amount of generations G. In this experiment, G was set to 50, 100, 150, 200, 300, 400, 600, 800, 1000, 1500, 2000, 3000, 4000, 5000, and 6000 when n was 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, and 16, respectively. For the PSO, DPSO, and FODPSO algorithms, the parameters were set as per the suggestion in [30] and is shown in Table 1. The other control parameter of the ABC algorithm,limit, was set to 50 [15]. The control parameters F and CR of the DE algorithms were set to 0.5 and 0.9, respectively [31, 32].Table 1
Essential parameters of the PSO, DPSO, and FODPSO taken from [30].
Parameter
PSO
DPSO
FODPSO
Population
50
50
50
ρ
1
1.5
1.5
1.5
ρ
2
1.5
1.5
1.5
W
1.2
1.2
1.2
V
max
⁡
2
2
2
V
min
⁡
−2
−2
−2
x
max
⁡
255
255
255
x
min
⁡
0
0
0
Min population
—
10
10
Max population
—
50
50
No. of swarms
—
4
4
Min swarms
—
2
2
Max swarms
—
6
6
Stagnancy
—
10
10
Fractional coefficient
—
—
0.75


## 5.2. Comparison Strategies and Metrics

To compare the performance of different algorithms, there are three metrics:(
1
) the convergence rate of algorithms was compared by the average of generations (NG
¯), a lower NG
¯ means a faster convergence rate; (
2
) the stability of algorithms was compared by the average of the success rate, (SRHM), a higher SRHM means higher stability; (
3
) the reliability was compared by the threshold value distortion measure (TVD), a lower TVD means higher reliability. The details of the three metrics are described as follows.When all 50 runs of an algorithm performing on an image with a specific number of thresholds are terminated, the outcomes will be analyzed. Runr’th is called a successful run if there is a generation of t
≤
G such that BCV
r
(
t
)
≥
VTR and the number of generations (NG) of the successful run is recorded. Thus, the number can be defined by
(16)
NG
r
=
ArgMin
t
(
BCV
r
(
t
)
≥
VTR
)
,
if
r
is
a
successful
run
and
otherwise
undefined
.
The average of N
G
r from those successful runs is represented by NG
¯ as follows:
(17)
NG
¯
=
1
number
of
successful
runs
∑
All
successful
runs
NG
r
.
The ratio of success rate (SR) for which the algorithm succeeds to reach the VTR for each subproblem is computed as
(18)
SR
=
number
of
successful
runs
total
number
of
runs
.
The experiments were conducted on 20 images. The arithmetic mean (AM) of NG
¯ (NG
¯
AM) over the entire set of images with a specific number of thresholds is calculated as
(19)
NG
¯
AM
=
1
N
∑
All
images
NG
¯
,
where N is the total number of images. NG
¯
AM is shown in Table 3. The worst-case scenario is that there is no successful run for a subproblem; this subproblem is called an “unsuccessful subproblem.” If an algorithm encounters this scenario, the subproblem will be grouped by its number of thresholds and the number of images in the group will be counted and assigned to x. These scenarios will be represented by NA
(
x
), as shown in Tables 3 and 4.The average of the success rate over the entire dataset with a specific number of thresholds (S
R
HM) is averaged by the Harmonic mean, HM, as follows:
(20)
S
R
HM
=
N
∑
All
images
(
1
/
SR
)
.TheS
R
HM is very important in measuring the stability of an algorithm and it means the ratio of runs that are achieving the target solution. Because the evolutionary methods are based on stochastic searching algorithms, the solutions are not the same in each run of the algorithm and depend on the search ability of the algorithm. Therefore, the S
R
HM is vital in evaluating the stability of the algorithms. The comparison of the stability gives us valuable information in terms of the ratio representing the success rates (S
R
HM). A higher S
R
HM means better stability of the algorithm.An algorithm producingS
R
HM
<
0.5 means that more than 50 percent of the independent runs of the algorithm cannot reach the global solution. Thus, the algorithm that yields S
R
HM
<
0.5 should not be selected to solve the problem. The experiments were conducted for the number of thresholds varying from 2 to 16. These experiments contained the maximum number of thresholds such that the algorithm yields S
R
HM
≥
0.5, which is represented by n
0.5 in Table 4. Furthermore, the experiments also contained the maximum number of thresholds that the algorithm can solve; and above this value there was the case such that all 50 runs of some subproblems missed the VTR. This number is represented by n
max
⁡. In this case the success rate was zero and the associated S
R
HM was zero too. And the definitions of the two values are presented in (21)
(21)
n
0.5
=
max
⁡
(
{
S
R
HM
n
∣
n
=
number
of
thresholds
that
has
S
R
HM
≥
0.5
S
R
HM
}
)
n
max
⁡
=
min
⁡
(
{
S
R
HM
n
∣
n
=
number
of
thresholds
that
has
S
R
HM
=
0
S
R
HM
}
)
-
1
.Letn be the number of thresholds. The reliability of a solution is measured by threshold value distortion measure (TVD) and is computed as
(22)
TVD
=
∑
All
images
∑
r
=
1
run
∑
i
=
1
n
|
T
r
i
*
-
T
r
i
m
|
1
+
∑
All
images
∑
r
=
1
run
∑
i
=
1
n
1
{
T
r
i
*
≠
T
r
i
m
}
×
(
1
-
SR
)
×
100
,
where T
* is the threshold value producing the VTR, T
m is the threshold value obtained from the algorithm, and 1
{
T
r
i
*
≠
T
r
i
m
} is the indicator function, which is equal to 1 when T
r
i
*
≠
T
r
i
m and is zero otherwise. TVD is zero if the algorithm can reach the VTR in every run. The lower the TVD the more reliable the algorithm is.


### 5.2.1. The Value to Reach (VTR)

Following the completion of all of the experiments the best values of the between-class variance and the corresponding thresholds were collected and are shown in Table6. The results are shown image by image and the numbers of thresholds vary from 2 to 16. The between-class variance values in column 3 are used as the VTR values.


### 5.2.2. Results Produced by Local Search Method

The multithresh function of the Matlab toolbox was conducted on the same images and number of thresholds as the other search methods. The capabilities of solving the optimal solution between a local search and a global search will be discussed here. This is the reason we focused on the global search, that is, the proposedO
(
0.0
)
R
-DE algorithm. Table 2 shows the between-class variances and threshold values of the “mountain” image. These values were the best outcomes of 50 runs produced by the multithresh function in the Matlab R2013a toolbox and by the proposed O
(
0.0
)
R
-DE algorithm. The terminated condition of the multithresh function was set by “MaxFunEvals” = 500000. That is the multithresh function performs more function calls than that of the O
(
0.0
)
R
-DE algorithm. It can be seen from columns 3 and 5 that all the BCVs produced by the O
(
0.0
)
R
-DE algorithm are better than the BCVs produced by the multithresh function; the difference of the BCVs is shown in column 7. The differences in the thresholds from the two algorithms, shown in column 8, tended to be large if the number of thresholds increased.Table 2
The best values of the between-class variance and thresholds of the “mountain” image produced by the multithresh function of the Matlab toolbox and by the proposedO(0.0)R-DE algorithm. The number of thresholds varies from 2 to 16.
Image(1)
n  (2)
Produced by multithresh
Produced by theO(0.0)R-DE algorithm
Objdiff
(
7
)
=
(
5
)
-
(
3
)
Threshdiff
(
8
)
=
∑
|
(
4
)
-
(
6
)
|
Between-class variance(3)
Thresholds(4)
Between-class variance(5)
Thresholds(6)
Mountain
2
2372.886
60, 127
2372.923
61, 128
0.037
2
3
2495.852
32, 76, 130
2496.113
33, 77, 131
0.261
3
4
2551.778
32, 72, 109, 145
2551.955
33, 73, 109, 147
0.177
4
5
2580.154
31, 68, 98, 124, 158
2580.336
32, 69, 99, 125, 159
0.182
5
6
2588.264
35, 75, 98, 118, 152, 175
2596.956
24, 46, 74, 101, 126, 160
8.692
122
7
2598.800
33, 70, 98, 118, 141, 166, 207
2608.807
24, 46, 73, 98, 119, 145, 175
10.007
153
8
2605.785
31, 68, 91, 105, 121, 140, 163, 192
2616.294
24, 46, 73, 97, 115, 135, 160, 191
10.509
70
9
2619.512
27, 52, 75, 95, 111, 129, 148, 168, 197
2622.314
20, 37, 54, 76, 98, 116, 136, 160, 191
2.802
114
10
2620.392
23, 46, 73, 97, 116, 135, 157, 175, 204, 232
2627.194
20, 36, 53, 74, 93, 106, 121, 140, 163, 194
6.802
258
11
2625.275
23, 45, 72, 93, 105, 119, 137, 156, 175, 204, 228
2630.496
20, 36, 53, 74, 93, 105, 118, 134, 152, 172, 201
5.221
199
12
2629.641
22, 39, 58, 78, 95, 109, 123, 139, 156, 177, 207, 247
2633.189
18, 31, 45, 59, 76, 93, 105, 118, 134, 152, 172, 201
3.548
246
13
2631.372
20, 38, 54, 74, 94, 107, 121, 139, 157, 173, 193, 224, 248
2635.290
18, 31, 45, 59, 76, 92, 103, 115, 129, 144, 161, 179, 207
3.918
283
14
2633.865
19, 36, 53, 74, 91, 103, 115, 129, 144, 160, 174, 190, 217, 245
2637.088
17, 29, 42, 56, 72, 86, 96, 106, 117, 130, 145, 161, 179, 207
3.223
307
15
2635.190
19, 33, 50, 69, 85, 96, 107, 121, 137, 152, 167, 178, 190, 212, 240
2638.449
17, 28, 39, 50, 61, 75, 88, 97, 107, 118, 131, 146, 162, 179, 207
3.259
351
16
2636.357
16, 30, 45, 59, 77, 93, 105, 118, 133, 148, 162, 176, 192, 209, 230, 244
2639.616
17, 27, 38, 49, 60, 74, 86, 95, 104, 113, 123, 135, 149, 164, 181, 209
3.259
415Table 3
The average of mean number of generation (NG
¯
AM) and ranks of the methods.
No.maxGen.
FODPSO
DPSO
PSO
ABC
Rank-DE
DE
O(rand)R-DE
O(0.9)R-DE
O(0.8)R-DE
O(0.7)R-DE
O(0.6)R-DE
O(0.5)R-DE
O(0.4)R-DE
O(0.3)R-DE
O(0.2)R-DE
O(0.1)R-DE
O(0.0)R-DE
n
=
2
G
=
50
NG
¯
AM
18.337
15.713
13.218
13.202
10.538
14.713
10.117
6.394
6.903
7.636
8.208
9.147
9.261
9.901
10.709
11.631
20.850
Rank
16
15
13
12
9
14
8
1
2
3
4
5
6
7
10
11
17
n
=
3
G
=
100
NG
¯
AM
35.200
36.210
27.479
33.210
19.488
29.049
18.958
10.492
12.296
13.587
14.807
16.969
17.107
18.100
19.546
21.749
44.192
Rank
15
16
12
14
9
13
8
1
2
3
4
5
6
7
10
11
17
n
=
4
G
=
150
NG
¯
AM
45.391
58.683
49.596
63.251
31.060
48.130
29.595
14.733
18.526
20.885
22.815
26.632
26.456
28.058
30.175
33.499
72.100
Rank
12
15
14
16
10
13
8
1
2
3
4
6
5
7
9
11
17
n
=
5
G
=
200
NG
¯
AM
55.664
81.877
87.441
108.131
44.157
69.088
41.283
19.508
24.524
28.097
30.963
36.935
36.748
38.647
41.455
45.631
100.693
Rank
12
14
15
17
10
13
8
1
2
3
4
6
5
7
9
11
16
n
=
6
G
=
300
NG
¯
AM
NA(2)
100.664
141.937
157.328
61.935
99.700
56.405
24.623
32.687
38.167
42.608
51.003
50.281
52.089
55.729
61.748
138.716
Rank
17
13
15
16
11
12
9
1
2
3
4
6
5
7
8
10
14
n
=
7
G
=
400
NG
¯
AM
NA(1)
NA(1)
192.267
236.928
86.938
142.096
75.340
29.763
41.691
49.408
56.193
67.619
67.063
69.696
74.389
83.326
184.371
Rank
17
16
14
15
11
12
9
1
2
3
4
6
5
7
8
10
13
n
=
8
G
=
600
NG
¯
AM
NA(7)
NA(6)
NA(2)
NA(1)
117.092
198.986
100.944
33.967
51.573
62.263
74.523
91.836
89.666
95.198
102.552
115.583
244.880
Rank
17
16
15
14
11
12
8
1
2
3
4
6
5
7
9
10
13
n
=
9
G
=
800
NG
¯
AM
NA(11)
NA(13)
NA(12)
NA(3)
151.004
264.253
125.299
41.228
63.742
79.432
90.201
113.530
110.416
114.397
122.555
137.538
299.492
Rank
17
16
15
14
11
12
9
1
2
3
4
6
5
7
8
10
13
n
=
10
G
=
1000
NG
¯
AM
NA(11)
NA(16)
NA(16)
NA(7)
NA(1)
321.373
147.770
NA(3)
73.564
91.121
105.020
132.968
128.999
132.691
141.825
157.997
356.546
Rank
17
16
15
14
12
10
8
13
1
2
3
6
4
5
7
9
11
n
=
11
G
=
1500
NG
¯
AM
NA(16)
NA(17)
NA(19)
NA(9)
NA(1)
405.090
185.020
NA(3)
NA(2)
NA(2)
127.135
162.822
157.184
160.123
175.152
198.522
434.877
Rank
17
16
15
14
12
8
6
13
10
10
1
4
2
3
5
7
9
n
=
12
G
=
2000
NG
¯
AM
NA(18)
NA(20)
NA(20)
NA(9)
278.743
535.542
229.965
NA(4)
NA(5)
135.266
160.047
205.266
197.386
203.814
222.094
251.535
574.104
Rank
17
16
15
14
12
8
6
13
11
10
1
4
2
3
5
7
9
n
=
13
G
=
3000
NG
¯
AM
NA(20)
NA(19)
NA(20)
NA(14)
NA(3)
NA(3)
275.860
NA(12)
NA(5)
NA(1)
NA(1)
NA(1)
NA(1)
261.522
278.082
316.468
726.521
Rank
17
16
15
14
12
9
2
13
11
10
6
6
6
1
3
4
5
n
=
14
G
=
4000
NG
¯
AM
NA(20)
NA(20)
NA(20)
NA(14)
NA(4)
NA(4)
330.194
NA(11)
NA(8)
NA(4)
NA(1)
NA(2)
296.414
NA(1)
331.626
385.951
889.711
Rank
17
16
15
14
12
9
1
13
11
10
7
8
6
5
2
3
4
n
=
15
G
=
5000
NG
¯
AM
NA(20)
NA(20)
NA(20)
NA(19)
NA(13)
NA(5)
399.228
NA(11)
NA(10)
NA(8)
NA(6)
363.506
362.665
383.788
417.958
481.830
1086.001
Rank
17
16
15
14
13
8
1
12
11
10
9
7
6
5
2
3
4
n
=
16
G
=
6000
NG
¯
AM
NA(20)
NA(20)
NA(20)
NA(18)
NA(10)
NA(8)
NA(4)
NA(18)
NA(11)
NA(9)
NA(5)
NA(1)
NA(2)
447.570
491.772
552.856
1374.438
Rank
17
16
15
14
12
9
4
13
11
10
8
6
7
5
1
2
3
Avg for
n
=
2 to 16
n
max
⁡,NA(x)
NG
¯
AM
5,NA(146), 38.648
6,NA(152), 58.629
7,NA(149), 85.323
7,NA(94), 102.008
9,NA(32), 88.995
12,NA(20), 193.456
15,NA(4), 144.713
9,NA(62), 22.589
10,NA(41), 36.167
10,NA(24), 52.586
12,NA(13), 66.593
12,NA(4), 106.519
12,NA(3), 119.204
13,NA(1), 143.971
16,NA(0), 167.708
16,NA(0), 190.391
16,NA(0), 436.499
Rank
17
16
15
14
12
9
4
13
11
10
8
7
6
5
1
2
3Table 4
The average success rate (SRHM) and ranks of the methods.
No
Multi-thresh
FODPSO
DPSO
PSO
ABC
Rank-DE
DE
O(rand)R-DE
O(0.9)R-DE
O(0.8)R-DE
O(0.7)R-DE
O(0.6)R-DE
O(0.5)R-DE
O(0.4)R-DE
O(0.3)R-DE
O(0.2)R-DE
O(0.1)R-DE
O(0.0)R-DE
2
SRHM
0
0.977
1
1
1
1
1
1
0.999
1
1
1
1
1
1
1
1
1
Rank
18
17
1
1
1
1
1
1
16
1
1
1
1
1
1
1
1
1
3
SRHM
0.000
0.635
0.972
0.991
0.979
0.998
1.000
0.997
0.986
0.993
0.999
0.997
0.998
0.999
1.000
1.000
1.000
1.000
Rank
18
17
16
13
15
8
1
10
14
12
6
10
8
6
1
1
1
1
4
SRHM
0.000
0.379
0.742
0.934
0.833
0.987
0.995
0.986
0.897
0.945
0.972
0.971
0.986
0.993
0.992
0.991
0.998
0.997
Rank
18
17
16
13
15
7
3
8
14
12
10
11
8
4
5
6
1
2
5
SRHM
0.000
0.185
0.391
0.767
0.706
0.936
0.977
0.960
0.782
0.878
0.874
0.944
0.950
0.959
0.968
0.960
0.985
1.000
Rank
18
17
16
14
15
10
3
5
13
11
12
9
8
7
4
5
2
1
6
SRHM
0.000
0.000
0.187
0.433
0.453
0.863
0.968
0.900
0.601
0.706
0.826
0.878
0.934
0.922
0.944
0.935
0.970
0.999
Rank
18
17
16
15
14
10
3
8
13
12
11
9
6
7
4
5
2
1
7
SRHM
0.000
0.000
0.000
0.153
0.164
0.696
0.884
0.835
0.366
0.561
0.675
0.741
0.840
0.851
0.883
0.914
0.949
0.998
Rank
18
17
16
15
14
10
4
8
13
12
11
9
7
6
5
3
2
1
8
SRHM
0.000
0.000
0.000
0.000
0.000
0.300
0.458
0.530
0.105
0.256
0.351
0.502
0.592
0.628
0.724
0.757
0.838
0.972
Rank
18
17
16
14
14
11
9
7
13
12
10
8
6
5
4
3
2
1
9
SRHM
0.000
0.000
0.000
0.000
0.000
0.495
0.695
0.681
0.083
0.274
0.372
0.548
0.721
0.725
0.769
0.815
0.852
0.999
Rank
18
17
16
14
14
10
7
8
13
12
11
9
6
5
4
3
2
1
10
SRHM
0.000
0.000
0.000
0.000
0.000
0.000
0.381
0.375
0.000
0.073
0.218
0.311
0.409
0.370
0.460
0.578
0.686
0.998
Rank
18
17
16
14
14
12
6
7
13
11
10
9
5
8
4
3
2
1
11
SRHM
0.000
0.000
0.000
0.000
0.000
0.000
0.271
0.267
0.000
0.000
0.000
0.144
0.407
0.316
0.424
0.584
0.704
0.999
Rank
18
17
16
14
14
12
7
8
13
10
10
9
5
6
4
3
2
1
12
SRHM
0.000
0.000
0.000
0.000
0.000
0.090
0.270
0.423
0.000
0.000
0.089
0.177
0.400
0.407
0.501
0.641
0.810
1.000
Rank
18
17
16
14
14
12
8
5
13
11
10
9
7
6
4
3
2
1
13
SRHM
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.151
0.000
0.000
0.000
0.000
0.000
0.000
0.254
0.267
0.381
0.974
Rank
18
17
16
14
14
12
6
5
13
10
10
6
6
6
4
3
2
1
14
SRHM
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.102
0.000
0.000
0.000
0.000
0.000
0.110
0.000
0.253
0.494
0.993
Rank
18
17
16
14
14
12
7
4
13
10
10
7
7
6
5
3
2
1
15
SRHM
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.090
0.000
0.000
0.000
0.000
0.095
0.117
0.149
0.288
0.466
1.000
Rank
18
17
16
14
14
12
8
4
13
10
10
8
7
6
5
3
2
1
16
SRHM
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.110
0.192
0.288
0.977
Rank
18
17
16
14
14
12
6
4
13
10
10
6
6
6
5
3
2
1
Avg. for
n
=
2 to 16
n
0.5,
n
max
⁡,  SRHM
<2,<2, 0.000
3,5, 0.376
4,6, 0.443
5,7, 0.454
5,7, 0.464
7,9, 0.665
9,12, 0.554
9,15, 0.304
6,9, 0.263
7,10, 0.313
7,10, 0.530
9,12, 0.420
9,12, 0.654
9,12, 0.625
9,13, 0.619
12,16, 0.498
12,16, 0.656
16,16, 0.994
Rank
18
17
16
15
14
12
8
4
13
11
10
9
6
7
5
3
2
1Figure2 shows the graph of the TVD of all the images and thresholds. These results are in the same pattern of the results of the “mountain” image in Table 2. That means the ability to search for the optimal solution of the proposed global search algorithm is higher than that of the multithresh function, especially when the number of thresholds is large. This goes to illustrate the difficulty of the problem. The problem with this kind is that it can be multimodal [33] or can be a nearly flat top surface [34]. The multithresh function solves the problem by performing the Nelder-Mead Simplex method [35], which is a local search method that cannot guarantee an optimal solution. Thus, its solutions are inferior to the solution produced by the algorithm using a global search.Figure 2
The threshold value distortion (TVD) of algorithms versus number of thresholds.


### 5.2.3. Convergence Rate Comparison

The number of generations (NG) is a measure used for the convergence rate comparisons. If the target value, VTR, is achieved in a lesser number of generations (NG), it means a faster convergence rate for the algorithm. Table3 shows the average of NG
¯ (NG
¯
AM) for each specific number of thresholds. The results of each algorithm are represented in the corresponding column’s name. In each column, the cell containing NG
¯
AM starts from the row associated with n
=
2 until the row associated with n
=
n
max
⁡. The cells associated with n
=
n
max
⁡
+
1 to the row associated with n
=
16 are filled by NA
(
x
). The second last row of the table is filled by the triple:
(23)
(
(
NG
¯
AM
of
n
=
2
until
n
=
n
max
⁡
)
n
max
⁡
,
NA
(
number
of
unsuccessful
subproblem
)
,
AM
(
NG
¯
AM
of
n
=
2
until
n
=
n
max
⁡
)
)
.The algorithm with the highestn
max
⁡, that is, the lowest number of unsuccessful subproblems and the lowest average of generation is the winner. The ranking of the algorithms depends on the ordering of (a
1, b
1, c
1) and (a
2, b
2, c
2) as follows.First, rank ona
1 and a
2. Since both a
1 and a
2 are numeric, the higher value has the higher rank.
Second, rank onb
1 and b2. If a
1 is 16, then b
1 must be NA(0). If a
1 is less than 16, then b
1 must be NA (number of unsuccessful subproblems) and b
2 has the same characteristics as b
1. Perform the order of the two numeric values in reverse; the lower value has the higher rank.
Third, rank on c1 and c
2. They are numeric but the lower is better; perform the order of the two numeric values in reverse. The lower value has the higher rank.When the ordering is finished, assign the numeric value of “1” to the object having the highest rank, assign the numeric value of “2” to the first runner up, and so on. These values are represented in the last row of the table.If the row havingn
=
m must be ranked, it can be done in the same manner as above with some minor modifications. If m is less than n
max
⁡, the values of the triple pair will be m, NA
(
0
), NG
¯
AM. If m is greater than n
max
⁡, the values of the triple pair will be (n
max
⁡, NA
(
x
), ∞). Thus the ranking can now be performed.From the ranking results, see the second last row of Table3; the convergence rate can be ranked from best to worst in the following order: O
(
0.2
)
R
-DE, O
(
0.1
)
R
-DE, O
(
0.0
)
R
-DE, O
(
rand
)
R
-DE, O
(
0.3
)
R
-DE, O
(
0.4
)
R
-DE, O
(
0.5
)
R
-DE, O
(
0.6
)
R
-DE, DE, O
(
0.7
)
R
-DE, O
(
0.8
)
R
-DE, rank-DE, O
(
0.9
)
R
-DE, ABC, PSO, DPSO, FODPSO.As can be seen from Table3, the DE algorithm cannot complete the task when n
>
12 and the rank-DE algorithm cannot complete the task when n
>
9. Thus, rank-DE cannot compete with DE on searching for global multilevel thresholding.In order forO
(
β
)
R
-DE to outperform DE then β must be in the range of [
0.1
,
0.6
] or set to rndreal
[
0,1
).


### 5.2.4. Stability Analysis

The harmonic mean of the success rate (S
R
HM) for each specific number of thresholds was computed and is presented in Table 4. The results of each algorithm are represented in the corresponding column’s name. The second row from the bottom shows the harmonic mean of the success rate of each algorithm for all threshold levels.In each column, the cells containingS
R
HM start from the row associated with n
=
2 to the row associated with n
=
n
0.5. The cells from the row associated with n
=
n
0.5
+
1 to the row associated with n
=
n
max
⁡ are filled by S
R
HM. The cells from the row associated with n
=
n
max
⁡
+
1 to the row associated with n
=
16 are the cells that have S
R
HM; these cells will be excluded from the comparison. The second last row of the table is filled with the triple:
(24)
(
n
0.5
,
n
max
⁡
,
HM
(
S
R
HM
of
n
=
2
until
n
=
n
max
⁡
)
)
.The algorithm with the highestn
0.5, the highest n
max
⁡, and the highest average success rate is the winner. The ranking of the algorithms depends on the ordering of (a1, b
1, c
1) and (a
2, b2, c
2) as follows.First, rank ona
1 and a2.
Second, rank onb
1 and b
2.
Third, rank onc
1 and c
2.Because they are numeric the higher value has the higher rank. When the ordering is finished, the numeric value of “1” is assigned to the object having the highest rank, the numeric value of “2” is assigned to the first runner up, and so on.If the row havingn
=
m must be ranked, it can be done in the same manner as above with some minor modifications. If m is less than or equal to n
0.5, the values of the triple pair will be (m, m, S
R
HM). If n
0.5
<
m
≤
n
max
⁡, then the values of the triple pair will be (n
0.5, m, S
R
HM). If m is greater than n
max
⁡, the values of the triple pair will be (n
0.5, n
max
⁡, S
R
HM). Thus, the ranking can now be performed.From the ranking results, see Table4, the success rate can be ranked from best to worst in the following order: O
(
0.0
)
R
-DE, O
(
0.1
)
R
-DE, O
(
0.2
)
R
-DE, O
(
rand
)
R
-DE, O
(
0.3
)
R
-DE, O
(
0.5
)
R
-DE, O
(
0.4
)
R
-DE, DE, O
(
0.6
)
R
-DE, O
(
0.7
)
R
-DE, O
(
0.8
)
R
-DE, rank-DE, O
(
0.9
)
R
-DE, ABC, PSO, DPSO, FODPSO, multithresh.As can be seen from Table4, the DE algorithm has an S
R
HM
≥
0.5 until n
=
9 and the rank-DE algorithm has an S
R
HM
≥
0.5 until n
=
7. This result confirms that rank-DE cannot compete with DE on searching for global multilevel thresholding. If the correct β is selected, the proposed algorithm can work very well. For β
≤
0.5, O
(
β
)
R
-DE has a higher rank than DE. The multithresh function cannot compete with any of the other algorithms. It can also be seen that the proposed O
(
0.0
)
R
-DE algorithm has the best stability because its S
R
HM is greater than 0.5 when n
=
2 to 16.


### 5.2.5. Reliability Comparison

The threshold value distortion a.k.a.TVD for each specific threshold is computed, shown in Table 5 and depicted in Figure 2. The results of each algorithm are illustrated in the corresponding column’s name. The second last row of Table 5 shows the slope or the approximated growth rate, δ, of the TVD of each algorithm for all threshold levels. The δ is the slope of the robust linear regression computed by the Matlab function “robustfit.” The lower slope exhibits the better reliability. The δ of each algorithm is sorted in descending order. From the results, the reliability can be ranked from best to worst in the following order: O
(
0.0
)
R
-DE, O
(
0.1
)
R
-DE, O
(
0.2
)
R
-DE, O
(
0.3
)
R
-DE, O
(
0.4
)
R
-DE, O
(
0.5
)
R
-DE, O
(
rand
)
R
-DE, DE, O
(
0.6
)
R
-DE, ABC, rank-DE, O
(
0.7
)
R
-DE, O
(
0.8
)
R
-DE, O
(
0.9
)
R
-DE, DPSO, FODPSO, PSO, multithresh.Table 5
The average of threshold value distortion (TVD) and ranks of the methods.
No
Multi-thresh
FODPSO
DPSO
PSO
ABC
Rank-DE
DE
O(rand)R-DE
O(0.9)R-DE
O(0.8)R-DE
O(0.7)R-DE
O(0.6)R-DE
O(0.5)R-DE
O(0.4)R-DE
O(0.3)R-DE
O(0.2)R-DE
O(0.1)R-DE
O(0.0)R-DE
2
TVD
74.153
2.217
0.000
0.000
0.000
0.000
0.000
0.000
0.050
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
Rank
18
17
1
1
1
1
1
1
16
1
1
1
1
1
1
1
1
1
3
TVD
163.333
41.753
2.968
7.424
1.510
0.150
0.000
0.270
1.004
0.519
0.075
0.246
0.171
0.075
0.000
0.000
0.000
0.000
Rank
18
17
15
16
14
8
1
11
13
12
6
10
9
6
1
1
1
1
4
TVD
257.083
98.848
34.770
36.891
8.716
7.340
1.428
2.184
25.499
13.609
7.318
7.360
7.398
4.303
1.672
0.728
0.130
0.257
Rank
18
17
15
16
12
9
4
6
14
13
8
10
11
7
5
3
1
2
5
TVD
516.582
148.344
97.216
62.319
21.565
5.373
1.864
3.241
20.441
10.064
10.422
4.584
3.904
3.289
2.441
3.385
1.195
0.000
Rank
18
17
16
15
14
10
3
5
13
11
12
9
8
6
4
7
2
1
6
TVD
534.226
219.437
187.620
166.775
45.728
19.536
2.528
7.761
89.938
43.666
30.923
13.010
6.524
6.023
4.136
6.129
2.421
0.050
Rank
18
17
16
15
13
10
3
8
14
12
11
9
7
5
4
6
2
1
7
TVD
599.250
265.029
242.194
285.862
66.259
58.877
27.976
35.481
125.040
75.011
49.851
44.610
28.039
29.841
26.075
17.950
4.776
0.150
Rank
18
16
15
17
12
11
5
8
14
13
10
9
6
7
4
3
2
1
8
TVD
475.458
382.076
380.124
309.635
72.763
120.734
90.287
89.406
218.266
143.225
130.898
99.991
76.353
82.886
57.251
57.665
26.822
2.103
Rank
18
17
16
15
5
11
9
8
14
13
12
10
6
7
3
4
2
1
9
TVD
862.458
394.744
375.229
410.710
113.892
82.099
44.407
46.539
250.470
154.662
96.417
68.197
44.223
35.333
26.627
23.406
12.757
0.780
Rank
18
16
15
17
12
10
7
8
14
13
11
9
6
5
4
3
2
1
10
TVD
847.703
458.536
445.765
498.328
117.664
84.541
43.991
48.474
249.147
136.935
93.935
72.288
43.883
44.110
36.700
27.978
20.213
0.164
Rank
18
16
15
17
12
10
6
8
14
13
11
9
5
7
4
3
2
1
11
TVD
977.976
536.258
528.329
531.211
137.941
87.267
44.944
44.973
296.513
152.195
104.092
75.115
44.461
47.564
34.944
26.231
18.782
0.083
Rank
18
17
15
16
12
10
6
7
14
13
11
9
5
8
4
3
2
1
12
TVD
1070.300
592.750
589.931
605.280
157.169
174.580
109.138
101.282
388.491
258.312
196.098
150.932
101.112
82.365
79.966
51.281
23.429
0.000
Rank
18
16
15
17
10
11
8
7
14
13
12
9
6
5
4
3
2
1
13
TVD
1142.808
567.442
579.495
711.382
183.451
204.665
147.251
123.438
466.508
299.040
216.656
178.394
114.244
114.491
96.104
81.076
52.578
1.975
Rank
18
15
16
17
10
11
8
7
14
13
12
9
5
6
4
3
2
1
14
TVD
1148.641
632.482
660.908
770.319
205.714
218.408
141.431
116.716
524.783
325.936
241.997
177.121
113.426
104.073
88.593
65.908
39.647
0.564
Rank
18
15
16
17
10
11
8
7
14
13
12
9
6
5
4
3
2
1
15
TVD
1245.529
707.835
722.126
816.158
256.477
307.873
219.556
182.464
605.623
428.370
343.439
261.522
174.700
166.309
138.710
100.296
62.940
0.000
Rank
18
15
16
17
9
11
8
7
14
13
12
10
6
5
4
3
2
1
16
TVD
1182.989
768.714
835.747
828.004
249.042
279.753
211.614
195.676
605.790
416.711
311.314
251.408
174.118
167.316
133.922
113.776
76.661
2.539
Rank
18
15
17
16
9
11
8
7
14
13
12
10
6
5
4
3
2
1
Avg. for
n
=
2  to 16
TVD
739.899
387.764
378.828
402.687
109.193
110.080
72.428
66.527
257.838
163.884
122.229
93.652
62.170
59.199
48.476
38.387
22.823
0.578
Growth rate (δ)
86.86
55.25
61.13
67.98
19.25
21.76
15.61
13.48
48.43
32.25
23.91
18.93
12.52
11.86
9.99
7.69
4.94
0.09
Rank
18
15
16
17
10
11
8
7
14
13
12
9
6
5
4
3
2
1Table 6
The between-class variance criterion and the best thresholds for test images.
Image
n
Between-class variance (VTR)
Thresholds
Starfish
2
2546.885
85, 157
3
2779.925
68, 119, 177
4
2865.707
60, 101, 138, 187
5
2912.859
52, 86, 117, 150, 194
6
2941.728
47, 77, 105, 132, 162, 201
7
2960.158
44, 71, 95, 118, 142, 170, 206
8
2972.356
43, 68, 90, 110, 131, 153, 180, 212
9
2981.138
38, 58, 78, 97, 116, 136, 157, 183, 214
10
2988.206
37, 56, 75, 93, 110, 128, 146, 167, 192, 219
11
2993.348
35, 53, 71, 88, 103, 119, 135, 152, 172, 196, 221
12
2997.352
34, 51, 68, 84, 98, 112, 127, 142, 158, 177, 200, 223
13
3000.480
33, 48, 64, 79, 93, 106, 119, 133, 147, 162, 181, 203, 225
14
3003.076
32, 47, 62, 76, 89, 101, 114, 127, 140, 154, 169, 187, 207, 227
15
3005.235
30, 43, 56, 70, 83, 95, 107, 119, 131, 143, 156, 171, 189, 209, 228
16
3007.060
30, 42, 55, 68, 80, 92, 103, 114, 126, 138, 150, 163, 178, 195, 213, 230
Mountain
2
2372.923
61, 128
3
2496.113
33, 77, 131
4
2551.955
33, 73, 109, 147
5
2580.336
32, 69, 99, 125, 159
6
2596.956
24, 46, 74, 101, 126, 160
7
2608.807
24, 46, 73, 98, 119, 145, 175
8
2616.294
24, 46, 73, 97, 115, 135, 160, 191
9
2622.314
20, 37, 54, 76, 98, 116, 136, 160, 191
10
2627.194
20, 36, 53, 74, 93, 106, 121, 140, 163, 194
11
2630.496
20, 36, 53, 74, 93, 105, 118, 134, 152, 172, 201
12
2633.189
18, 31, 45, 59, 76, 93, 105, 118, 134, 152, 172, 201
13
2635.290
18, 31, 45, 59, 76, 92, 103, 115, 129, 144, 161, 179, 207
14
2637.088
17, 29, 42, 56, 72, 86, 96, 106, 117, 130, 145, 161, 179, 207
15
2638.449
17, 28, 39, 50, 61, 75, 88, 97, 107, 118, 131, 146, 162, 179, 207
16
2639.616
17, 27, 38, 49, 60, 74, 86, 95, 104, 113, 123, 135, 149, 164, 181, 209
Cactus
2
1816.448
73, 151
3
1970.112
64, 106, 173
4
2042.275
55, 87, 125, 187
5
2080.884
49, 76, 102, 138, 196
6
2104.147
46, 70, 93, 119, 157, 208
7
2119.670
44, 65, 85, 105, 131, 168, 215
8
2129.358
42, 60, 77, 94, 113, 138, 174, 218
9
2136.162
41, 58, 74, 89, 105, 125, 152, 186, 224
10
2141.579
40, 56, 71, 85, 99, 115, 135, 161, 193, 228
11
2145.397
39, 53, 66, 79, 91, 104, 119, 139, 165, 196, 229
12
2148.282
38, 51, 64, 76, 88, 100, 114, 131, 152, 176, 204, 233
13
2150.740
37, 49, 61, 72, 83, 94, 105, 118, 135, 155, 178, 205, 233
14
2152.626
36, 47, 58, 68, 78, 88, 98, 109, 122, 138, 158, 181, 207, 234
15
2154.162
36, 46, 56, 66, 76, 85, 94, 104, 115, 128, 144, 164, 187, 212, 237
16
2155.471
36, 46, 56, 66, 75, 84, 93, 102, 112, 124, 138, 155, 174, 195, 217, 239
Butterfly
2
3873.222
84, 155
3
3990.855
81, 144, 199
4
4051.357
77, 121, 167, 207
5
4092.929
60, 89, 129, 172, 209
6
4119.448
59, 87, 122, 161, 193, 221
7
4135.937
53, 74, 98, 128, 165, 195, 222
8
4148.556
53, 73, 96, 123, 156, 183, 203, 228
9
4156.528
52, 72, 94, 118, 144, 170, 190, 207, 231
10
4163.794
48, 63, 80, 99, 121, 147, 172, 191, 208, 231
11
4168.489
47, 62, 79, 98, 118, 141, 164, 183, 198, 213, 234
12
4172.517
46, 59, 73, 89, 104, 122, 144, 167, 185, 199, 214, 235
13
4175.487
46, 59, 72, 87, 102, 118, 137, 157, 175, 189, 202, 216, 236
14
4177.893
44, 55, 66, 79, 93, 106, 121, 141, 161, 178, 191, 203, 217, 237
15
4179.908
44, 55, 66, 78, 92, 105, 119, 137, 156, 173, 186, 197, 208, 221, 239
16
4181.559
42, 52, 61, 71, 83, 95, 107, 121, 139, 157, 173, 186, 197, 208, 221, 239
Circus
2
1651.257
118, 172
3
1760.512
105, 150, 187
4
1817.487
93, 132, 165, 195
5
1850.243
87, 122, 152, 177, 203
6
1870.083
82, 113, 141, 164, 185, 208
7
1883.966
77, 104, 129, 151, 171, 190, 212
8
1893.450
73, 98, 122, 143, 161, 178, 195, 216
9
1900.592
70, 92, 114, 134, 152, 168, 183, 199, 219
10
1905.992
68, 89, 109, 128, 145, 160, 174, 188, 203, 222
11
1909.887
66, 86, 105, 123, 139, 153, 166, 179, 192, 206, 225
12
1913.079
63, 81, 98, 114, 130, 145, 158, 170, 182, 194, 208, 226
13
1915.676
62, 79, 95, 111, 126, 140, 152, 164, 175, 186, 197, 210, 228
14
1917.756
61, 78, 94, 109, 123, 136, 148, 159, 170, 180, 190, 201, 214, 231
15
1919.524
59, 74, 88, 102, 115, 128, 140, 151, 161, 171, 181, 191, 202, 214, 231
16
1920.958
58, 73, 87, 100, 113, 126, 138, 149, 159, 169, 178, 187, 196, 206, 218, 234
Snow
2
5261.705
80, 169
3
5624.289
71, 139, 207
4
5729.116
50, 92, 144, 208
5
5785.138
49, 91, 140, 192, 231
6
5819.333
45, 81, 111, 148, 194, 232
7
5835.770
43, 76, 101, 127, 159, 196, 232
8
5850.190
30, 55, 83, 107, 133, 163, 197, 233
9
5862.437
30, 55, 82, 106, 129, 157, 185, 211, 237
10
5870.284
29, 52, 75, 94, 111, 132, 159, 186, 212, 237
11
5875.817
29, 52, 75, 94, 111, 132, 158, 183, 206, 227, 244
12
5880.335
29, 52, 74, 93, 109, 127, 149, 169, 188, 209, 228, 244
13
5884.513
23, 39, 57, 76, 94, 110, 128, 150, 170, 188, 209, 228, 244
14
5887.355
22, 37, 54, 70, 84, 98, 112, 129, 151, 170, 188, 209, 228, 244
15
5889.953
21, 36, 53, 69, 83, 97, 110, 124, 141, 159, 175, 192, 211, 229, 245
16
5891.998
21, 36, 53, 69, 83, 97, 110, 124, 141, 159, 174, 189, 207, 222, 236, 248
Palace
2
2623.440
99, 165
3
2791.488
84, 132, 186
4
2860.98
70, 103, 143, 191
5
2908.165
69, 101, 138, 177, 218
6
2934.330
64, 89, 117, 147, 181, 220
7
2953.745
54, 75, 99, 126, 153, 183, 220
8
2966.250
50, 69, 89, 111, 134, 158, 185, 221
9
2974.719
47, 65, 81, 100, 120, 141, 163, 188, 222
10
2981.875
47, 64, 80, 98, 117, 137, 157, 178, 199, 226
11
2986.898
45, 61, 75, 90, 106, 123, 141, 159, 179, 200, 227
12
2990.579
43, 58, 69, 82, 96, 111, 126, 143, 160, 180, 201, 227
13
2993.809
43, 58, 69, 81, 95, 110, 125, 141, 157, 173, 190, 208, 231
14
2996.112
43, 57, 68, 80, 94, 108, 123, 138, 153, 167, 182, 199, 218, 239
15
2998.213
42, 56, 66, 77, 88, 100, 113, 126, 140, 154, 167, 182, 199, 218, 239
16
2999.918
42, 55, 65, 75, 86, 98, 110, 123, 136, 149, 162, 176, 191, 205, 222, 241
Flower
2
1489.281
61, 130
3
1627.897
39, 77, 141
4
1685.956
36, 67, 105, 160
5
1715.220
28, 49, 75, 111, 164
6
1736.423
27, 47, 71, 102, 143, 192
7
1752.512
26, 43, 61, 83, 111, 151, 199
8
1761.968
25, 42, 58, 77, 99, 126, 161, 204
9
1768.354
24, 40, 54, 70, 88, 109, 135, 167, 207
10
1772.903
23, 37, 48, 60, 75, 92, 112, 138, 169, 208
11
1776.470
23, 36, 47, 58, 72, 87, 104, 124, 149, 177, 211
12
1779.106
22, 34, 44, 54, 65, 78, 92, 108, 128, 152, 179, 212
13
1781.251
20, 30, 39, 48, 58, 70, 83, 96, 112, 131, 154, 180, 213
14
1783.049
19, 29, 38, 47, 56, 66, 78, 90, 104, 120, 139, 161, 185, 215
15
1784.478
19, 29, 38, 46, 54, 63, 74, 85, 97, 111, 128, 147, 168, 190, 218
16
1785.641
19, 28, 37, 45, 53, 62, 72, 83, 94, 106, 120, 136, 155, 175, 196, 222
Wherry
2
3313.161
108, 189
3
3543.272
102, 161, 218
4
3599.924
83, 121, 163, 218
5
3634.708
81, 118, 152, 184, 224
6
3656.048
72, 103, 130, 156, 186, 225
7
3668.313
68, 94, 120, 139, 161, 189, 226
8
3678.216
60, 83, 110, 132, 152, 175, 198, 230
9
3686.001
56, 77, 100, 122, 138, 156, 179, 201, 231
10
3691.730
56, 76, 98, 120, 136, 153, 174, 194, 219, 243
11
3696.416
55, 74, 94, 114, 130, 142, 158, 178, 197, 222, 245
12
3699.866
54, 72, 91, 110, 126, 138, 151, 168, 185, 202, 225, 246
13
3702.619
52, 68, 83, 100, 117, 130, 140, 153, 169, 185, 202, 225, 246
14
3705.033
52, 68, 83, 100, 117, 130, 140, 152, 167, 182, 197, 215, 235, 249
15
3706.937
51, 66, 79, 94, 110, 123, 133, 142, 154, 169, 184, 199, 216, 235, 249
16
3708.484
49, 63, 75, 89, 104, 118, 129, 138, 147, 159, 173, 186, 200, 217, 235, 249
Waterfall
2
4512.801
88, 170
3
4646.137
72, 115, 182
4
4711.019
67, 103, 150, 204
5
4752.267
58, 87, 119, 164, 212
6
4777.063
53, 78, 104, 134, 176, 217
7
4793.510
48, 70, 92, 116, 147, 186, 221
8
4805.756
46, 66, 86, 107, 131, 163, 199, 226
9
4814.097
43, 61, 79, 97, 117, 141, 173, 205, 228
10
4820.724
40, 57, 73, 90, 108, 128, 152, 182, 210, 230
11
4825.739
38, 54, 69, 84, 100, 117, 137, 162, 191, 215, 232
12
4829.689
37, 53, 67, 81, 96, 111, 128, 148, 173, 198, 218, 233
13
4832.826
35, 50, 63, 76, 89, 103, 117, 133, 153, 177, 201, 220, 234
14
4835.371
32, 46, 58, 70, 82, 95, 108, 122, 138, 158, 182, 204, 221, 234
15
4837.537
32, 46, 57, 68, 80, 92, 104, 117, 131, 148, 169, 191, 210, 224, 236
16
4839.226
30, 43, 54, 64, 74, 85, 96, 108, 120, 134, 151, 172, 193, 211, 225, 236
Bird
2
901.450
71, 122
3
975.230
64, 111, 140
4
1027.509
61, 104, 131, 164
5
1051.482
54, 93, 119, 138, 169
6
1067.992
47, 82, 108, 127, 142, 172
7
1077.304
40, 70, 94, 113, 129, 143, 173
8
1086.005
39, 69, 93, 112, 128, 141, 159, 192
9
1091.341
37, 65, 88, 105, 119, 131, 142, 160, 193
10
1095.355
37, 64, 86, 103, 117, 129, 139, 149, 167, 200
11
1098.475
33, 56, 77, 93, 107, 119, 130, 140, 150, 169, 202
12
1100.749
33, 56, 77, 93, 107, 119, 129, 138, 146, 158, 178, 209
13
1102.639
31, 52, 71, 87, 100, 111, 121, 130, 139, 147, 159, 179, 210
14
1104.132
29, 48, 66, 82, 95, 107, 118, 127, 134, 141, 149, 161, 181, 211
15
1105.446
28, 45, 62, 77, 90, 101, 111, 120, 128, 135, 142, 150, 162, 182, 212
16
1106.500
28, 45, 62, 77, 90, 101, 111, 120, 128, 135, 141, 148, 157, 171, 191, 219
Police
2
3647.353
74, 150
3
3844.314
70, 135, 192
4
3966.225
63, 112, 158, 209
5
4013.875
61, 104, 140, 174, 214
6
4047.198
32, 67, 106, 141, 175, 214
7
4067.996
32, 67, 104, 133, 161, 186, 219
8
4084.933
29, 52, 78, 106, 134, 162, 187, 219
9
4094.705
29, 52, 78, 103, 125, 147, 169, 190, 220
10
4101.702
29, 52, 78, 102, 123, 143, 165, 184, 203, 228
11
4108.018
28, 49, 71, 90, 107, 126, 146, 166, 185, 204, 229
12
4112.304
28, 49, 71, 89, 105, 122, 139, 157, 174, 189, 207, 231
13
4115.165
28, 46, 62, 78, 92, 106, 123, 140, 158, 174, 189, 207, 231
14
4117.926
28, 46, 62, 78, 91, 105, 120, 135, 151, 166, 180, 193, 210, 232
15
4120.045
28, 46, 62, 78, 91, 103, 116, 129, 143, 158, 172, 185, 197, 214, 235
16
4121.861
28, 46, 62, 78, 91, 103, 116, 128, 142, 156, 170, 182, 193, 206, 223, 240
Ostrich
2
1073.452
75, 135
3
1139.260
69, 101, 149
4
1178.650
65, 92, 125, 176
5
1203.749
56, 78, 100, 131, 179
6
1218.643
47, 65, 85, 103, 133, 181
7
1228.925
47, 64, 83, 100, 122, 152, 192
8
1236.023
45, 59, 75, 90, 104, 125, 155, 194
9
1240.756
40, 52, 65, 80, 94, 107, 128, 157, 195
10
1244.909
40, 52, 64, 78, 91, 103, 119, 141, 168, 201
11
1247.879
40, 51, 62, 75, 87, 97, 108, 125, 148, 174, 205
12
1250.313
37, 48, 57, 68, 80, 91, 101, 112, 128, 150, 175, 206
13
1252.298
31, 44, 53, 63, 75, 86, 95, 105, 117, 134, 154, 178, 207
14
1253.956
29, 42, 50, 58, 68, 79, 89, 98, 108, 120, 137, 157, 181, 209
15
1255.366
29, 42, 50, 58, 67, 77, 86, 94, 102, 111, 124, 141, 160, 183, 210
16
1256.490
28, 41, 49, 57, 66, 76, 85, 93, 101, 110, 122, 137, 155, 174, 196, 219
Viaduct
2
7920.458
77, 180
3
8117.991
54, 109, 193
4
8203.807
42, 84, 131, 203
5
8246.806
35, 68, 103, 146, 210
6
8272.775
31, 59, 88, 120, 160, 216
7
8287.714
28, 53, 77, 103, 132, 169, 220
8
8298.322
27, 51, 75, 100, 128, 164, 212, 246
9
8308.240
24, 45, 66, 88, 112, 139, 172, 216, 247
10
8315.133
22, 41, 60, 79, 99, 121, 146, 178, 219, 247
11
8320.032
20, 37, 54, 71, 89, 108, 128, 152, 183, 221, 248
12
8323.799
20, 36, 52, 68, 84, 101, 119, 139, 162, 190, 224, 248
13
8326.793
18, 33, 48, 62, 77, 92, 108, 125, 144, 167, 195, 226, 248
14
8329.119
17, 31, 44, 57, 70, 84, 98, 113, 129, 148, 170, 197, 227, 248
15
8330.983
17, 31, 44, 57, 70, 84, 98, 113, 129, 147, 169, 195, 224, 243, 251
16
8332.744
16, 29, 42, 54, 66, 78, 91, 104, 118, 133, 151, 172, 196, 224, 243, 251
Fish
2
3593.389
64, 148
3
3870.456
44, 104, 177
4
3972.731
34, 81, 127, 188
5
4024.885
27, 63, 101, 139, 194
6
4054.836
24, 56, 90, 123, 156, 205
7
4075.236
22, 49, 78, 107, 135, 168, 213
8
4088.300
20, 44, 68, 93, 118, 142, 173, 216
9
4097.101
19, 40, 62, 85, 107, 128, 149, 178, 218
10
4103.194
18, 38, 58, 78, 98, 118, 137, 158, 185, 222
11
4107.954
16, 34, 51, 69, 88, 107, 125, 143, 163, 190, 224
12
4111.678
15, 31, 47, 64, 82, 100, 117, 133, 149, 169, 195, 227
13
4114.783
14, 28, 43, 58, 74, 90, 106, 121, 136, 152, 172, 198, 228
14
4116.954
14, 28, 43, 58, 73, 88, 103, 117, 131, 145, 160, 179, 203, 231
15
4118.931
13, 25, 38, 51, 64, 78, 92, 106, 120, 133, 147, 162, 181, 205, 232
16
4120.522
13, 25, 37, 49, 62, 75, 89, 102, 115, 127, 139, 152, 168, 188, 211, 235
Houses
2
2543.788
56, 116
3
2627.230
53, 105, 150
4
2663.698
42, 69, 110, 152
5
2694.902
39, 65, 96, 130, 158
6
2708.655
39, 65, 95, 127, 150, 169
7
2720.263
35, 55, 74, 98, 128, 151, 170
8
2726.676
35, 55, 74, 98, 127, 148, 164, 184
9
2732.853
32, 48, 65, 80, 101, 128, 148, 164, 184
10
2736.510
30, 45, 60, 74, 87, 106, 129, 149, 164, 184
11
2739.566
30, 45, 60, 73, 86, 105, 128, 145, 156, 168, 187
12
2742.148
29, 42, 55, 68, 79, 94, 112, 130, 145, 156, 168, 187
13
2744.071
28, 39, 51, 63, 74, 84, 98, 115, 131, 145, 156, 168, 187
14
2745.542
28, 39, 51, 63, 74, 84, 98, 115, 131, 145, 155, 165, 176, 193
15
2746.817
27, 37, 47, 58, 68, 77, 86, 100, 116, 131, 145, 155, 165, 176, 193
16
2747.991
27, 37, 47, 57, 67, 76, 85, 98, 113, 127, 139, 147, 156, 165, 176, 193
Mushroom
2
1988.328
76, 145
3
2153.037
65, 110, 174
4
2237.441
59, 93, 135, 193
5
2277.427
52, 78, 106, 145, 199
6
2301.629
48, 71, 94, 122, 157, 205
7
2317.538
46, 67, 87, 110, 138, 171, 213
8
2328.526
44, 62, 79, 97, 118, 145, 177, 216
9
2335.812
42, 58, 74, 90, 107, 127, 152, 181, 218
10
2340.981
41, 56, 70, 84, 99, 116, 136, 159, 186, 220
11
2345.160
39, 52, 65, 78, 92, 107, 124, 144, 166, 191, 223
12
2348.413
38, 51, 63, 75, 87, 100, 115, 132, 152, 174, 199, 227
13
2350.988
38, 50, 62, 74, 85, 97, 110, 125, 142, 160, 180, 203, 229
14
2353.119
37, 48, 59, 69, 79, 89, 100, 113, 127, 144, 162, 181, 204, 230
15
2354.748
36, 46, 56, 66, 75, 84, 94, 105, 117, 130, 146, 163, 182, 205, 230
16
2356.113
36, 46, 56, 65, 74, 83, 92, 102, 113, 125, 139, 154, 169, 187, 208, 232
Snow mountain
2
1912.613
85, 149
3
2135.274
79, 137, 197
4
2234.669
70, 119, 154, 204
5
2288.799
55, 96, 129, 160, 206
6
2317.224
52, 90, 118, 142, 167, 209
7
2333.078
50, 85, 111, 132, 153, 174, 212
8
2344.629
45, 76, 100, 120, 139, 158, 177, 214
9
2352.951
44, 74, 97, 116, 133, 150, 167, 187, 220
10
2359.175
41, 68, 90, 108, 124, 140, 156, 172, 193, 225
11
2364.304
38, 62, 84, 102, 117, 132, 146, 160, 175, 196, 227
12
2367.971
34, 56, 78, 96, 111, 125, 139, 152, 165, 179, 199, 228
13
2371.208
33, 53, 73, 90, 104, 117, 130, 143, 155, 167, 180, 200, 229
14
2373.567
32, 52, 72, 89, 103, 115, 127, 139, 150, 161, 172, 185, 205, 232
15
2375.586
31, 49, 68, 84, 98, 109, 120, 131, 142, 153, 164, 175, 188, 208, 233
16
2377.255
26, 42, 59, 75, 89, 101, 111, 121, 132, 143, 154, 165, 176, 189, 209, 234
Snake
2
1118.615
87, 134
3
1231.320
76, 114, 154
4
1286.555
69, 101, 129, 166
5
1317.027
63, 91, 115, 140, 175
6
1336.172
59, 84, 105, 126, 149, 182
7
1348.933
55, 78, 97, 115, 133, 155, 187
8
1357.665
52, 73, 91, 107, 123, 140, 161, 192
9
1364.159
50, 70, 87, 102, 116, 131, 148, 169, 198
10
1368.955
47, 65, 81, 95, 108, 121, 135, 151, 172, 200
11
1372.626
46, 63, 78, 91, 103, 115, 127, 140, 156, 176, 203
12
1375.513
45, 61, 75, 88, 99, 110, 121, 133, 146, 162, 182, 208
13
1377.847
43, 58, 72, 84, 95, 106, 116, 127, 138, 151, 166, 185, 211
14
1379.736
42, 56, 69, 81, 92, 102, 112, 122, 132, 143, 155, 170, 189, 214
15
1381.282
41, 54, 66, 77, 87, 97, 106, 115, 124, 134, 145, 157, 172, 191, 215
16
1382.603
40, 52, 63, 74, 84, 93, 102, 111, 120, 129, 139, 150, 162, 177, 195, 219
Riosanpablo
2
2667.020
95, 160
3
2818.660
75, 121, 177
4
2892.439
68, 102, 143, 189
5
2931.654
62, 89, 121, 158, 197
6
2957.018
58, 82, 107, 138, 171, 204
7
2973.269
53, 74, 95, 119, 147, 177, 207
8
2984.972
50, 69, 87, 108, 133, 159, 185, 212
9
2993.359
48, 66, 83, 101, 122, 145, 168, 191, 215
10
2999.615
46, 63, 78, 93, 110, 130, 151, 173, 195, 217
11
3004.374
45, 61, 75, 89, 104, 121, 140, 160, 180, 200, 220
12
3008.082
44, 59, 72, 84, 97, 112, 129, 147, 166, 185, 203, 222
13
3011.101
43, 57, 69, 81, 93, 107, 122, 138, 155, 172, 189, 206, 224
14
3013.465
41, 54, 66, 77, 88, 100, 113, 128, 144, 160, 176, 192, 208, 225
15
3015.423
40, 53, 64, 74, 84, 95, 107, 121, 135, 150, 165, 180, 195, 210, 226
16
3017.081
39, 51, 62, 72, 82, 92, 103, 115, 128, 142, 156, 170, 184, 198, 212, 227We can see from these results thatrank-DE has a higher approximated TVD growth rate than DE. O
(
β
)
R
-DE with β
≤
0.5 and O
(
rand
)
R
-DE are still better than DE. O
(
0.0
)
R
-DE produced the best result with a very flat slope and a very low y-intercept. The multithresh function yielded a higher growth rate of solution distortion and a higher y-intercept than the other algorithms. The higher growth rate of solution distortion means the quality of solution drops very fast if the number of thresholds increases. The higher y-intercept means that the solution distortion at the lowest number of thresholds is high. Thus, the global optimization algorithm is required for solving the multilevel thresholding.


## 5.2.1. The Value to Reach (VTR)

Following the completion of all of the experiments the best values of the between-class variance and the corresponding thresholds were collected and are shown in Table6. The results are shown image by image and the numbers of thresholds vary from 2 to 16. The between-class variance values in column 3 are used as the VTR values.


## 5.2.2. Results Produced by Local Search Method

The multithresh function of the Matlab toolbox was conducted on the same images and number of thresholds as the other search methods. The capabilities of solving the optimal solution between a local search and a global search will be discussed here. This is the reason we focused on the global search, that is, the proposedO
(
0.0
)
R
-DE algorithm. Table 2 shows the between-class variances and threshold values of the “mountain” image. These values were the best outcomes of 50 runs produced by the multithresh function in the Matlab R2013a toolbox and by the proposed O
(
0.0
)
R
-DE algorithm. The terminated condition of the multithresh function was set by “MaxFunEvals” = 500000. That is the multithresh function performs more function calls than that of the O
(
0.0
)
R
-DE algorithm. It can be seen from columns 3 and 5 that all the BCVs produced by the O
(
0.0
)
R
-DE algorithm are better than the BCVs produced by the multithresh function; the difference of the BCVs is shown in column 7. The differences in the thresholds from the two algorithms, shown in column 8, tended to be large if the number of thresholds increased.Table 2
The best values of the between-class variance and thresholds of the “mountain” image produced by the multithresh function of the Matlab toolbox and by the proposedO(0.0)R-DE algorithm. The number of thresholds varies from 2 to 16.
Image(1)
n  (2)
Produced by multithresh
Produced by theO(0.0)R-DE algorithm
Objdiff
(
7
)
=
(
5
)
-
(
3
)
Threshdiff
(
8
)
=
∑
|
(
4
)
-
(
6
)
|
Between-class variance(3)
Thresholds(4)
Between-class variance(5)
Thresholds(6)
Mountain
2
2372.886
60, 127
2372.923
61, 128
0.037
2
3
2495.852
32, 76, 130
2496.113
33, 77, 131
0.261
3
4
2551.778
32, 72, 109, 145
2551.955
33, 73, 109, 147
0.177
4
5
2580.154
31, 68, 98, 124, 158
2580.336
32, 69, 99, 125, 159
0.182
5
6
2588.264
35, 75, 98, 118, 152, 175
2596.956
24, 46, 74, 101, 126, 160
8.692
122
7
2598.800
33, 70, 98, 118, 141, 166, 207
2608.807
24, 46, 73, 98, 119, 145, 175
10.007
153
8
2605.785
31, 68, 91, 105, 121, 140, 163, 192
2616.294
24, 46, 73, 97, 115, 135, 160, 191
10.509
70
9
2619.512
27, 52, 75, 95, 111, 129, 148, 168, 197
2622.314
20, 37, 54, 76, 98, 116, 136, 160, 191
2.802
114
10
2620.392
23, 46, 73, 97, 116, 135, 157, 175, 204, 232
2627.194
20, 36, 53, 74, 93, 106, 121, 140, 163, 194
6.802
258
11
2625.275
23, 45, 72, 93, 105, 119, 137, 156, 175, 204, 228
2630.496
20, 36, 53, 74, 93, 105, 118, 134, 152, 172, 201
5.221
199
12
2629.641
22, 39, 58, 78, 95, 109, 123, 139, 156, 177, 207, 247
2633.189
18, 31, 45, 59, 76, 93, 105, 118, 134, 152, 172, 201
3.548
246
13
2631.372
20, 38, 54, 74, 94, 107, 121, 139, 157, 173, 193, 224, 248
2635.290
18, 31, 45, 59, 76, 92, 103, 115, 129, 144, 161, 179, 207
3.918
283
14
2633.865
19, 36, 53, 74, 91, 103, 115, 129, 144, 160, 174, 190, 217, 245
2637.088
17, 29, 42, 56, 72, 86, 96, 106, 117, 130, 145, 161, 179, 207
3.223
307
15
2635.190
19, 33, 50, 69, 85, 96, 107, 121, 137, 152, 167, 178, 190, 212, 240
2638.449
17, 28, 39, 50, 61, 75, 88, 97, 107, 118, 131, 146, 162, 179, 207
3.259
351
16
2636.357
16, 30, 45, 59, 77, 93, 105, 118, 133, 148, 162, 176, 192, 209, 230, 244
2639.616
17, 27, 38, 49, 60, 74, 86, 95, 104, 113, 123, 135, 149, 164, 181, 209
3.259
415Table 3
The average of mean number of generation (NG
¯
AM) and ranks of the methods.
No.maxGen.
FODPSO
DPSO
PSO
ABC
Rank-DE
DE
O(rand)R-DE
O(0.9)R-DE
O(0.8)R-DE
O(0.7)R-DE
O(0.6)R-DE
O(0.5)R-DE
O(0.4)R-DE
O(0.3)R-DE
O(0.2)R-DE
O(0.1)R-DE
O(0.0)R-DE
n
=
2
G
=
50
NG
¯
AM
18.337
15.713
13.218
13.202
10.538
14.713
10.117
6.394
6.903
7.636
8.208
9.147
9.261
9.901
10.709
11.631
20.850
Rank
16
15
13
12
9
14
8
1
2
3
4
5
6
7
10
11
17
n
=
3
G
=
100
NG
¯
AM
35.200
36.210
27.479
33.210
19.488
29.049
18.958
10.492
12.296
13.587
14.807
16.969
17.107
18.100
19.546
21.749
44.192
Rank
15
16
12
14
9
13
8
1
2
3
4
5
6
7
10
11
17
n
=
4
G
=
150
NG
¯
AM
45.391
58.683
49.596
63.251
31.060
48.130
29.595
14.733
18.526
20.885
22.815
26.632
26.456
28.058
30.175
33.499
72.100
Rank
12
15
14
16
10
13
8
1
2
3
4
6
5
7
9
11
17
n
=
5
G
=
200
NG
¯
AM
55.664
81.877
87.441
108.131
44.157
69.088
41.283
19.508
24.524
28.097
30.963
36.935
36.748
38.647
41.455
45.631
100.693
Rank
12
14
15
17
10
13
8
1
2
3
4
6
5
7
9
11
16
n
=
6
G
=
300
NG
¯
AM
NA(2)
100.664
141.937
157.328
61.935
99.700
56.405
24.623
32.687
38.167
42.608
51.003
50.281
52.089
55.729
61.748
138.716
Rank
17
13
15
16
11
12
9
1
2
3
4
6
5
7
8
10
14
n
=
7
G
=
400
NG
¯
AM
NA(1)
NA(1)
192.267
236.928
86.938
142.096
75.340
29.763
41.691
49.408
56.193
67.619
67.063
69.696
74.389
83.326
184.371
Rank
17
16
14
15
11
12
9
1
2
3
4
6
5
7
8
10
13
n
=
8
G
=
600
NG
¯
AM
NA(7)
NA(6)
NA(2)
NA(1)
117.092
198.986
100.944
33.967
51.573
62.263
74.523
91.836
89.666
95.198
102.552
115.583
244.880
Rank
17
16
15
14
11
12
8
1
2
3
4
6
5
7
9
10
13
n
=
9
G
=
800
NG
¯
AM
NA(11)
NA(13)
NA(12)
NA(3)
151.004
264.253
125.299
41.228
63.742
79.432
90.201
113.530
110.416
114.397
122.555
137.538
299.492
Rank
17
16
15
14
11
12
9
1
2
3
4
6
5
7
8
10
13
n
=
10
G
=
1000
NG
¯
AM
NA(11)
NA(16)
NA(16)
NA(7)
NA(1)
321.373
147.770
NA(3)
73.564
91.121
105.020
132.968
128.999
132.691
141.825
157.997
356.546
Rank
17
16
15
14
12
10
8
13
1
2
3
6
4
5
7
9
11
n
=
11
G
=
1500
NG
¯
AM
NA(16)
NA(17)
NA(19)
NA(9)
NA(1)
405.090
185.020
NA(3)
NA(2)
NA(2)
127.135
162.822
157.184
160.123
175.152
198.522
434.877
Rank
17
16
15
14
12
8
6
13
10
10
1
4
2
3
5
7
9
n
=
12
G
=
2000
NG
¯
AM
NA(18)
NA(20)
NA(20)
NA(9)
278.743
535.542
229.965
NA(4)
NA(5)
135.266
160.047
205.266
197.386
203.814
222.094
251.535
574.104
Rank
17
16
15
14
12
8
6
13
11
10
1
4
2
3
5
7
9
n
=
13
G
=
3000
NG
¯
AM
NA(20)
NA(19)
NA(20)
NA(14)
NA(3)
NA(3)
275.860
NA(12)
NA(5)
NA(1)
NA(1)
NA(1)
NA(1)
261.522
278.082
316.468
726.521
Rank
17
16
15
14
12
9
2
13
11
10
6
6
6
1
3
4
5
n
=
14
G
=
4000
NG
¯
AM
NA(20)
NA(20)
NA(20)
NA(14)
NA(4)
NA(4)
330.194
NA(11)
NA(8)
NA(4)
NA(1)
NA(2)
296.414
NA(1)
331.626
385.951
889.711
Rank
17
16
15
14
12
9
1
13
11
10
7
8
6
5
2
3
4
n
=
15
G
=
5000
NG
¯
AM
NA(20)
NA(20)
NA(20)
NA(19)
NA(13)
NA(5)
399.228
NA(11)
NA(10)
NA(8)
NA(6)
363.506
362.665
383.788
417.958
481.830
1086.001
Rank
17
16
15
14
13
8
1
12
11
10
9
7
6
5
2
3
4
n
=
16
G
=
6000
NG
¯
AM
NA(20)
NA(20)
NA(20)
NA(18)
NA(10)
NA(8)
NA(4)
NA(18)
NA(11)
NA(9)
NA(5)
NA(1)
NA(2)
447.570
491.772
552.856
1374.438
Rank
17
16
15
14
12
9
4
13
11
10
8
6
7
5
1
2
3
Avg for
n
=
2 to 16
n
max
⁡,NA(x)
NG
¯
AM
5,NA(146), 38.648
6,NA(152), 58.629
7,NA(149), 85.323
7,NA(94), 102.008
9,NA(32), 88.995
12,NA(20), 193.456
15,NA(4), 144.713
9,NA(62), 22.589
10,NA(41), 36.167
10,NA(24), 52.586
12,NA(13), 66.593
12,NA(4), 106.519
12,NA(3), 119.204
13,NA(1), 143.971
16,NA(0), 167.708
16,NA(0), 190.391
16,NA(0), 436.499
Rank
17
16
15
14
12
9
4
13
11
10
8
7
6
5
1
2
3Table 4
The average success rate (SRHM) and ranks of the methods.
No
Multi-thresh
FODPSO
DPSO
PSO
ABC
Rank-DE
DE
O(rand)R-DE
O(0.9)R-DE
O(0.8)R-DE
O(0.7)R-DE
O(0.6)R-DE
O(0.5)R-DE
O(0.4)R-DE
O(0.3)R-DE
O(0.2)R-DE
O(0.1)R-DE
O(0.0)R-DE
2
SRHM
0
0.977
1
1
1
1
1
1
0.999
1
1
1
1
1
1
1
1
1
Rank
18
17
1
1
1
1
1
1
16
1
1
1
1
1
1
1
1
1
3
SRHM
0.000
0.635
0.972
0.991
0.979
0.998
1.000
0.997
0.986
0.993
0.999
0.997
0.998
0.999
1.000
1.000
1.000
1.000
Rank
18
17
16
13
15
8
1
10
14
12
6
10
8
6
1
1
1
1
4
SRHM
0.000
0.379
0.742
0.934
0.833
0.987
0.995
0.986
0.897
0.945
0.972
0.971
0.986
0.993
0.992
0.991
0.998
0.997
Rank
18
17
16
13
15
7
3
8
14
12
10
11
8
4
5
6
1
2
5
SRHM
0.000
0.185
0.391
0.767
0.706
0.936
0.977
0.960
0.782
0.878
0.874
0.944
0.950
0.959
0.968
0.960
0.985
1.000
Rank
18
17
16
14
15
10
3
5
13
11
12
9
8
7
4
5
2
1
6
SRHM
0.000
0.000
0.187
0.433
0.453
0.863
0.968
0.900
0.601
0.706
0.826
0.878
0.934
0.922
0.944
0.935
0.970
0.999
Rank
18
17
16
15
14
10
3
8
13
12
11
9
6
7
4
5
2
1
7
SRHM
0.000
0.000
0.000
0.153
0.164
0.696
0.884
0.835
0.366
0.561
0.675
0.741
0.840
0.851
0.883
0.914
0.949
0.998
Rank
18
17
16
15
14
10
4
8
13
12
11
9
7
6
5
3
2
1
8
SRHM
0.000
0.000
0.000
0.000
0.000
0.300
0.458
0.530
0.105
0.256
0.351
0.502
0.592
0.628
0.724
0.757
0.838
0.972
Rank
18
17
16
14
14
11
9
7
13
12
10
8
6
5
4
3
2
1
9
SRHM
0.000
0.000
0.000
0.000
0.000
0.495
0.695
0.681
0.083
0.274
0.372
0.548
0.721
0.725
0.769
0.815
0.852
0.999
Rank
18
17
16
14
14
10
7
8
13
12
11
9
6
5
4
3
2
1
10
SRHM
0.000
0.000
0.000
0.000
0.000
0.000
0.381
0.375
0.000
0.073
0.218
0.311
0.409
0.370
0.460
0.578
0.686
0.998
Rank
18
17
16
14
14
12
6
7
13
11
10
9
5
8
4
3
2
1
11
SRHM
0.000
0.000
0.000
0.000
0.000
0.000
0.271
0.267
0.000
0.000
0.000
0.144
0.407
0.316
0.424
0.584
0.704
0.999
Rank
18
17
16
14
14
12
7
8
13
10
10
9
5
6
4
3
2
1
12
SRHM
0.000
0.000
0.000
0.000
0.000
0.090
0.270
0.423
0.000
0.000
0.089
0.177
0.400
0.407
0.501
0.641
0.810
1.000
Rank
18
17
16
14
14
12
8
5
13
11
10
9
7
6
4
3
2
1
13
SRHM
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.151
0.000
0.000
0.000
0.000
0.000
0.000
0.254
0.267
0.381
0.974
Rank
18
17
16
14
14
12
6
5
13
10
10
6
6
6
4
3
2
1
14
SRHM
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.102
0.000
0.000
0.000
0.000
0.000
0.110
0.000
0.253
0.494
0.993
Rank
18
17
16
14
14
12
7
4
13
10
10
7
7
6
5
3
2
1
15
SRHM
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.090
0.000
0.000
0.000
0.000
0.095
0.117
0.149
0.288
0.466
1.000
Rank
18
17
16
14
14
12
8
4
13
10
10
8
7
6
5
3
2
1
16
SRHM
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.110
0.192
0.288
0.977
Rank
18
17
16
14
14
12
6
4
13
10
10
6
6
6
5
3
2
1
Avg. for
n
=
2 to 16
n
0.5,
n
max
⁡,  SRHM
<2,<2, 0.000
3,5, 0.376
4,6, 0.443
5,7, 0.454
5,7, 0.464
7,9, 0.665
9,12, 0.554
9,15, 0.304
6,9, 0.263
7,10, 0.313
7,10, 0.530
9,12, 0.420
9,12, 0.654
9,12, 0.625
9,13, 0.619
12,16, 0.498
12,16, 0.656
16,16, 0.994
Rank
18
17
16
15
14
12
8
4
13
11
10
9
6
7
5
3
2
1Figure2 shows the graph of the TVD of all the images and thresholds. These results are in the same pattern of the results of the “mountain” image in Table 2. That means the ability to search for the optimal solution of the proposed global search algorithm is higher than that of the multithresh function, especially when the number of thresholds is large. This goes to illustrate the difficulty of the problem. The problem with this kind is that it can be multimodal [33] or can be a nearly flat top surface [34]. The multithresh function solves the problem by performing the Nelder-Mead Simplex method [35], which is a local search method that cannot guarantee an optimal solution. Thus, its solutions are inferior to the solution produced by the algorithm using a global search.Figure 2
The threshold value distortion (TVD) of algorithms versus number of thresholds.


## 5.2.3. Convergence Rate Comparison

The number of generations (NG) is a measure used for the convergence rate comparisons. If the target value, VTR, is achieved in a lesser number of generations (NG), it means a faster convergence rate for the algorithm. Table3 shows the average of NG
¯ (NG
¯
AM) for each specific number of thresholds. The results of each algorithm are represented in the corresponding column’s name. In each column, the cell containing NG
¯
AM starts from the row associated with n
=
2 until the row associated with n
=
n
max
⁡. The cells associated with n
=
n
max
⁡
+
1 to the row associated with n
=
16 are filled by NA
(
x
). The second last row of the table is filled by the triple:
(23)
(
(
NG
¯
AM
of
n
=
2
until
n
=
n
max
⁡
)
n
max
⁡
,
NA
(
number
of
unsuccessful
subproblem
)
,
AM
(
NG
¯
AM
of
n
=
2
until
n
=
n
max
⁡
)
)
.The algorithm with the highestn
max
⁡, that is, the lowest number of unsuccessful subproblems and the lowest average of generation is the winner. The ranking of the algorithms depends on the ordering of (a
1, b
1, c
1) and (a
2, b
2, c
2) as follows.First, rank ona
1 and a
2. Since both a
1 and a
2 are numeric, the higher value has the higher rank.
Second, rank onb
1 and b2. If a
1 is 16, then b
1 must be NA(0). If a
1 is less than 16, then b
1 must be NA (number of unsuccessful subproblems) and b
2 has the same characteristics as b
1. Perform the order of the two numeric values in reverse; the lower value has the higher rank.
Third, rank on c1 and c
2. They are numeric but the lower is better; perform the order of the two numeric values in reverse. The lower value has the higher rank.When the ordering is finished, assign the numeric value of “1” to the object having the highest rank, assign the numeric value of “2” to the first runner up, and so on. These values are represented in the last row of the table.If the row havingn
=
m must be ranked, it can be done in the same manner as above with some minor modifications. If m is less than n
max
⁡, the values of the triple pair will be m, NA
(
0
), NG
¯
AM. If m is greater than n
max
⁡, the values of the triple pair will be (n
max
⁡, NA
(
x
), ∞). Thus the ranking can now be performed.From the ranking results, see the second last row of Table3; the convergence rate can be ranked from best to worst in the following order: O
(
0.2
)
R
-DE, O
(
0.1
)
R
-DE, O
(
0.0
)
R
-DE, O
(
rand
)
R
-DE, O
(
0.3
)
R
-DE, O
(
0.4
)
R
-DE, O
(
0.5
)
R
-DE, O
(
0.6
)
R
-DE, DE, O
(
0.7
)
R
-DE, O
(
0.8
)
R
-DE, rank-DE, O
(
0.9
)
R
-DE, ABC, PSO, DPSO, FODPSO.As can be seen from Table3, the DE algorithm cannot complete the task when n
>
12 and the rank-DE algorithm cannot complete the task when n
>
9. Thus, rank-DE cannot compete with DE on searching for global multilevel thresholding.In order forO
(
β
)
R
-DE to outperform DE then β must be in the range of [
0.1
,
0.6
] or set to rndreal
[
0,1
).


## 5.2.4. Stability Analysis

The harmonic mean of the success rate (S
R
HM) for each specific number of thresholds was computed and is presented in Table 4. The results of each algorithm are represented in the corresponding column’s name. The second row from the bottom shows the harmonic mean of the success rate of each algorithm for all threshold levels.In each column, the cells containingS
R
HM start from the row associated with n
=
2 to the row associated with n
=
n
0.5. The cells from the row associated with n
=
n
0.5
+
1 to the row associated with n
=
n
max
⁡ are filled by S
R
HM. The cells from the row associated with n
=
n
max
⁡
+
1 to the row associated with n
=
16 are the cells that have S
R
HM; these cells will be excluded from the comparison. The second last row of the table is filled with the triple:
(24)
(
n
0.5
,
n
max
⁡
,
HM
(
S
R
HM
of
n
=
2
until
n
=
n
max
⁡
)
)
.The algorithm with the highestn
0.5, the highest n
max
⁡, and the highest average success rate is the winner. The ranking of the algorithms depends on the ordering of (a1, b
1, c
1) and (a
2, b2, c
2) as follows.First, rank ona
1 and a2.
Second, rank onb
1 and b
2.
Third, rank onc
1 and c
2.Because they are numeric the higher value has the higher rank. When the ordering is finished, the numeric value of “1” is assigned to the object having the highest rank, the numeric value of “2” is assigned to the first runner up, and so on.If the row havingn
=
m must be ranked, it can be done in the same manner as above with some minor modifications. If m is less than or equal to n
0.5, the values of the triple pair will be (m, m, S
R
HM). If n
0.5
<
m
≤
n
max
⁡, then the values of the triple pair will be (n
0.5, m, S
R
HM). If m is greater than n
max
⁡, the values of the triple pair will be (n
0.5, n
max
⁡, S
R
HM). Thus, the ranking can now be performed.From the ranking results, see Table4, the success rate can be ranked from best to worst in the following order: O
(
0.0
)
R
-DE, O
(
0.1
)
R
-DE, O
(
0.2
)
R
-DE, O
(
rand
)
R
-DE, O
(
0.3
)
R
-DE, O
(
0.5
)
R
-DE, O
(
0.4
)
R
-DE, DE, O
(
0.6
)
R
-DE, O
(
0.7
)
R
-DE, O
(
0.8
)
R
-DE, rank-DE, O
(
0.9
)
R
-DE, ABC, PSO, DPSO, FODPSO, multithresh.As can be seen from Table4, the DE algorithm has an S
R
HM
≥
0.5 until n
=
9 and the rank-DE algorithm has an S
R
HM
≥
0.5 until n
=
7. This result confirms that rank-DE cannot compete with DE on searching for global multilevel thresholding. If the correct β is selected, the proposed algorithm can work very well. For β
≤
0.5, O
(
β
)
R
-DE has a higher rank than DE. The multithresh function cannot compete with any of the other algorithms. It can also be seen that the proposed O
(
0.0
)
R
-DE algorithm has the best stability because its S
R
HM is greater than 0.5 when n
=
2 to 16.


## 5.2.5. Reliability Comparison

The threshold value distortion a.k.a.TVD for each specific threshold is computed, shown in Table 5 and depicted in Figure 2. The results of each algorithm are illustrated in the corresponding column’s name. The second last row of Table 5 shows the slope or the approximated growth rate, δ, of the TVD of each algorithm for all threshold levels. The δ is the slope of the robust linear regression computed by the Matlab function “robustfit.” The lower slope exhibits the better reliability. The δ of each algorithm is sorted in descending order. From the results, the reliability can be ranked from best to worst in the following order: O
(
0.0
)
R
-DE, O
(
0.1
)
R
-DE, O
(
0.2
)
R
-DE, O
(
0.3
)
R
-DE, O
(
0.4
)
R
-DE, O
(
0.5
)
R
-DE, O
(
rand
)
R
-DE, DE, O
(
0.6
)
R
-DE, ABC, rank-DE, O
(
0.7
)
R
-DE, O
(
0.8
)
R
-DE, O
(
0.9
)
R
-DE, DPSO, FODPSO, PSO, multithresh.Table 5
The average of threshold value distortion (TVD) and ranks of the methods.
No
Multi-thresh
FODPSO
DPSO
PSO
ABC
Rank-DE
DE
O(rand)R-DE
O(0.9)R-DE
O(0.8)R-DE
O(0.7)R-DE
O(0.6)R-DE
O(0.5)R-DE
O(0.4)R-DE
O(0.3)R-DE
O(0.2)R-DE
O(0.1)R-DE
O(0.0)R-DE
2
TVD
74.153
2.217
0.000
0.000
0.000
0.000
0.000
0.000
0.050
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
Rank
18
17
1
1
1
1
1
1
16
1
1
1
1
1
1
1
1
1
3
TVD
163.333
41.753
2.968
7.424
1.510
0.150
0.000
0.270
1.004
0.519
0.075
0.246
0.171
0.075
0.000
0.000
0.000
0.000
Rank
18
17
15
16
14
8
1
11
13
12
6
10
9
6
1
1
1
1
4
TVD
257.083
98.848
34.770
36.891
8.716
7.340
1.428
2.184
25.499
13.609
7.318
7.360
7.398
4.303
1.672
0.728
0.130
0.257
Rank
18
17
15
16
12
9
4
6
14
13
8
10
11
7
5
3
1
2
5
TVD
516.582
148.344
97.216
62.319
21.565
5.373
1.864
3.241
20.441
10.064
10.422
4.584
3.904
3.289
2.441
3.385
1.195
0.000
Rank
18
17
16
15
14
10
3
5
13
11
12
9
8
6
4
7
2
1
6
TVD
534.226
219.437
187.620
166.775
45.728
19.536
2.528
7.761
89.938
43.666
30.923
13.010
6.524
6.023
4.136
6.129
2.421
0.050
Rank
18
17
16
15
13
10
3
8
14
12
11
9
7
5
4
6
2
1
7
TVD
599.250
265.029
242.194
285.862
66.259
58.877
27.976
35.481
125.040
75.011
49.851
44.610
28.039
29.841
26.075
17.950
4.776
0.150
Rank
18
16
15
17
12
11
5
8
14
13
10
9
6
7
4
3
2
1
8
TVD
475.458
382.076
380.124
309.635
72.763
120.734
90.287
89.406
218.266
143.225
130.898
99.991
76.353
82.886
57.251
57.665
26.822
2.103
Rank
18
17
16
15
5
11
9
8
14
13
12
10
6
7
3
4
2
1
9
TVD
862.458
394.744
375.229
410.710
113.892
82.099
44.407
46.539
250.470
154.662
96.417
68.197
44.223
35.333
26.627
23.406
12.757
0.780
Rank
18
16
15
17
12
10
7
8
14
13
11
9
6
5
4
3
2
1
10
TVD
847.703
458.536
445.765
498.328
117.664
84.541
43.991
48.474
249.147
136.935
93.935
72.288
43.883
44.110
36.700
27.978
20.213
0.164
Rank
18
16
15
17
12
10
6
8
14
13
11
9
5
7
4
3
2
1
11
TVD
977.976
536.258
528.329
531.211
137.941
87.267
44.944
44.973
296.513
152.195
104.092
75.115
44.461
47.564
34.944
26.231
18.782
0.083
Rank
18
17
15
16
12
10
6
7
14
13
11
9
5
8
4
3
2
1
12
TVD
1070.300
592.750
589.931
605.280
157.169
174.580
109.138
101.282
388.491
258.312
196.098
150.932
101.112
82.365
79.966
51.281
23.429
0.000
Rank
18
16
15
17
10
11
8
7
14
13
12
9
6
5
4
3
2
1
13
TVD
1142.808
567.442
579.495
711.382
183.451
204.665
147.251
123.438
466.508
299.040
216.656
178.394
114.244
114.491
96.104
81.076
52.578
1.975
Rank
18
15
16
17
10
11
8
7
14
13
12
9
5
6
4
3
2
1
14
TVD
1148.641
632.482
660.908
770.319
205.714
218.408
141.431
116.716
524.783
325.936
241.997
177.121
113.426
104.073
88.593
65.908
39.647
0.564
Rank
18
15
16
17
10
11
8
7
14
13
12
9
6
5
4
3
2
1
15
TVD
1245.529
707.835
722.126
816.158
256.477
307.873
219.556
182.464
605.623
428.370
343.439
261.522
174.700
166.309
138.710
100.296
62.940
0.000
Rank
18
15
16
17
9
11
8
7
14
13
12
10
6
5
4
3
2
1
16
TVD
1182.989
768.714
835.747
828.004
249.042
279.753
211.614
195.676
605.790
416.711
311.314
251.408
174.118
167.316
133.922
113.776
76.661
2.539
Rank
18
15
17
16
9
11
8
7
14
13
12
10
6
5
4
3
2
1
Avg. for
n
=
2  to 16
TVD
739.899
387.764
378.828
402.687
109.193
110.080
72.428
66.527
257.838
163.884
122.229
93.652
62.170
59.199
48.476
38.387
22.823
0.578
Growth rate (δ)
86.86
55.25
61.13
67.98
19.25
21.76
15.61
13.48
48.43
32.25
23.91
18.93
12.52
11.86
9.99
7.69
4.94
0.09
Rank
18
15
16
17
10
11
8
7
14
13
12
9
6
5
4
3
2
1Table 6
The between-class variance criterion and the best thresholds for test images.
Image
n
Between-class variance (VTR)
Thresholds
Starfish
2
2546.885
85, 157
3
2779.925
68, 119, 177
4
2865.707
60, 101, 138, 187
5
2912.859
52, 86, 117, 150, 194
6
2941.728
47, 77, 105, 132, 162, 201
7
2960.158
44, 71, 95, 118, 142, 170, 206
8
2972.356
43, 68, 90, 110, 131, 153, 180, 212
9
2981.138
38, 58, 78, 97, 116, 136, 157, 183, 214
10
2988.206
37, 56, 75, 93, 110, 128, 146, 167, 192, 219
11
2993.348
35, 53, 71, 88, 103, 119, 135, 152, 172, 196, 221
12
2997.352
34, 51, 68, 84, 98, 112, 127, 142, 158, 177, 200, 223
13
3000.480
33, 48, 64, 79, 93, 106, 119, 133, 147, 162, 181, 203, 225
14
3003.076
32, 47, 62, 76, 89, 101, 114, 127, 140, 154, 169, 187, 207, 227
15
3005.235
30, 43, 56, 70, 83, 95, 107, 119, 131, 143, 156, 171, 189, 209, 228
16
3007.060
30, 42, 55, 68, 80, 92, 103, 114, 126, 138, 150, 163, 178, 195, 213, 230
Mountain
2
2372.923
61, 128
3
2496.113
33, 77, 131
4
2551.955
33, 73, 109, 147
5
2580.336
32, 69, 99, 125, 159
6
2596.956
24, 46, 74, 101, 126, 160
7
2608.807
24, 46, 73, 98, 119, 145, 175
8
2616.294
24, 46, 73, 97, 115, 135, 160, 191
9
2622.314
20, 37, 54, 76, 98, 116, 136, 160, 191
10
2627.194
20, 36, 53, 74, 93, 106, 121, 140, 163, 194
11
2630.496
20, 36, 53, 74, 93, 105, 118, 134, 152, 172, 201
12
2633.189
18, 31, 45, 59, 76, 93, 105, 118, 134, 152, 172, 201
13
2635.290
18, 31, 45, 59, 76, 92, 103, 115, 129, 144, 161, 179, 207
14
2637.088
17, 29, 42, 56, 72, 86, 96, 106, 117, 130, 145, 161, 179, 207
15
2638.449
17, 28, 39, 50, 61, 75, 88, 97, 107, 118, 131, 146, 162, 179, 207
16
2639.616
17, 27, 38, 49, 60, 74, 86, 95, 104, 113, 123, 135, 149, 164, 181, 209
Cactus
2
1816.448
73, 151
3
1970.112
64, 106, 173
4
2042.275
55, 87, 125, 187
5
2080.884
49, 76, 102, 138, 196
6
2104.147
46, 70, 93, 119, 157, 208
7
2119.670
44, 65, 85, 105, 131, 168, 215
8
2129.358
42, 60, 77, 94, 113, 138, 174, 218
9
2136.162
41, 58, 74, 89, 105, 125, 152, 186, 224
10
2141.579
40, 56, 71, 85, 99, 115, 135, 161, 193, 228
11
2145.397
39, 53, 66, 79, 91, 104, 119, 139, 165, 196, 229
12
2148.282
38, 51, 64, 76, 88, 100, 114, 131, 152, 176, 204, 233
13
2150.740
37, 49, 61, 72, 83, 94, 105, 118, 135, 155, 178, 205, 233
14
2152.626
36, 47, 58, 68, 78, 88, 98, 109, 122, 138, 158, 181, 207, 234
15
2154.162
36, 46, 56, 66, 76, 85, 94, 104, 115, 128, 144, 164, 187, 212, 237
16
2155.471
36, 46, 56, 66, 75, 84, 93, 102, 112, 124, 138, 155, 174, 195, 217, 239
Butterfly
2
3873.222
84, 155
3
3990.855
81, 144, 199
4
4051.357
77, 121, 167, 207
5
4092.929
60, 89, 129, 172, 209
6
4119.448
59, 87, 122, 161, 193, 221
7
4135.937
53, 74, 98, 128, 165, 195, 222
8
4148.556
53, 73, 96, 123, 156, 183, 203, 228
9
4156.528
52, 72, 94, 118, 144, 170, 190, 207, 231
10
4163.794
48, 63, 80, 99, 121, 147, 172, 191, 208, 231
11
4168.489
47, 62, 79, 98, 118, 141, 164, 183, 198, 213, 234
12
4172.517
46, 59, 73, 89, 104, 122, 144, 167, 185, 199, 214, 235
13
4175.487
46, 59, 72, 87, 102, 118, 137, 157, 175, 189, 202, 216, 236
14
4177.893
44, 55, 66, 79, 93, 106, 121, 141, 161, 178, 191, 203, 217, 237
15
4179.908
44, 55, 66, 78, 92, 105, 119, 137, 156, 173, 186, 197, 208, 221, 239
16
4181.559
42, 52, 61, 71, 83, 95, 107, 121, 139, 157, 173, 186, 197, 208, 221, 239
Circus
2
1651.257
118, 172
3
1760.512
105, 150, 187
4
1817.487
93, 132, 165, 195
5
1850.243
87, 122, 152, 177, 203
6
1870.083
82, 113, 141, 164, 185, 208
7
1883.966
77, 104, 129, 151, 171, 190, 212
8
1893.450
73, 98, 122, 143, 161, 178, 195, 216
9
1900.592
70, 92, 114, 134, 152, 168, 183, 199, 219
10
1905.992
68, 89, 109, 128, 145, 160, 174, 188, 203, 222
11
1909.887
66, 86, 105, 123, 139, 153, 166, 179, 192, 206, 225
12
1913.079
63, 81, 98, 114, 130, 145, 158, 170, 182, 194, 208, 226
13
1915.676
62, 79, 95, 111, 126, 140, 152, 164, 175, 186, 197, 210, 228
14
1917.756
61, 78, 94, 109, 123, 136, 148, 159, 170, 180, 190, 201, 214, 231
15
1919.524
59, 74, 88, 102, 115, 128, 140, 151, 161, 171, 181, 191, 202, 214, 231
16
1920.958
58, 73, 87, 100, 113, 126, 138, 149, 159, 169, 178, 187, 196, 206, 218, 234
Snow
2
5261.705
80, 169
3
5624.289
71, 139, 207
4
5729.116
50, 92, 144, 208
5
5785.138
49, 91, 140, 192, 231
6
5819.333
45, 81, 111, 148, 194, 232
7
5835.770
43, 76, 101, 127, 159, 196, 232
8
5850.190
30, 55, 83, 107, 133, 163, 197, 233
9
5862.437
30, 55, 82, 106, 129, 157, 185, 211, 237
10
5870.284
29, 52, 75, 94, 111, 132, 159, 186, 212, 237
11
5875.817
29, 52, 75, 94, 111, 132, 158, 183, 206, 227, 244
12
5880.335
29, 52, 74, 93, 109, 127, 149, 169, 188, 209, 228, 244
13
5884.513
23, 39, 57, 76, 94, 110, 128, 150, 170, 188, 209, 228, 244
14
5887.355
22, 37, 54, 70, 84, 98, 112, 129, 151, 170, 188, 209, 228, 244
15
5889.953
21, 36, 53, 69, 83, 97, 110, 124, 141, 159, 175, 192, 211, 229, 245
16
5891.998
21, 36, 53, 69, 83, 97, 110, 124, 141, 159, 174, 189, 207, 222, 236, 248
Palace
2
2623.440
99, 165
3
2791.488
84, 132, 186
4
2860.98
70, 103, 143, 191
5
2908.165
69, 101, 138, 177, 218
6
2934.330
64, 89, 117, 147, 181, 220
7
2953.745
54, 75, 99, 126, 153, 183, 220
8
2966.250
50, 69, 89, 111, 134, 158, 185, 221
9
2974.719
47, 65, 81, 100, 120, 141, 163, 188, 222
10
2981.875
47, 64, 80, 98, 117, 137, 157, 178, 199, 226
11
2986.898
45, 61, 75, 90, 106, 123, 141, 159, 179, 200, 227
12
2990.579
43, 58, 69, 82, 96, 111, 126, 143, 160, 180, 201, 227
13
2993.809
43, 58, 69, 81, 95, 110, 125, 141, 157, 173, 190, 208, 231
14
2996.112
43, 57, 68, 80, 94, 108, 123, 138, 153, 167, 182, 199, 218, 239
15
2998.213
42, 56, 66, 77, 88, 100, 113, 126, 140, 154, 167, 182, 199, 218, 239
16
2999.918
42, 55, 65, 75, 86, 98, 110, 123, 136, 149, 162, 176, 191, 205, 222, 241
Flower
2
1489.281
61, 130
3
1627.897
39, 77, 141
4
1685.956
36, 67, 105, 160
5
1715.220
28, 49, 75, 111, 164
6
1736.423
27, 47, 71, 102, 143, 192
7
1752.512
26, 43, 61, 83, 111, 151, 199
8
1761.968
25, 42, 58, 77, 99, 126, 161, 204
9
1768.354
24, 40, 54, 70, 88, 109, 135, 167, 207
10
1772.903
23, 37, 48, 60, 75, 92, 112, 138, 169, 208
11
1776.470
23, 36, 47, 58, 72, 87, 104, 124, 149, 177, 211
12
1779.106
22, 34, 44, 54, 65, 78, 92, 108, 128, 152, 179, 212
13
1781.251
20, 30, 39, 48, 58, 70, 83, 96, 112, 131, 154, 180, 213
14
1783.049
19, 29, 38, 47, 56, 66, 78, 90, 104, 120, 139, 161, 185, 215
15
1784.478
19, 29, 38, 46, 54, 63, 74, 85, 97, 111, 128, 147, 168, 190, 218
16
1785.641
19, 28, 37, 45, 53, 62, 72, 83, 94, 106, 120, 136, 155, 175, 196, 222
Wherry
2
3313.161
108, 189
3
3543.272
102, 161, 218
4
3599.924
83, 121, 163, 218
5
3634.708
81, 118, 152, 184, 224
6
3656.048
72, 103, 130, 156, 186, 225
7
3668.313
68, 94, 120, 139, 161, 189, 226
8
3678.216
60, 83, 110, 132, 152, 175, 198, 230
9
3686.001
56, 77, 100, 122, 138, 156, 179, 201, 231
10
3691.730
56, 76, 98, 120, 136, 153, 174, 194, 219, 243
11
3696.416
55, 74, 94, 114, 130, 142, 158, 178, 197, 222, 245
12
3699.866
54, 72, 91, 110, 126, 138, 151, 168, 185, 202, 225, 246
13
3702.619
52, 68, 83, 100, 117, 130, 140, 153, 169, 185, 202, 225, 246
14
3705.033
52, 68, 83, 100, 117, 130, 140, 152, 167, 182, 197, 215, 235, 249
15
3706.937
51, 66, 79, 94, 110, 123, 133, 142, 154, 169, 184, 199, 216, 235, 249
16
3708.484
49, 63, 75, 89, 104, 118, 129, 138, 147, 159, 173, 186, 200, 217, 235, 249
Waterfall
2
4512.801
88, 170
3
4646.137
72, 115, 182
4
4711.019
67, 103, 150, 204
5
4752.267
58, 87, 119, 164, 212
6
4777.063
53, 78, 104, 134, 176, 217
7
4793.510
48, 70, 92, 116, 147, 186, 221
8
4805.756
46, 66, 86, 107, 131, 163, 199, 226
9
4814.097
43, 61, 79, 97, 117, 141, 173, 205, 228
10
4820.724
40, 57, 73, 90, 108, 128, 152, 182, 210, 230
11
4825.739
38, 54, 69, 84, 100, 117, 137, 162, 191, 215, 232
12
4829.689
37, 53, 67, 81, 96, 111, 128, 148, 173, 198, 218, 233
13
4832.826
35, 50, 63, 76, 89, 103, 117, 133, 153, 177, 201, 220, 234
14
4835.371
32, 46, 58, 70, 82, 95, 108, 122, 138, 158, 182, 204, 221, 234
15
4837.537
32, 46, 57, 68, 80, 92, 104, 117, 131, 148, 169, 191, 210, 224, 236
16
4839.226
30, 43, 54, 64, 74, 85, 96, 108, 120, 134, 151, 172, 193, 211, 225, 236
Bird
2
901.450
71, 122
3
975.230
64, 111, 140
4
1027.509
61, 104, 131, 164
5
1051.482
54, 93, 119, 138, 169
6
1067.992
47, 82, 108, 127, 142, 172
7
1077.304
40, 70, 94, 113, 129, 143, 173
8
1086.005
39, 69, 93, 112, 128, 141, 159, 192
9
1091.341
37, 65, 88, 105, 119, 131, 142, 160, 193
10
1095.355
37, 64, 86, 103, 117, 129, 139, 149, 167, 200
11
1098.475
33, 56, 77, 93, 107, 119, 130, 140, 150, 169, 202
12
1100.749
33, 56, 77, 93, 107, 119, 129, 138, 146, 158, 178, 209
13
1102.639
31, 52, 71, 87, 100, 111, 121, 130, 139, 147, 159, 179, 210
14
1104.132
29, 48, 66, 82, 95, 107, 118, 127, 134, 141, 149, 161, 181, 211
15
1105.446
28, 45, 62, 77, 90, 101, 111, 120, 128, 135, 142, 150, 162, 182, 212
16
1106.500
28, 45, 62, 77, 90, 101, 111, 120, 128, 135, 141, 148, 157, 171, 191, 219
Police
2
3647.353
74, 150
3
3844.314
70, 135, 192
4
3966.225
63, 112, 158, 209
5
4013.875
61, 104, 140, 174, 214
6
4047.198
32, 67, 106, 141, 175, 214
7
4067.996
32, 67, 104, 133, 161, 186, 219
8
4084.933
29, 52, 78, 106, 134, 162, 187, 219
9
4094.705
29, 52, 78, 103, 125, 147, 169, 190, 220
10
4101.702
29, 52, 78, 102, 123, 143, 165, 184, 203, 228
11
4108.018
28, 49, 71, 90, 107, 126, 146, 166, 185, 204, 229
12
4112.304
28, 49, 71, 89, 105, 122, 139, 157, 174, 189, 207, 231
13
4115.165
28, 46, 62, 78, 92, 106, 123, 140, 158, 174, 189, 207, 231
14
4117.926
28, 46, 62, 78, 91, 105, 120, 135, 151, 166, 180, 193, 210, 232
15
4120.045
28, 46, 62, 78, 91, 103, 116, 129, 143, 158, 172, 185, 197, 214, 235
16
4121.861
28, 46, 62, 78, 91, 103, 116, 128, 142, 156, 170, 182, 193, 206, 223, 240
Ostrich
2
1073.452
75, 135
3
1139.260
69, 101, 149
4
1178.650
65, 92, 125, 176
5
1203.749
56, 78, 100, 131, 179
6
1218.643
47, 65, 85, 103, 133, 181
7
1228.925
47, 64, 83, 100, 122, 152, 192
8
1236.023
45, 59, 75, 90, 104, 125, 155, 194
9
1240.756
40, 52, 65, 80, 94, 107, 128, 157, 195
10
1244.909
40, 52, 64, 78, 91, 103, 119, 141, 168, 201
11
1247.879
40, 51, 62, 75, 87, 97, 108, 125, 148, 174, 205
12
1250.313
37, 48, 57, 68, 80, 91, 101, 112, 128, 150, 175, 206
13
1252.298
31, 44, 53, 63, 75, 86, 95, 105, 117, 134, 154, 178, 207
14
1253.956
29, 42, 50, 58, 68, 79, 89, 98, 108, 120, 137, 157, 181, 209
15
1255.366
29, 42, 50, 58, 67, 77, 86, 94, 102, 111, 124, 141, 160, 183, 210
16
1256.490
28, 41, 49, 57, 66, 76, 85, 93, 101, 110, 122, 137, 155, 174, 196, 219
Viaduct
2
7920.458
77, 180
3
8117.991
54, 109, 193
4
8203.807
42, 84, 131, 203
5
8246.806
35, 68, 103, 146, 210
6
8272.775
31, 59, 88, 120, 160, 216
7
8287.714
28, 53, 77, 103, 132, 169, 220
8
8298.322
27, 51, 75, 100, 128, 164, 212, 246
9
8308.240
24, 45, 66, 88, 112, 139, 172, 216, 247
10
8315.133
22, 41, 60, 79, 99, 121, 146, 178, 219, 247
11
8320.032
20, 37, 54, 71, 89, 108, 128, 152, 183, 221, 248
12
8323.799
20, 36, 52, 68, 84, 101, 119, 139, 162, 190, 224, 248
13
8326.793
18, 33, 48, 62, 77, 92, 108, 125, 144, 167, 195, 226, 248
14
8329.119
17, 31, 44, 57, 70, 84, 98, 113, 129, 148, 170, 197, 227, 248
15
8330.983
17, 31, 44, 57, 70, 84, 98, 113, 129, 147, 169, 195, 224, 243, 251
16
8332.744
16, 29, 42, 54, 66, 78, 91, 104, 118, 133, 151, 172, 196, 224, 243, 251
Fish
2
3593.389
64, 148
3
3870.456
44, 104, 177
4
3972.731
34, 81, 127, 188
5
4024.885
27, 63, 101, 139, 194
6
4054.836
24, 56, 90, 123, 156, 205
7
4075.236
22, 49, 78, 107, 135, 168, 213
8
4088.300
20, 44, 68, 93, 118, 142, 173, 216
9
4097.101
19, 40, 62, 85, 107, 128, 149, 178, 218
10
4103.194
18, 38, 58, 78, 98, 118, 137, 158, 185, 222
11
4107.954
16, 34, 51, 69, 88, 107, 125, 143, 163, 190, 224
12
4111.678
15, 31, 47, 64, 82, 100, 117, 133, 149, 169, 195, 227
13
4114.783
14, 28, 43, 58, 74, 90, 106, 121, 136, 152, 172, 198, 228
14
4116.954
14, 28, 43, 58, 73, 88, 103, 117, 131, 145, 160, 179, 203, 231
15
4118.931
13, 25, 38, 51, 64, 78, 92, 106, 120, 133, 147, 162, 181, 205, 232
16
4120.522
13, 25, 37, 49, 62, 75, 89, 102, 115, 127, 139, 152, 168, 188, 211, 235
Houses
2
2543.788
56, 116
3
2627.230
53, 105, 150
4
2663.698
42, 69, 110, 152
5
2694.902
39, 65, 96, 130, 158
6
2708.655
39, 65, 95, 127, 150, 169
7
2720.263
35, 55, 74, 98, 128, 151, 170
8
2726.676
35, 55, 74, 98, 127, 148, 164, 184
9
2732.853
32, 48, 65, 80, 101, 128, 148, 164, 184
10
2736.510
30, 45, 60, 74, 87, 106, 129, 149, 164, 184
11
2739.566
30, 45, 60, 73, 86, 105, 128, 145, 156, 168, 187
12
2742.148
29, 42, 55, 68, 79, 94, 112, 130, 145, 156, 168, 187
13
2744.071
28, 39, 51, 63, 74, 84, 98, 115, 131, 145, 156, 168, 187
14
2745.542
28, 39, 51, 63, 74, 84, 98, 115, 131, 145, 155, 165, 176, 193
15
2746.817
27, 37, 47, 58, 68, 77, 86, 100, 116, 131, 145, 155, 165, 176, 193
16
2747.991
27, 37, 47, 57, 67, 76, 85, 98, 113, 127, 139, 147, 156, 165, 176, 193
Mushroom
2
1988.328
76, 145
3
2153.037
65, 110, 174
4
2237.441
59, 93, 135, 193
5
2277.427
52, 78, 106, 145, 199
6
2301.629
48, 71, 94, 122, 157, 205
7
2317.538
46, 67, 87, 110, 138, 171, 213
8
2328.526
44, 62, 79, 97, 118, 145, 177, 216
9
2335.812
42, 58, 74, 90, 107, 127, 152, 181, 218
10
2340.981
41, 56, 70, 84, 99, 116, 136, 159, 186, 220
11
2345.160
39, 52, 65, 78, 92, 107, 124, 144, 166, 191, 223
12
2348.413
38, 51, 63, 75, 87, 100, 115, 132, 152, 174, 199, 227
13
2350.988
38, 50, 62, 74, 85, 97, 110, 125, 142, 160, 180, 203, 229
14
2353.119
37, 48, 59, 69, 79, 89, 100, 113, 127, 144, 162, 181, 204, 230
15
2354.748
36, 46, 56, 66, 75, 84, 94, 105, 117, 130, 146, 163, 182, 205, 230
16
2356.113
36, 46, 56, 65, 74, 83, 92, 102, 113, 125, 139, 154, 169, 187, 208, 232
Snow mountain
2
1912.613
85, 149
3
2135.274
79, 137, 197
4
2234.669
70, 119, 154, 204
5
2288.799
55, 96, 129, 160, 206
6
2317.224
52, 90, 118, 142, 167, 209
7
2333.078
50, 85, 111, 132, 153, 174, 212
8
2344.629
45, 76, 100, 120, 139, 158, 177, 214
9
2352.951
44, 74, 97, 116, 133, 150, 167, 187, 220
10
2359.175
41, 68, 90, 108, 124, 140, 156, 172, 193, 225
11
2364.304
38, 62, 84, 102, 117, 132, 146, 160, 175, 196, 227
12
2367.971
34, 56, 78, 96, 111, 125, 139, 152, 165, 179, 199, 228
13
2371.208
33, 53, 73, 90, 104, 117, 130, 143, 155, 167, 180, 200, 229
14
2373.567
32, 52, 72, 89, 103, 115, 127, 139, 150, 161, 172, 185, 205, 232
15
2375.586
31, 49, 68, 84, 98, 109, 120, 131, 142, 153, 164, 175, 188, 208, 233
16
2377.255
26, 42, 59, 75, 89, 101, 111, 121, 132, 143, 154, 165, 176, 189, 209, 234
Snake
2
1118.615
87, 134
3
1231.320
76, 114, 154
4
1286.555
69, 101, 129, 166
5
1317.027
63, 91, 115, 140, 175
6
1336.172
59, 84, 105, 126, 149, 182
7
1348.933
55, 78, 97, 115, 133, 155, 187
8
1357.665
52, 73, 91, 107, 123, 140, 161, 192
9
1364.159
50, 70, 87, 102, 116, 131, 148, 169, 198
10
1368.955
47, 65, 81, 95, 108, 121, 135, 151, 172, 200
11
1372.626
46, 63, 78, 91, 103, 115, 127, 140, 156, 176, 203
12
1375.513
45, 61, 75, 88, 99, 110, 121, 133, 146, 162, 182, 208
13
1377.847
43, 58, 72, 84, 95, 106, 116, 127, 138, 151, 166, 185, 211
14
1379.736
42, 56, 69, 81, 92, 102, 112, 122, 132, 143, 155, 170, 189, 214
15
1381.282
41, 54, 66, 77, 87, 97, 106, 115, 124, 134, 145, 157, 172, 191, 215
16
1382.603
40, 52, 63, 74, 84, 93, 102, 111, 120, 129, 139, 150, 162, 177, 195, 219
Riosanpablo
2
2667.020
95, 160
3
2818.660
75, 121, 177
4
2892.439
68, 102, 143, 189
5
2931.654
62, 89, 121, 158, 197
6
2957.018
58, 82, 107, 138, 171, 204
7
2973.269
53, 74, 95, 119, 147, 177, 207
8
2984.972
50, 69, 87, 108, 133, 159, 185, 212
9
2993.359
48, 66, 83, 101, 122, 145, 168, 191, 215
10
2999.615
46, 63, 78, 93, 110, 130, 151, 173, 195, 217
11
3004.374
45, 61, 75, 89, 104, 121, 140, 160, 180, 200, 220
12
3008.082
44, 59, 72, 84, 97, 112, 129, 147, 166, 185, 203, 222
13
3011.101
43, 57, 69, 81, 93, 107, 122, 138, 155, 172, 189, 206, 224
14
3013.465
41, 54, 66, 77, 88, 100, 113, 128, 144, 160, 176, 192, 208, 225
15
3015.423
40, 53, 64, 74, 84, 95, 107, 121, 135, 150, 165, 180, 195, 210, 226
16
3017.081
39, 51, 62, 72, 82, 92, 103, 115, 128, 142, 156, 170, 184, 198, 212, 227We can see from these results thatrank-DE has a higher approximated TVD growth rate than DE. O
(
β
)
R
-DE with β
≤
0.5 and O
(
rand
)
R
-DE are still better than DE. O
(
0.0
)
R
-DE produced the best result with a very flat slope and a very low y-intercept. The multithresh function yielded a higher growth rate of solution distortion and a higher y-intercept than the other algorithms. The higher growth rate of solution distortion means the quality of solution drops very fast if the number of thresholds increases. The higher y-intercept means that the solution distortion at the lowest number of thresholds is high. Thus, the global optimization algorithm is required for solving the multilevel thresholding.


## 6. Conclusions

The differential evolution with onlooker(
β
) ranking-based mutation operator O
(
β
)
R
-DE algorithm was proposed and applied to the multilevel image thresholding problem. The objective of this proposed algorithm was to increase the ability for adjusting the balance of the exploitation and exploration abilities. Its concept is a combination of the ranking-difference evolution and onlooker selection of the ABC algorithm. The experiments compared the proposed O
(
β
)
R
-DE algorithm with six existing algorithms: PSO, DPSO, FODPSO, ABC, DE, and rank-DE on 20 real images of the Berkeley Segmentation Dataset and Benchmark and a satellite image. The stability analysis, convergence speed, and the reliability were measured. The results signified that the proposed O
(
β
)
R
-DE algorithm is more efficient than the six tested algorithms. The onlooker ranking-based mutation operator is able to enhance the performance of the proposed algorithm. The O
(
β
)
R
-DE not only obtained more stability analysis, but it also achieved faster convergence rates to reach the target BCV, if a proper value of β is set.For future work based on this paper, the proposedO
(
β
)
R-DE algorithm has one parameter to be set by a user; the mechanism to automatically adapt this parameter is not presented but is required.



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*Source: 974024-2014-03-20.xml*