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mathematics
Article
Modified Artificial Ecosystem-Based Optimization for
Multilevel Thresholding Image Segmentation
Ahmed A. Ewees 1,2, Laith Abualigah 3,4 , Dalia Yousri 5, Ahmed T. Sahlol 2, Mohammed A. A. Al-qaness 6,
Samah Alshathri 7,* and Mohamed Abd Elaziz 8,9,10
Citation: Ewees, A.A.; Abualigah, L.;
Yousri, D.; Sahlol, A.T.; Al-qaness,
M.A.A.; Alshathri, S.; Elaziz, M.A.
Modified Artificial Ecosystem-Based
Optimization for Multilevel
Thresholding Image Segmentation.
Mathematics 2021,9, 2363.
https://doi.org/10.3390/
math9192363
Academic Editor: Dumitru Baleanu
Received: 12 July 2021
Accepted: 15 September 2021
Published: 23 September 2021
Publisher’s Note: MDPI stays neutral
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Copyright: © 2021 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
1Department of e-Systems, University of Bisha, Bisha 61922, Saudi Arabia; ewees@du.edu.eg
2Department of Computer, Damietta University, Damietta 34511, Egypt; atsegypt@du.edu.eg
3Faculty of Computer Sciences and Informatics, Amman Arab University, Amman 11953, Jordan;
aligah.2020@gmail.com
4School of Computer Sciences, Universiti Sains Malaysia, Pulau Pinang 11800, Malaysia
5Electrical Engineering Department, Faculty of Engineering, Fayoum University, Faiyum 63514, Egypt;
day01@fayoum.edu.eg
6State Key Laboratory for Information Engineering in Surveying, Mapping and Remote Sensing,
Wuhan University, Wuhan 430079, China; alqaness@whu.edu.cn
7
Department of Information Technology, College of Computer and Information Sciences, Princess Nourah Bint
Abdulrahman University, Riyadh 84428, Saudi Arabia
8Department of Mathematics, Faculty of Science, Zagazig University, Zagazig 44519, Egypt;
abd_el_aziz_m@yahoo.com
9Artificial Intelligence Research Center (AIRC), Ajman University, Ajman 346, United Arab Emirates
10 School of Computer Science and Robotics, Tomsk Polytechnic University, 634050 Tomsk, Russia
*Correspondence: sealshathry@pnu.edu.sa
Abstract:
Multilevel thresholding is one of the most effective image segmentation methods, due to
its efficiency and easy implementation. This study presents a new multilevel thresholding method
based on a modified artificial ecosystem-based optimization (AEO). The differential evolution (DE) is
applied to overcome the shortcomings of the original AEO. The main idea of the proposed method,
artificial ecosystem-based optimization differential evolution (AEODE), is to employ the operators
of the DE as a local search of the AEO to improve the ecosystem of solutions. We used benchmark
images to test the performance of the AEODE, and we compared it to several existing approaches.
The proposed AEODE achieved a high performance when evaluated by the structural similarity index
(SSIM), peak signal-to-noise ratio (PSNR), and fitness values. Moreover, the AEODE outperformed
the basic version of the AEO concerning SSIM and PSNR by 78% and 82%, respectively, which
reserves the best features for each of AEO and DE.
Keywords:
image segmentation; multilevel thresholding; artificial ecosystem-based optimization
(AEO); differential evolution (DE); optimization algorithms
1. Introduction
Image segmentation is one of the primary and essential operations for pattern recogni-
tion and image analysis. Over the past years, various segmentation techniques have been
introduced to improve the segmentation processes, such as fuzzy c-means, fuzzy clustering
algorithms, and mean-shift analysis. The image thresholding approach is a successful and
standard image segmentation method among other similar segmentation techniques. This
is because of its clarity, robustness, and efficiency, which transforms a gray level image
into a binary level image [
1
]. It is the process of partitioning an image into its component
objects or regions based on color, size, shape, position, or texture [2].
There are several image segmentation techniques. However, multilevel thresholding
is the most successful technique employed for segmentation purposes of different types
of images out of all the current techniques [
3
]. The motivations behind the widespread
Mathematics 2021,9, 2363. https://doi.org/10.3390/math9192363 https://www.mdpi.com/journal/mathematics
Mathematics 2021,9, 2363 2 of 25
application of multilevel thresholding are due to its simplicity and computation efficiency.
It is employed to recognize and obtain a target from the image background depending on
the allocation of gray levels or texture in image components (objects). So, understanding
or interpretation of a given image needs to precisely segment it into vital regions [
4
].
Therefore, it plays a vital role in computer vision domains.
Thresholding methods are divided into local and global methods. The global methods
are arranged in terms of the clustering-based information into spatial-based, entropy-based,
object attribute-based, and histogram shape-based methods [
5
]. These methods are usually
divided into two types: bi-level and multilevel [
6
]. In the bi-level, the image is divided
into two categories [
7
]. In contrast, multilevel is utilized for detailed image segmentation
because it provides multiple levels, such as quad-level or tri-level, and breaking pixels into
multilevel depending on the image’s pixels. Therefore, image segmentation by multilevel
thresholding has drawn more consideration because of its various applications in different
fields [
8
]. However, determining the optimal threshold value is still the most challenging
problem encountered in thresholding techniques, and it needs further investigation.
For a precise description of the thresholds, different variance measures as the entropy
are employed. Two conventional techniques are the intra-class variance [
9
] and Kapur’s
entropy [
10
]. These techniques are beneficial in obtaining a single threshold. However,
conventional techniques suffer from some limitations, such as computational time. In ad-
dition to that, the complexity rises considerably when dividing the image into more than
two classes is necessary. The applications of meta-heuristic optimization methods were
proposed in many studies to address this problem, which presented good results [11].
Optimization algorithms are promising mechanisms to perform complicated optimiza-
tion tasks with a high level of accuracy [
12
]. In the published literature, several methods
were proposed for multilevel thresholding, such as the stochastic fractal search (SFS) algo-
rithm, spherical search optimization (SSO) algorithm, artificial bee colony (ABC) algorithm,
sine–cosine algorithm (SCA), differential evolution (DE) algorithm, multi-verse optimiza-
tion (MVO) algorithm, particle swarm optimization (PSO) algorithm, whale optimization
algorithm (WOA), genetic algorithm (GA), social spider optimization (SSO), honey bee
mating optimization (HBMO), flower pollination algorithm (FPA) algorithm, crow search
algorithm (CSA), and world cup optimization (WCO) algorithm [
13
]. However, as every
image is considered an optimization problem, not all optimization algorithms give precise
results by applying the Otsu and Kapur as objective functions. In particular, an algorithm
can obtain the optimal configuration of thresholds for a particular image but not for all
given images.
Zhao et al. in [
14
] proposed a new optimization algorithm inspired by nature, called
artificial ecosystem-based optimization (AEO). This algorithm is a warm-based search
method driven by the flow of power in an environment on the ground, and it simulates
three different styles of living organisms: production, consumption, and decomposition.
The authors experimented using different test functions and optimization tasks. The results
showed that the AEO has an advantage over other equivalents. AEO is superior to other
comparative algorithms in both computational cost and convergence speed, particularly for
engineering applications. In general, the AEO is prevalent, which motivates us to employ
it for solving multilevel thresholding problems.
This paper develops an efficient image segmentation method depending on a modified
version of the AEO, using the differential evolution (DE) strategy. The proposed method,
called AEODE, intends to evade the shortcomings of the conventional AEO by applying
the operators of DE as an exploitation strategy (local search) to enhance the ecosystem
of solutions. The power of these combined algorithms is employed for enhancing the
performance of the multilevel thresholds image segmentation. DE is a search method
that optimizes the given problem by iteratively working to enhance a solution concerning
the fitness function [
15
]. This method was used before in many engineering problems
and applications, such as engineering problems, numerical function optimization, feature
selection, big data optimization problems, text clustering, reactive power dispatch problem,
Mathematics 2021,9, 2363 3 of 25
solar photovoltaic and wind energy, multilevel image thresholding, optimal power flow,
and continuous optimization problems.
Generally, the proposed AEODE method works on calculating the histogram of the
provided image. Then, it creates random candidate solutions subject to the minimum and
maximum histogram values. Each candidate solution includes a collection of position
values that design the threshold values. The fuzzy entropy is employed as a fitness function
to assess the quality of the candidate solutions, and the best-obtained solution has the
most significant fitness value. After that, the current candidate solutions are updated
according to a condition with a random value to operate the AEO or DE. After completing
the updating procedure for the given solutions, the terminal criteria are checked to verify
whether they are satisfied or not, so the preceding steps are executed again. We implement
extensive experiments, using benchmark images to evaluate the AEODE method; then, we
compare it to many existing optimization methods.
Our main contribution can be summarized as follows:
•
Propose an alternative multilevel thresholding image segmentation method based on
hybrid AEO and DE algorithms.
•
Blend the strength of the AEO and the DE to develop the proposed AEODE method
for optimizing the thresholding value.
• Evaluate the efficiency of the AEODE, using a set of benchmark images.
•
Compare the proposed image segmentation method with the state-of-the-art methods
published in that domain.
The remainder of this paper is presented as follows. Section 2presents the related
works that studied the image segmentation problems, using optimization techniques.
Section 3
presents the preliminaries of the methods. The description of the AEODE method
is presented in Section 4. The experiment and comparisons are described in Section 5.
Section 6concludes the paper.
2. Literature Review
One of the most vital techniques utilized in the computer vision domain is multilevel
thresholding. Conventional segmentation techniques suffer from difficulties in settling
the proper threshold values; hence, meta-heuristic (MH) algorithms are successfully ap-
plied to address these difficulties. Generally, meta-heuristic algorithms are introduced by
mimicking natural behaviors of groups in the environment, such as wild animals, flying
animals, science, theories, and others. In [
16
], a new multilevel thresholding approach
was presented based on a new version of the spherical search optimization (SSO) algo-
rithms, called SSOSCA. The proposed method used operators from the components of
the conventional SCA to improve the exploitation searchability of the conventional SSO
algorithm. Fuzzy entropy was employed as the leading fitness function to assess the nature
of candidate solutions of the SSOSCA. In the experiments, several images were taken from
Berkeley benchmark datasets to analyze and test the performance of the SSOSCA. The
results showed that the SSOSCA achieved competitive performance, compared to several
existing algorithms based on various image segmentation metrics.
Horng in [
17
] presented a multilevel maximum entropy thresholding method, using
the ABC algorithm, called (MEABCT). Four approaches were compared to the MEABCT
algorithm: the PSO algorithm, cooperative–comprehensive PSO, HBMO, and Fast Otsu’s
approach. The evaluation results showed that the MEABCT could explore multiple thresh-
old values near the optimal value observed by the optimization method. The segmentation
results of the MEABCT algorithm are better than the other methods; nevertheless, the com-
putation time by utilizing the proposed MEABCT is less than that of the other comparative
algorithms.
Yousri et al. [
18
] proposed a new method to improve the search abilities of the FPA.
Accordingly, fractional-order (FO) calculus characteristics were utilized to improve the
exploitation searchability of the conventional FPA and adaptively enhance the harmoniza-
tion parameter between the FPA exploitation and exploration search. It was tested on
Mathematics 2021,9, 2363 4 of 25
different test functions with various dimensions. In addition, it was compared with similar
established techniques in the literature, using different statistical evaluation measures and
non-parametric analyses. Moreover, it was implemented for a real-word image segmenta-
tion purposes, and the obtained results were compared to several existing methods.
Forouzanfar et al. in [
19
] proposed a study to examine the potential of the GA and PSO
algorithms to determine the optimum value of the degree of attraction of the segmentation
processes. The GA showed better performance in determining a close-optimal solution;
however, it had an issue in obtaining an exact solution, while the PSO algorithm enhanced
the exploration for finding the optimal solution. Consequently, vital changes are expected
by utilizing a hybrid method by combining the PSO with GA algorithms. In this way, the
hybrid GA-PSO algorithm is used to find the optimum degree of attraction in the image
segmentation process. The quantitative and qualitative analyses are conducted on image
benchmark datasets and brain MR images with various noise shapes. The results showed
unique enhancements in segmentation results matched to other fuzzy c-means methods.
In [
20
], the authors presented a clustering approach, using PSO for image segmenta-
tion, called divergent–convergent PSO (DCPSO). The DCPSO automatically manages the
number of groups and concurrently gathers the dataset with the smallest user interference.
The DCPSO begins by clustering the dataset into many clusters to minimize the impacts
of initial shapes. The optimal number of clusters is selected by using the binary PSO.
Then, the K-means clustering algorithm perfects the centers of the determined clusters.
The proposed algorithm was tested on both natural and synthetic image datasets. The
results revealed that the DCPSO algorithm obtained the optimal number of clusters on
the experimented image datasets, compared to other similar methods, such as GA and the
conventional PSO algorithm.
In [
21
], a new image segmentation method is presented, using many-objective opti-
mization: knee evolutionary algorithm (KnEA) is one of the best multi-objective methods
utilized to determine solutions, using seven functions to improve the effectiveness of
the segmentation process. The KnEA was assessed, utilizing several images tested at six
various threshold levels, and the results were compared to several existing many-objective
optimization techniques. Comparison of the results showed that the KnEA performs better
in estimating the optimal solutions than other methods in terms of the computational time,
and the segmented quality, such as SSIM, PSNR, coverage, hypervolume, and spacing
indicators.
Smith in [
22
] proposed a segmentation method, using the RFC algorithm. The pro-
posed method quantitatively assesses various possible image segmentation alternatives
to distinguish the segmentation scale(s). The selection process of the image segmentation
rule was utilized to distinguish between three basic image object rules after combination.
Then, the RFC algorithm was applied to determine the land cover classes to 11 gray-scale
images of SPOT satellite imagery. The results showed that the RFC algorithm achieved
better performance with an average accuracy of 85.2%. In another study, Shahrezaee [
23
]
proposed an image segmentation approach based on the WCO algorithm. The proposed
approach used the WCO to analyze the original pixels of images into various groups. The
performance of the WCO was evaluated and compared to several existing approaches,
such as Otsu, GA, and PSO, and it achieved good performance.
In [
24
], the authors proposed a multi-objective optimization method, using the MVO
algorithm for gray-scale image segmentation by multi-thresholding values. An approxi-
mate Pareto-optimal set was involved in increasing the Otsu and Kapur objective functions
in their method. These functions are usually employed in solving image segmentation
through bi-level and multi-level thresholding. Nevertheless, each function has specific
conditions and properties. Several optimization algorithms were introduced to optimize
these functions in terms of efficiency independently. In contrast, just a few multi-objective
methods have investigated the advantages of utilizing a combination of Otsu and Kapur.
However, the computational damage of Otsu and Kapur is extensive, and their efficiency
needs to be enhanced. Eleven gray-scale images were utilized to test the multi-objective
Mathematics 2021,9, 2363 5 of 25
MVO. Moreover, it was compared with three different multi-objective methods, and it
obtained significant performance.
In conclusion, we notice that optimization-based multilevel thresholding image seg-
mentation is considered an emerging research field with new and exciting theories and
strategies. It is also recognized as a critical problem to employ various techniques with
various objective functions over varying models of images.
3. Background
3.1. Problem Definition
In this section, we briefly define the problem of multilevel thresholding. If we have
I
gray-scale image, it will have
K+
1 classes. For dividing the
I
image into classes, we need
the values of kthresholds {tk,k=1, 2, K}, and this can be formulated as follows [25]:
C0={Iij |0≤Iij ≤t1−1},
C1={Iij |t1≤Iij ≤t2−1},
. . .
CK={Iij |tK≤Iij ≤L−1}
(1)
in which
L
is the maximum gray levels, and
CK
is a
k
th class of the
I
. The
tk
is the
k
-th
threshold, where
Iij
is a gray level at
(i
,
j)
-th pixel. Moreover, a multilevel threshold-
ing problem is defined as a maximization problem that is employed to find the optimal
threshold, as given by Equation (2):
t∗
1,t∗
2, . . . , t∗
K=arg max
t1,...,tK
Fit(t1, . . . , tK)(2)
where
Fit
represents the objective function. In this paper, we use fuzzy entropy [
9
] as the
objective function. Fuzzy entropy was used in various segmentation approaches [
26
,
27
],
and it can be formulated as follows:
Fit(t1, . . . , tK) =
K
∑
k=1
Hi(3)
Hk=−
L−1
∑
i=0
pi×µk(i)
Pk
×ln(pi×µk(i)
Pk
), (4)
Pk=
L−1
∑
i=0
pi×µk(i)(5)
µ1(l) =
1l≤a1
l−c1
a1−c1a1≤l≤c1
0l>c1
(6)
µK(l) =
1l≤aK−1
l−aK
cK−aKaK−1<l≤cK−1
0l>cK−1
where
pi
is the probability distribution, which is computed as
pi=h(i)/Np
(0
<i<L−
1).
h(i)
and
Np
represent pixel numbers for the corresponding gray level
L
and total pixel
numbers of the
I
. Furthermore,
a1
,
c1
,
. . .
,
ak−1
,
ck−1
represent the fuzzy parameters, and
0
≤a1≤c1≤ · · · ≤ aK−1≤cK−1
. Then
t1=a1+c1
2
,
t2=a2+c2
2
,
. . .
,
tK−1=aK−1+cK−1
2
. The
relevant fitness value is the highest one.
Mathematics 2021,9, 2363 6 of 25
3.2. Differential Evolution (DE)
Storn and Price [
15
] presented the DE as a first attempt in 1997 for solving several
optimization problems. DE is distinguished by its flexibility, short execution time, rabid
acceleration trend, and its fast and precise local operator for selection. The optimization
process in DE starts with a random set of solutions for discovering most of the points in
the search space (initialization phase). After that, the solutions can be modified based on a
set of operators, which are mutation and crossover, and then the agent solution is upgraded
when the newly generated solution achieves a better objective value. The mathematical
model of the mutation operator
Zt
i
can be implemented for the current individual
XDi
as
below:
Zt
i=XDt
rand1+F×(XDt
rand2−XDt
rand3), (7)
where
rand1
,
rand2
, and
rand3
are random indexes that are varied from the current index
i
.
F
is the mutation scaling factor, and it has a value greater than 0. The symbol of
t
refers to
the iteration number.
For the crossover operator, Equation (8), below, illustrates that during the crossover
process, a new solution vector
Vi
is created based on the mutated individual
Zi
and the
solutions vector
XDi
. The crossover process is accounted for as a mixture between vectors
Ziand XDi, where the diversity of the agents is enhanced.
Vt
i=(ziti f rand ≤Cr
XDt
iotherwise (8)
where rand is a random value in the [0, 1] interval, and Cris the crossover probability.
The final phase in the DE algorithm is upgrading the agents’ solutions based on the
attained objective values, where the generated individual
Vt
i
is exchanged with the current
individual if it has a better objective as follows:
XDt+1
i=(Vt
ii f f (vt
i)<f(XDt
i)
XDt
iotherwise (9)
These steps are repeated until the stopping criteria are met.
3.3. Artificial Ecosystem-Based Optimization
AEO is a new optimization method inspired by the chain energy transfer among living
organisms [
14
]. This process happens through three sequential processes: production, con-
sumption, and decomposition. Zhao et al. [
14
] represented these processes mathematically
to achieve optimal solutions for several optimization problems; thus, the AEO optimizer
was proposed. The production process is responsible for improving the balance among
the diversification and intensification perspectives in AEO, whereas for discovering the
total search space, the consumption operator is implemented (diversification perspective).
The intensification perspective can be excluded in the decomposition process. In AEO,
the producer (plants) and decomposer (fungi and bacteria) are only one agent, while the
rest of the agents can be considered consumers that can be classified based on their diet.
These animal types are (1) herbivores (eat plants), (2) omnivores (eat plants and other
animals), and (3) carnivores (eat only other animals). The agents in this hierarchy update
their positions, according to the following equations.
1.
Production process: Zhao et al. [
14
] considered the producer as the worst agent in
the populations. In AEO, the producer updates its location depending on randomly
selected individual in the search space, as well as the decomposed (best agent) as
presented below:
X1(t+1) = (1−d)Xn(t) + d·Xrand(t)(10)
d= (1−t
Tmax
)rand1(11)
Mathematics 2021,9, 2363 7 of 25
Xrand(t) = rand2() ·(Imax −Imin) + lb, (12)
where
Imax
and
Imin
are the upper and lower boundaries of the search space, respec-
tively.
t
and
Tmax
refer to the iteration number and the maximum number of iterations,
respectively.
rand1
and
rand2
represent random values withdrawn from the [0, 1]
interval, and drepresents a weight coefficient.
2.
Consumption process: in this stage, the consumer feeds on another consumer with
a lower level of energy or on a producer. For the consumer classes, carnivores,
herbivores, and omnivores, each one has its specific tactic for updating the location as
illustrated below:
(a)
Herbivore consumers update their locations depending on the producers only
(feed on the producers).
Xi(t+1) = Xi(t) + G·(Xi(t)−X1(t)) (13)
where producer
X1
represents the producer location, and
G
represents a
consumption factor, which is computed using levy flight as follows:
G=1
2
u
v,u∈Norm(0, 1),v∈Norm(0, 1)(14)
where
Norm(
0, 1
)
represents the normal distribution with zero mean and unit
variance.
(b)
Carnivore consumers update their locations using a random consumer with
high-level energy with an index
(ind)
, as carnivores feed on other animals
only as mentioned earlier. The carnivore location can be formulated as follows:
Xi(t+1) = Xi(t) + G·(Xi(t)−Xind (t)) (15)
ind =randi([2i−1]),i=3, . . . , N, (16)
(c)
Omnivore consumers update their locations depending on the producer and
randomly selected consumer with index
(ind)
with a higher level of energy as
modeled below:
Xi(t+1) = Xi(t) + G·Xra +Xr a2, (17)
Xra = (rand3·(Xi(t)−X1(t))), (18)
Xra2(1−rand3)·(Xi(t)−Xind(t)), (19)
ind =randi([2i−1]),i=3, . . . , N, (20)
3.
Decomposition process: this is the final process of the ecosystem as each individual in
the agent dies, and the decomposer starts breaking down its remains.
Zhao et al. [14]
considered it to be the intensification phase of AEO and modeled it as follows:
Xi(t+1) = Xi(t) + D·(e·Xn(t)−q·Xi(t)), (21)
D=3u,u∈N(0, 1)(22)
e=rand4·randi([1 2]) −1, (23)
q=2·rand5−1, (24)
where the decomposition factor is represented by
D
, and weight parameters are represented
by eand q.
4. The Proposed AEODE
The structure of the proposed method based on modified the AEO using DE is given
in Figure 1. The developed AEODE uses the operators of DE to enhance the exploitation of
Mathematics 2021,9, 2363 8 of 25
AEO. It has the largest effect on AEO performance and provides it with suitable operators
to avoid the attractiveness to a local point.
End
Termination criteria is
met?
Display the Optimsl parameters
Yes
No
Update the solutions for production
phase of Eq.(10)
Sort the solutions and assign the best
Compute the histogram
Start
Set the number of iterations, population
size (N), the threshold levels (K)
Set the Initial population using Eq.(16)
Compute the initial values of µ1, µk, Pk
using Eqs. (6,5) then calculate the Hk
using Eq. (4) then calculate the fitness
function using Eq. (3)
Evaluate the objective function and determine the
best
1/3 < t & Rand
< 2 /3
Yes
No
Rand < 1/3 No
Herbivore consumers
location update using
Eq.(11)
Carnivore consumers
location update using
Eq.(13)
Omnivore consumers
location update using
Eq.(14)
Computer probability (Pr) using Eq.(18)
and Control operator (rs) using Eq.(19)
Yes
Pr > rs UPdate The solutions
via DE operator using
Eq.(9)
Update the solutions in
decomposition process
using Eq.(15)
Yes No
Compute the initial values of µ1, µk, Pk using Eqs. (6,5) then
calculate the Hk using Eq. (4) then calculate the fitness
function using Eq. (3)
For each agent
Figure 1. The flowchart of the AEODE.
In general, the developed AEODE begins by using Equation (25) to generate the initial
population Xas shown in the following equation:
Xi,j=Imin,j+r1×(Ima x,j−Imin,j),i=1, 2, . . . , N(25)
In Equation (25), the value of
Imax,j
and
Imin,j
represent the maximum and minimum gray
value of
I
at dimension
j
, respectively, whereas
Dim =
2
K
is the dimension of each solution
(here
K
refers to the threshold levels used to segment
I
). Therefore, the best solution
Xb
is
allocated, followed by updating the value of other solutions.
The process of updating solutions is implemented, using the operators of the tradi-
tional AEO algorithm during the exploration. However, in the case of the solutions that go
through the exploitation phase, they will be updated using either the operators of AEO or
DE, using the following equation:
Xi=o perato rs o f AEO Pri>rs
o perato rs o f DE otherwise (26)
In Equation (26),
Pri
is the probability of each
Xi
and it depends on the fitness value (
Fiti
,
which is defined in Equation (3). The formulation of Priis given as follows:
Pri=Fiti
∑N
i=1Fiti
(27)
where
rs=min(Pri) + rand ×(max(Pri)−min(Pri)),rand ∈[0, 1](28)
The main objective of using
rs
is a variable that controls the process of using the operators
of DE and AEO. To avoid the problem of making it a constant value since it is expected
Mathematics 2021,9, 2363 9 of 25
that the value of
Pri
is increased with excess iterations, the operators of AEO are used
only, especially at the end of iterations. Therefore, we update dynamically the value of
rs
according to the probability of each solution. This gives the developed AEODE high
flexibility in switching between AEO and DE.
The next process in AEODE is to check the terminal criteria and return by the best
solution when they are reached (i.e., here, the maximum number of iterations), followed by
extracting the threshold values from Xbas tk=Xk
b+Xk+1
b
2,k=1 : 2 : K−1.
Complexity of AEODE
The complexity of the AEODE, in general, depends on the complexity of traditional
AEO, quick sort (QS), and DE. Since the complexity of DE is given as
O(DE) = O((N×D+N×D+N)×tm ax )(29)
Therefore, the complexity of AEODE is given as follows:
O(AEODE) = O(tmax ×(Term1+Term2)) (30)
Term1= (QS +N×D+N×D+N)
Term2= ((NK×D+NK×D+NK) + (N−NK)×D)
In the best case of QS, the complexity of AEODE is given as follows:
O(AEODE) = O(tmax ×(NlogN +N×D+NK×D+NK))(31)
In the worst case of QS, the complexity of AEODE is given as follows:
O(AEODE) = Otmax ×(N2+N×D+NK×D+NK)(32)
where NKis the number of solutions updated, using DE.
5. Evaluation Experiment
We compare the proposed AEODE method with relevant approaches to segment 10
standard test images, as shown in Figure 2. In addition, the histogram of each image is
given in Figure 2which indicates the characteristic of each image. Swarm parameters are
adjusted to their original implementations [
28
]. The number of population is set to 20,
while the dimension equals the threshold level. Moreover, the iteration number is set to
100, and all parameters are selected, depending on the recommendation by the authors
of [29].
Several threshold values were adopted to test the proposed approach (i.e., 6, 8, 15, 17,
19, and 25). The performance of the AEODE was obtained by applying it on several images
that are variant in shape, morphology, and contents. The experiments were implemented,
using Matlab 2014b on a computer “Core i5 and 8 GB of RAM running on MS Windows 10”.
Mathematics 2021,9, 2363 10 of 25
(a) I1 (b) Histogram of I1 (c) I2 (d) Histogram of I2
(e) I3 (f) Histogram of I3 (g) I4 (h) Histogram of I4
(i) I5 (j) Histogram of I5 (k) I6 (l) Histogram of I6
(m) I7 (n) Histogram of I7 (o) I8 (p) Histogram of I8
(q) I9 (r) Histogram of I9 (s) I10 (t) Histogram of I10
Figure 2. Tested images with their histograms.
5.1. Performance Measures
We evaluate the performance of the AEODE using three performance measures, called
the fitness function value, the structural similarity index (SSIM), and the peak signal-to-
noise ratio (PSNR). SSIM and PSNR are computed by the following equations:
SSIM(I,IS) = (2µIµIS+c1)(2σI,IS+c2)
(µ2
I+µ2
IS+c1)(σ2
I+σ2
IS+c2)(33)
where
µIS
(
σIS
) and
µI
(
σI
) represent images’ mean intensity of
IS
and
I
, respectively.
σI,IS
defines the covariance of
I
and
IS
.
c1
and
c2
are equal to 6.5025 and 58.52252, respec-
tively [30].
PSNR =20l og10(255
RM SE ), (34)
RMSE =s∑Nr
i=1∑Nc
j=1(Ii,j−ISi,j)2
Nr×Nc
Mathematics 2021,9, 2363 11 of 25
where RMSE refers to the root mean-squared error.
5.2. Results and Discussion
The proposed AEODE method is tested besides other optimization algorithms, such
as the basic artificial ecosystem-based optimization (AEO), marine predators algorithm
(MPA), gray wolf optimization (GWO), spherical search optimization (SSO), cuckoo search
(CS), and grasshopper optimization algorithm (GOA). The results can be divided mainly
into three main categories, as follows.
5.2.1. Performance Measure by Structural Similarity Index (SSIM)
Figure 3and Table 1show the results of our AEODE method, compared to the most
recent optimization algorithms based on SSIM measure.
Figure 3. SSIM at different threshold levels.
Figure 3shows that the AEODE performs better in both of the low thresholding levels
(i.e., 5 and 6) and also in the higher thresholding levels (i.e., 19 and 25), while GOA shows
the lowest performance among other optimization algorithms.
Table 1shows the SSIM values performed by each optimization algorithm for each
image with different threshold levels. From Table 1, we see that the AEODE method
allocates the first rank (the highest SSIM values at 25 cases), followed by MPA and CS
algorithms (eight highest SSIM values for each), which provide better results than others.
Additionally, AEODE achieved good SSIM values in all threshold levels.
Table 1. SSIM results for all algorithms (bold means the best value).
Threshold Image AEO AEODE MPA CS GWO SSA GOA SSO
6
I1 0.53396 0.5618 0.50579 0.52352 0.51025 0.39938 0.48972 0.53913
I2 0.44281 0.43436 0.41918 0.40398 0.40233 0.52873 0.38488 0.40887
I3 0.6135 0.62192 0.59835 0.61624 0.60716 0.64755 0.61252 0.63662
I4 0.52059 0.52952 0.48349 0.5448 0.55125 0.52551 0.53858 0.53506
I5 0.39568 0.4004 0.37672 0.29943 0.31531 0.39835 0.24689 0.35574
I6 0.4401 0.45909 0.41752 0.3415 0.36174 0.39093 0.30873 0.38448
I7 0.45267 0.46214 0.4291 0.41906 0.4188 0.54524 0.39571 0.42966
I8 0.56102 0.56591 0.54474 0.59211 0.57265 0.64572 0.54101 0.57902
I9 0.7241 0.73382 0.71886 0.57796 0.70238 0.71535 0.56442 0.54372
I10 0.75823 0.75492 0.73026 0.66034 0.66138 0.77956 0.62704 0.68312
Mathematics 2021,9, 2363 12 of 25
Table 1. Cont.
Threshold Image AEO AEODE MPA CS GWO SSA GOA SSO
8
I1 0.72614 0.72068 0.71035 0.71459 0.70441 0.54141 0.68052 0.70587
I2 0.55282 0.56874 0.58633 0.45404 0.45667 0.52761 0.40373 0.46437
I3 0.72347 0.72196 0.70095 0.76111 0.75279 0.65808 0.75203 0.77642
I4 0.645 0.65213 0.65855 0.60049 0.58887 0.55473 0.57312 0.60482
I5 0.63042 0.63415 0.63535 0.55223 0.56527 0.52105 0.53659 0.53372
I6 0.6327 0.63537 0.6475 0.51139 0.5342 0.41109 0.44601 0.53234
I7 0.61356 0.61623 0.6379 0.58497 0.56852 0.58721 0.53671 0.58961
I8 0.72496 0.73439 0.71537 0.64796 0.70713 0.64018 0.63638 0.63361
I9 0.77189 0.76749 0.77213 0.80557 0.80464 0.74337 0.78055 0.80571
I10 0.8359 0.83796 0.81993 0.77708 0.76321 0.79199 0.73836 0.76977
15
I1 0.84257 0.84348 0.83516 0.8378 0.81281 0.6043 0.79505 0.83572
I2 0.70787 0.72741 0.72224 0.67424 0.70364 0.58825 0.58646 0.66409
I3 0.82551 0.82882 0.78965 0.85413 0.84984 0.7061 0.82652 0.8465
I4 0.76668 0.76218 0.76217 0.73951 0.74846 0.58438 0.68076 0.76032
I5 0.80116 0.80893 0.798 0.76274 0.78447 0.55193 0.67289 0.76257
I6 0.76882 0.76403 0.75681 0.74085 0.72203 0.43352 0.61737 0.75585
I7 0.76895 0.77076 0.78581 0.76759 0.79293 0.61501 0.64608 0.76416
I8 0.84272 0.83542 0.84811 0.8248 0.83539 0.68349 0.76642 0.82585
I9 0.86649 0.87485 0.8676 0.84894 0.8542 0.8011 0.82481 0.83208
I10 0.90939 0.91455 0.9152 0.86198 0.84617 0.81927 0.82345 0.84518
17
I1 0.86894 0.86292 0.85717 0.87164 0.8434 0.59045 0.83058 0.86417
I2 0.7478 0.76251 0.75533 0.73452 0.75615 0.54872 0.61554 0.72061
I3 0.84686 0.83269 0.82665 0.86869 0.86657 0.72056 0.85029 0.87188
I4 0.78076 0.80353 0.79274 0.77364 0.77225 0.56542 0.72636 0.77225
I5 0.83757 0.84195 0.83266 0.81014 0.82748 0.60601 0.74483 0.80114
I6 0.79235 0.79536 0.78603 0.77734 0.77474 0.47968 0.67667 0.80482
I7 0.80524 0.81419 0.79872 0.78759 0.81934 0.62238 0.72879 0.7812
I8 0.86742 0.87129 0.87037 0.84137 0.8635 0.68178 0.79223 0.85152
I9 0.86965 0.87394 0.87281 0.85917 0.85488 0.8118 0.83079 0.85502
I10 0.91904 0.91465 0.92399 0.88497 0.85498 0.83222 0.85329 0.87835
19
I1 0.87635 0.88501 0.87613 0.883 0.86528 0.59886 0.84743 0.88062
I2 0.76872 0.77978 0.78149 0.7644 0.78806 0.59244 0.66842 0.74173
I3 0.85051 0.84382 0.83837 0.87621 0.87408 0.69963 0.85775 0.87997
I4 0.81724 0.83029 0.82357 0.79278 0.80193 0.55642 0.73432 0.78558
I5 0.86194 0.86551 0.84376 0.85091 0.85249 0.57686 0.78763 0.82448
I6 0.81372 0.81319 0.81469 0.8361 0.80929 0.49202 0.71495 0.81491
I7 0.82006 0.82438 0.83027 0.83388 0.84005 0.64693 0.76148 0.82062
I8 0.87385 0.88711 0.87839 0.87664 0.88092 0.72755 0.80646 0.86929
I9 0.88497 0.88746 0.88334 0.87026 0.87113 0.81153 0.83718 0.86986
I10 0.90722 0.91623 0.92831 0.90501 0.88699 0.82973 0.87028 0.87877
25
I1 0.91122 0.91809 0.91088 0.91508 0.90582 0.64155 0.8951 0.91449
I2 0.82578 0.83129 0.82393 0.83724 0.8647 0.63701 0.79231 0.8200
I3 0.86761 0.88327 0.88309 0.90407 0.90137 0.7196 0.88303 0.89804
I4 0.86909 0.87565 0.85928 0.86056 0.85436 0.60861 0.82264 0.85181
I5 0.90929 0.91231 0.90371 0.91259 0.91112 0.60574 0.86639 0.89367
I6 0.86516 0.87165 0.86501 0.87474 0.88643 0.57681 0.82227 0.8918
I7 0.86834 0.87462 0.86578 0.87983 0.87767 0.66865 0.84386 0.87017
I8 0.91503 0.91964 0.91417 0.9118 0.91455 0.75603 0.88377 0.90359
I9 0.90205 0.90818 0.90161 0.90348 0.8934 0.81819 0.87403 0.89324
I10 0.91263 0.91849 0.91732 0.92418 0.92226 0.8431 0.9025 0.92624
Mathematics 2021,9, 2363 13 of 25
Figure 4summarizes the average SSIM values with different threshold levels. From
Figure 4, we notice that the AEODE outperforms other algorithms, such as AEO, MPA, CS,
GWO, GOA, and SSO, by achieving the highest average of SSIM with different threshold
levels, with a slight advantage over MPA and SSO.
Figure 4. Average SSIM values for all algorithms’ overall images.
5.2.2. Performance Measure by Peak Signal-to-Noise Ratio (PSNR)
Table 2and Figure 5present the performance of the AEODE, compared to other recent
optimization algorithms based on PSNR measured with different threshold values.
Figure 5. PSNR at different threshold levels.
It can be noticed from Figure 3that the proposed AEODE outperforms other optimiza-
tion algorithms at most threshold levels (except at level 8). Table 2shows the PSNR values
performed by the proposed method and other optimization algorithms for each image with
all threshold levels.
Mathematics 2021,9, 2363 14 of 25
Table 2. PSNR results for all algorithms (bold means the best value).
Threshold Image AEO AEODE MPA CS GWO GOA SSO
6
I1 14.3825 14.8805 14.0022 14.2535 14.123 13.5623 14.5985
I2 16.776 16.5781 16.25 15.8815 15.6123 15.4548 15.9642
I3 10.301 10.4584 10.0393 12.8807 12.6054 11.3303 13.4835
I4 15.5356 15.6956 14.7613 16.2105 16.3312 16.0247 16.0098
I5 13.002 13.0219 12.7031 11.6661 11.9035 10.8805 12.5634
I6 13.8393 14.1526 13.4173 11.9244 12.183 11.5066 12.6031
I7 12.1668 12.2929 11.7442 11.983 11.8217 11.6867 12.3337
I8 13.8981 13.9791 13.5196 14.4894 14.0186 13.3968 14.1717
I9 11.5561 12.0803 11.0956 10.1514 10.5988 9.38632 10.1912
I10 11.394 11.3348 10.7158 14.212 14.4238 13.0733 14.7021
8
I1 18.0812 18.0815 17.6837 18.1512 17.7062 17.0512 18.1623
I2 19.713 19.895 20.1166 16.8944 16.5392 15.6426 17.0666
I3 12.554 12.6388 11.8524 15.7199 15.9636 14.909 16.5606
I4 18.3923 18.567 18.5216 17.6977 17.0637 16.7765 17.7364
I5 16.5871 16.7741 16.2219 16.0126 16.1572 15.7477 15.7234
I6 17.726 17.9671 18.0062 15.1837 15.5847 14.0664 15.7411
I7 14.8103 14.9134 14.6142 15.9958 15.5439 15.1388 16.2394
I8 17.7434 18.0346 17.2219 15.1528 16.8989 14.7088 15.0617
I9 13.7439 13.7056 12.8298 15.5042 15.4239 14.2372 15.6629
I10 13.6444 13.8088 12.5811 19.1078 19.316 18.1821 18.3384
15
I1 22.9214 22.977 22.2853 23.0132 21.5091 20.8354 22.8682
I2 23.5558 23.861 23.5189 22.437 22.187 20.0346 22.4572
I3 19.1193 19.4112 16.7735 21.5282 19.6672 19.2986 21.9266
I4 22.3949 22.249 22.0038 21.6674 21.6847 19.8822 22.5474
I5 21.7982 22.1718 21.3887 21.1647 21.2952 18.609 21.1493
I6 22.6571 22.4339 21.9557 21.1505 20.3099 17.7507 21.9507
I7 21.2712 22.3375 20.2573 21.3237 20.229 18.4218 21.5466
I8 22.219 22.1627 22.2886 21.8225 21.2988 18.7217 21.6013
I9 20.1963 20.8863 18.9355 20.9695 18.0963 17.7748 19.9499
I10 21.2212 21.558 19.7074 21.4588 21.4665 19.492 21.4161
17
I1 24.0811 23.8706 23.5962 24.5295 23.0746 22.3151 24.2334
I2 24.7625 25.1301 24.5872 24.1457 24.0483 20.8547 23.8381
I3 20.8615 21.862 19.2265 23.327 20.6577 20.9033 23.3563
I4 22.9854 23.9083 23.2481 22.8942 22.4866 20.9846 22.8827
I5 23.4436 23.4242 22.3991 22.6849 22.8679 20.3653 22.2991
I6 23.6858 23.9178 23.1132 22.2126 22.1551 19.2307 23.9446
I7 23.0405 23.2971 21.5097 22.6135 21.4138 20.1451 22.1639
I8 23.5485 23.847 23.4845 22.6809 22.8869 19.9435 23.2374
I9 21.3322 22.7893 20.6068 22.7039 19.356 18.9158 21.635
I10 22.9969 23.5863 21.6966 23.1547 21.9301 20.5416 23.0256
19
I1 24.4745 25.1133 24.517 25.2359 24.251 23.0769 25.1513
I2 25.6186 25.4214 25.5213 25.35 24.9708 22.2735 24.5691
I3 22.5205 23.8711 20.6197 24.743 21.7863 21.5832 25.1239
I4 24.6237 25.2183 24.561 23.7095 23.9133 21.4378 23.342
I5 24.2058 24.3305 23.3839 24.1541 23.8568 21.7525 23.1785
I6 24.6762 24.7183 24.4006 24.8508 23.6233 20.3269 24.0407
I7 23.6407 23.9977 23.3394 24.5317 22.6665 21.2738 24.2728
I8 23.8093 24.8603 24.0163 24.1517 23.8789 20.4655 24.1552
I9 24.225 22.7931 21.2062 22.5229 20.8636 19.7875 22.4684
I10 24.7213 24.4432 22.0929 24.3168 22.7555 21.4517 24.1257
Mathematics 2021,9, 2363 15 of 25
Table 2. Cont.
Threshold Image AEO AEODE MPA CS GWO GOA SSO
25
I1 26.9248 27.4147 26.7551 27.4008 26.7323 25.7589 27.4094
I2 27.912 28.2136 27.5864 28.2266 28.058 26.214 27.7472
I3 26.8428 27.1681 24.4243 26.8027 23.9304 23.9079 27.4464
I4 27.1151 27.5947 26.5531 26.7518 26.2573 24.955 26.3363
I5 26.8594 27.1087 26.1684 27.3946 26.9059 24.8264 26.3298
I6 27.1197 27.4133 26.8836 26.7446 27.1805 23.7763 28.3197
I7 26.7169 27.0784 25.6628 27.4061 25.9714 24.7308 26.7924
I8 27.1753 27.3743 26.6731 27.2032 26.7091 24.6398 26.6686
I9 25.9341 26.2033 24.8043 26.5647 24.4355 23.3068 25.7296
I10 26.8821 27.2701 26.1785 27.6637 25.9562 24.6613 27.6004
From Table 2, it is noticed that the proposed AEODE method allocates the first rank
with the highest PSNR values at 23 cases, while SSO and CS are in second and third ranks
with the highest 13 and 12 PSNR values, respectively. According to Table 2, the proposed
AEODE method performs better with higher thresholding levels (at level 17 and higher)
than lower levels.
Figure 6summarizes the average PSNR values with all threshold levels. Figure 6
shows that our proposed method outperforms other optimization algorithms by achieving
the highest average of PSNR with different threshold levels for all images. GOA achieves
the worst performance, putting it last.
Figure 6. PSNR values of AEODE and other optimization algorithms.
5.2.3. Performance Measure by Fitness Function
Figure 7and Table 3show the fitness function values of AEODE method and other
optimization algorithms with different threshold levels.
Based on Figure 7, the proposed AEODE method outperforms other optimization
algorithms in all threshold levels, while GOA shows the lowest fitness values in all thresh-
old levels.
Table 3shows that all optimization algorithms achieve acceptable fitness values with
all threshold levels. There is a slight advantage (in decimal levels) between each one.
All optimization algorithms achieve higher fitness values along with higher thresholding
levels. The higher the thresholding value, the better the fitness value obtained. Applying
the 6, 8, 15, 17, 19, and 25 threshold levels achieves fitness values of 17, 20, 29, 32, 34, and
41, respectively. Figure 8summarizes the average fitness values with all threshold levels.
Mathematics 2021,9, 2363 16 of 25
Based on Figure 8, AEODE, MPA, and AEO came in the first, second, and third ranks,
respectively, while GOA is ranked last.
Figure 7. Fitness values of AEODE and other optimization algorithms.
Table 3. Fitness function results for all algorithms (bold means the best value).
Threshold Image AEO AEODE MPA CS GWO SSA GOA SSO
6
I1 17.5265 17.5167 17.5348 17.5163 17.5245 14.555 17.5398 17.4546
I2 17.5215 17.5209 17.5372 17.2918 17.2894 15.5694 17.3161 17.2726
I3 17.5249 17.52 17.5372 17.0874 17.0818 13.9699 17.1016 17.0616
I4 17.5298 17.5217 17.5364 17.5522 17.5704 15.3916 17.5896 17.5283
I5 17.5273 17.5211 17.5362 15.5982 15.5925 12.7276 15.6182 15.6395
I6 17.531 17.5214 17.5366 15.0703 15.0806 11.5213 15.1272 15.0172
I7 17.5305 17.5192 17.5379 17.6206 17.6236 15.135 17.3165 17.4764
I8 17.5323 17.5327 17.5334 17.5738 17.5903 15.4892 17.6009 17.5415
I9 17.5272 17.5201 17.5359 17.4772 17.5094 14.9546 17.5371 17.4722
I10 17.5233 17.5177 17.5364 16.7679 16.7749 14.0574 16.7988 16.7677
8
I1 20.8258 20.8222 20.8474 20.7724 20.8195 15.6432 20.8393 20.6894
I2 20.8179 20.818 20.8467 20.7772 20.8182 16.2007 20.914 20.6923
I3 20.8335 20.8203 20.8516 20.443 20.4538 14.4673 20.5345 20.3825
I4 20.8315 20.8183 20.8546 20.9136 20.9509 16.3078 21.0092 20.8526
I5 20.8266 20.816 20.8484 18.2622 18.322 14.1362 18.3769 18.2611
I6 20.8341 20.8238 20.8486 17.3866 17.4261 11.5681 17.5021 17.2807
I7 20.8299 20.8306 20.8422 20.8701 20.9101 15.3929 20.9493 20.8268
I8 20.842 20.8136 20.8474 20.8738 20.8353 15.306 20.9881 20.8589
I9 20.8349 20.8287 20.8386 20.9832 21.0398 15.6711 21.0546 20.9873
I10 20.826 20.8282 20.8418 19.9763 20.0173 14.7812 20.0601 19.9178
15
I1 29.6442 29.6133 29.6279 29.3907 29.4684 16.2101 29.8008 29.2784
I2 29.6674 29.6322 29.6724 29.6823 29.7556 16.0706 28.5566 29.6875
I3 29.6503 29.5867 29.5937 29.2606 29.2624 14.0884 28.5465 29.131
I4 29.6753 29.5881 29.6813 29.5308 29.6343 15.9021 30.0178 29.5538
I5 29.6461 29.6588 29.6368 25.204 25.2145 13.7662 25.7182 25.2165
I6 29.6671 29.6193 29.6527 23.6303 23.6167 11.6248 24.2313 23.1818
I7 29.6416 29.6049 29.6934 29.4742 29.5973 14.9086 28.6135 29.4155
I8 29.6577 29.6269 29.6843 30.0662 30.1392 15.5166 28.6444 30.0349
I9 29.6793 29.6249 29.6955 29.748 30.0096 15.2901 28.5204 29.9023
I10 29.6519 29.5961 29.6896 28.8684 28.9454 14.781 29.281 28.8635
Mathematics 2021,9, 2363 17 of 25
Table 3. Cont.
Threshold Image AEO AEODE MPA CS GWO SSA GOA SSO
17
I1 32.3488 32.2296 32.3095 31.9577 31.9444 16.0424 31.0762 31.8416
I2 32.2875 32.2371 32.3061 32.3915 32.4331 15.7114 33.0069 32.4197
I3 32.3439 32.214 32.3012 31.7861 31.7911 15.0282 32.4284 31.6946
I4 32.2961 32.2027 32.2791 32.1339 32.1396 16.4578 32.7554 32.1749
I5 32.3251 32.2187 32.3283 27.1632 27.2144 13.1787 27.7373 27.2223
I6 32.3018 32.1828 32.2808 25.2824 25.287 11.5259 26.1223 24.6364
I7 32.2647 32.2341 32.3332 32.1072 32.1944 15.0693 32.6285 32.099
I8 32.3107 32.215 32.3415 32.6783 32.71 15.2613 33.3399 32.6571
I9 32.347 32.2675 32.2917 32.4441 32.5326 14.957 30.9936 32.4632
I10 32.3163 32.1987 32.3126 31.4601 31.5808 14.6868 31.0726 31.5033
19
I1 34.8139 34.8003 34.8714 34.3638 34.2342 16.3417 33.2823 34.2175
I2 34.8117 34.7727 34.8053 34.9751 34.9733 16.3809 33.3096 35.0698
I3 34.8028 34.6484 34.7831 34.2196 34.14 14.9697 35.0649 34.0982
I4 34.8114 34.7633 34.8187 34.6671 34.6481 15.7964 35.3929 34.6825
I5 34.8495 34.7301 34.826 28.9961 29.036 13.6795 29.684 29.1493
I6 34.8331 34.7356 34.8254 26.7512 26.5405 11.5293 27.536 25.9821
I7 34.82 34.7292 34.8609 34.6364 34.73 15.3604 35.3216 34.5569
I8 34.808 34.7364 34.799 35.2033 35.228 15.4885 35.9805 35.2658
I9 34.8603 34.7427 34.8427 34.9619 35.0218 15.9977 33.319 35.0619
I10 34.8632 34.768 34.8505 33.9221 34.0155 15.1945 33.3191 34.0152
25
I1 41.8453 41.7239 41.6587 41.0703 40.6435 17.1863 39.5637 40.9583
I2 41.8725 41.6996 41.7333 42.1871 41.8686 16.6713 42.9219 42.1336
I3 41.8546 41.756 41.7622 40.6073 40.2479 15.2268 41.681 40.417
I4 41.8022 41.6242 41.7986 41.5551 41.2241 17.3623 42.4607 41.6875
I5 41.8499 41.7091 41.724 33.8368 33.7225 14.7999 34.7461 33.9924
I6 41.8209 41.6941 41.7224 30.4701 29.6229 12.2152 32.0503 29.2935
I7 41.8718 41.7143 41.6714 41.5924 41.4857 16.0446 39.5462 41.5544
I8 41.8436 41.6505 41.671 42.3413 42.1102 16.375 39.7254 42.3516
I9 41.8299 41.632 41.6478 41.8899 41.9941 16.2916 39.5576 42.1339
I10 41.8737 41.7306 41.7884 40.8165 40.4964 15.9225 39.7665 40.7734
Figure 8. Average fitness values of AEODE and other optimization algorithms.
Figures 9and 10 show the segmented image I10 and image I7, respectively, while
their histograms at threshold value 19 are given in Figures 11 and 12 respectively. Both
figures show that the proposed method has successfully determined the best threshold
Mathematics 2021,9, 2363 18 of 25
value to segment several types of images. Moreover, Figure 13 illustrates the quality of
segmented image I1 at level 19. It can be noticed from this figure that the quality of AEODE
is decreased in this case since its obtained threshold values do not cover the whole search
space (as shown in Figure 14) and its feasible region does not contain the optimal threshold
values, which affects the performance of the final output.
(a) CS (b) GWO (c) SSO (d) MPA
(e) GOA (f) AEO (g) AEODE
Figure 9. Segmentation results at threshold level 19 for the images I10.
(a) CS (b) GWO (c) SSO (d) MPA
(e) GOA (f) AEO (g) AEODE
Figure 10. Segmented image I7 at threshold level 19.
Mathematics 2021,9, 2363 19 of 25
(a) CS (b) GWO (c) SSO (d) MPA
(e) GOA (f) AEO (g) AEODE
Figure 11. The threshold values at level 19 over the histogram of I10.
(a) CS (b) GWO (c) SSO (d) MPA
(e) GOA (f) AEO (g) AEODE
Figure 12. The threshold values at level 19 over the histogram of image I7.
Moreover, the improvement rate between the proposed method and other methods
according to each performance measure is computed as follows:
IR(%) = (S−S0)
S0∗100 (35)
where
S0
and
S
denote the values obtained by AEODE and values obtained by other
methods, respectively.
Mathematics 2021,9, 2363 20 of 25
(a) CS (b) GWO (c) SSO (d) MPA
(e) GOA (f) AEO (g) AEODE
Figure 14. The threshold values at level 19 over the histogram of image I1.
(a) CS (b) GWO (c) SSO (d) MPA
(e) GOA (f) AEO (g) AEODE
Figure 13. Segmented image I1 at threshold level 19.
Table 4shows the IR for the three measures (i.e., PSNR, SSIM, and fitness value). From
this table, it can be noticed that AEODE provides a high IR rate in terms of PSNR, which is
better than AEO, MPA, CS, GWO, GOA, and SSO with 47, 46, 44, 44, 55, and 46. In terms of
SSIM as well, it provides results better than AEO, MPA, CS, GWO, GOA, and SSO with
49, 56, 37, 51, 53, and 39. However, the IR of the developed AEODE in terms of the fitness
value is better than AEO and MPA with only 5 and 3 cases. However, with other methods,
it still provides better fitness values of 41, 37, 37, and 42 for CS, GWO, GOA, and SSO.
Mathematics 2021,9, 2363 21 of 25
Table 4. Improvement ratio for developed method with other methods (bold means the best value).
PSNR SSIM Fitness
T I AEO MPA CS GWO GOA SSO AEO MPA CS GWO GOA SSO AEO MPA CS GWO GOA SSO
6
I1 −5.0 −10.0 −6.8 −9.2 −12.8 −4.0 −3.3 −5.9 −4.2 −5.1 −8.9 −1.9 0.1 0.1 −0.0 0.0 0.1 −0.4
I2 1.9 −3.5 −7.0 −7.4 −11.4 −5.9 1.2 −2.0 −4.2 −5.8 −6.8 −3.7 0.0 0.1 −1.3 −1.3 −1.2 −1.4
I3 −1.4 −3.8 −0.9 −2.4 −1.5 2.4 −1.5 −4.0 23.2 20.5 8.3 28.9 0.0 0.1 −2.5 −2.5 −2.4 −2.6
I4 −1.7 −8.7 2.9 4.1 1.7 1.0 −1.0 −6.0 3.3 4.0 2.1 2.0 0.0 0.1 0.2 0.3 0.4 0.0
I5 −1.2 −5.9 −25.2 −21.3 −38.3 −11.2 −0.2 −2.4 −10.4 −8.6 −16.4 −3.5 0.0 0.1 −11.0 −11.0 −10.9 −10.7
I6 −4.1 −9.1 −25.6 −21.2 −32.8 −16.3 −2.2 −5.2 −15.7 −13.9 −18.7 −10.9 0.1 0.1 −14.0 −13.9 −13.7 −14.3
I7 −2.0 −7.1 −9.3 −9.4 −14.4 −7.0 −1.0 −4.5 −2.5 −3.8 −4.9 0.3 0.1 0.1 0.6 0.6 −1.2 −0.2
I8 −0.9 −3.7 4.6 1.2 −4.4 2.3 −0.6 −3.3 3.7 0.3 −4.2 1.4 −0.0 0.0 0.2 0.3 0.4 0.1
I9 −1.3 −2.0 −21.2 −4.3 −23.1 −25.9 −4.3 −8.2 −16.0 −12.3 −22.3 −15.6 0.0 0.1 −0.2 −0.1 0.1 −0.3
I10 0.4 −3.3 −12.5 −12.4 −16.9 −9.5 0.5 −5.5 25.4 27.3 15.3 29.7 0.0 0.1 −4.3 −4.2 −4.1 −4.3
8
I1 0.8 −1.4 −0.8 −2.3 −5.6 −2.1 −0.0 −2.2 0.4 −2.1 −5.7 0.4 0.0 0.1 −0.2 −0.0 0.1 −0.6
I2 −2.8 3.1 −20.2 −19.7 −29.0 −18.4 −0.9 1.1 −15.1 −16.9 −21.4 −14.2 −0.0 0.1 −0.2 0.0 0.5 −0.6
I3 0.2 −2.9 5.4 4.3 4.2 7.5 −0.7 −6.2 24.4 26.3 18.0 31.0 0.1 0.2 −1.8 −1.8 −1.4 −2.1
I4 −1.1 1.0 −7.9 −9.7 −12.1 −7.3 −0.9 −0.2 −4.7 −8.1 −9.6 −4.5 0.1 0.2 0.5 0.6 0.9 0.2
I5 −0.6 0.2 −12.9 −10.9 −15.4 −15.8 −1.1 −3.3 −4.5 −3.7 −6.1 −6.3 0.1 0.2 −12.3 −12.0 −11.7 −12.3
I6 −0.4 1.9 −19.5 −15.9 −29.8 −16.2 −1.3 0.2 −15.5 −13.3 −21.7 −12.4 0.0 0.1 −16.5 −16.3 −16.0 −17.0
I7 −0.4 3.5 −5.1 −7.7 −12.9 −4.3 −0.7 −2.0 7.3 4.2 1.5 8.9 −0.0 0.1 0.2 0.4 0.6 −0.0
I8 −1.3 −2.6 −11.8 −3.7 −13.3 −13.7 −1.6 −4.5 −16.0 −6.3 −18.4 −16.5 0.1 0.2 0.3 0.1 0.8 0.2
I9 0.6 0.6 5.0 4.8 1.7 5.0 0.3 −6.4 13.1 12.5 3.9 14.3 0.0 0.0 0.7 1.0 1.1 0.8
I10 −0.2 −2.2 −7.3 −8.9 −11.9 −8.1 −1.2 −8.9 38.4 39.9 31.7 32.8 −0.0 0.1 −4.1 −3.9 −3.7 −4.4
15
I1 −0.1 −1.0 −0.7 −3.6 −5.7 −0.9 −0.2 −3.0 0.2 −6.4 −9.3 −0.5 0.1 0.0 −0.8 −0.5 0.6 −1.1
I2 −2.7 −0.7 −7.3 −3.3 −19.4 −8.7 −1.3 −1.4 −6.0 −7.0 −16.0 −5.9 0.1 0.1 0.2 0.4 −3.6 0.2
I3 −0.4 −4.7 3.1 2.5 −0.3 2.1 −1.5 −13.6 10.9 1.3 −0.6 13.0 0.2 0.0 −1.1 −1.1 −3.5 −1.5
I4 0.6 −0.0 −3.0 −1.8 −10.7 −0.2 0.7 −1.1 −2.6 −2.5 −10.6 1.3 0.3 0.3 −0.2 0.2 1.5 −0.1
I5 −1.0 −1.4 −5.7 −3.0 −16.8 −5.7 −1.7 −3.5 −4.5 −4.0 −16.1 −4.6 −0.0 −0.1 −15.0 −15.0 −13.3 −15.0
I6 0.6 −0.9 −3.0 −5.5 −19.2 −1.1 1.0 −2.1 −5.7 −9.5 −20.9 −2.2 0.2 0.1 −20.2 −20.3 −18.2 −21.7
I7 −0.2 2.0 −0.4 2.9 −16.2 −0.9 −4.8 −9.3 −4.5 −9.4 −17.5 −3.5 0.1 0.3 −0.4 −0.0 −3.3 −0.6
I8 0.9 1.5 −1.3 −0.0 −8.3 −1.1 0.3 0.6 −1.5 −3.9 −15.5 −2.5 0.1 0.2 1.5 1.7 −3.3 1.4
I9 −1.0 −0.8 −3.0 −2.4 −5.7 −4.9 −3.3 −9.3 0.4 −13.4 −14.9 −4.5 0.2 0.2 0.4 1.3 −3.7 0.9
I10 −0.6 0.1 −5.7 −7.5 −10.0 −7.6 −1.6 −8.6 −0.5 −0.4 −9.6 −0.7 0.2 0.3 −2.5 −2.2 −1.1 −2.5
Mathematics 2021,9, 2363 22 of 25
Table 4. Cont.
PSNR SSIM Fitness
T I AEO MPA CS GWO GOA SSO AEO MPA CS GWO GOA SSO AEO MPA CS GWO GOA SSO
17
I1 0.7 −0.7 1.0 −2.3 −3.7 0.1 0.9 −1.1 2.8 −3.3 −6.5 1.5 0.4 0.2 −0.8 −0.9 −3.6 −1.2
I2 −1.9 −0.9 −3.7 −0.8 −19.3 −5.5 −1.5 −2.2 −3.9 −4.3 −17.0 −5.1 0.2 0.2 0.5 0.6 2.4 0.6
I3 1.7 −0.7 4.3 4.1 2.1 4.7 −4.6 −12.1 6.7 −5.5 −4.4 6.8 0.4 0.3 −1.3 −1.3 0.7 −1.6
I4 −2.8 −1.3 −3.7 −3.9 −9.6 −3.9 −3.9 −2.8 −4.2 −5.9 −12.2 −4.3 0.3 0.2 −0.2 −0.2 1.7 −0.1
I5 −0.5 −1.1 −3.8 −1.7 −11.5 −4.8 0.1 −4.4 −3.2 −2.4 −13.1 −4.8 0.3 0.3 −15.7 −15.5 −13.9 −15.5
I6 −0.4 −1.2 −2.3 −2.6 −14.9 1.2 −1.0 −3.4 −7.1 −7.4 −19.6 0.1 0.4 0.3 −21.4 −21.4 −18.8 −23.4
I7 −1.1 −1.9 −3.3 0.6 −10.5 −4.1 −1.1 −7.7 −2.9 −8.1 −13.5 −4.9 0.1 0.3 −0.4 −0.1 1.2 −0.4
I8 −0.4 −0.1 −3.4 −0.9 −9.1 −2.3 −1.3 −1.5 −4.9 −4.0 −16.4 −2.6 0.3 0.4 1.4 1.5 3.5 1.4
I9 −0.5 −0.1 −1.7 −2.2 −4.9 −2.2 −6.4 −9.6 −0.4 −15.1 −17.0 −5.1 0.2 0.1 0.5 0.8 −3.9 0.6
I10 0.5 1.0 −3.2 −6.5 −6.7 −4.0 −2.5 −8.0 −1.8 −7.0 −12.9 −2.4 0.4 0.4 −2.3 −1.9 −3.5 −2.2
19
I1 −1.0 −1.0 −0.2 −2.2 −4.2 −0.5 −2.5 −2.4 0.5 −3.4 −8.1 0.2 0.0 0.2 −1.3 −1.6 −4.4 −1.7
I2 −1.4 0.2 −2.0 1.1 −14.3 −4.9 0.8 0.4 −0.3 −1.8 −12.4 −3.4 0.1 0.1 0.6 0.6 −4.2 0.9
I3 0.8 −0.6 3.8 3.6 1.7 4.3 −5.7 −13.6 3.7 −8.7 −9.6 5.2 0.4 0.4 −1.2 −1.5 1.2 −1.6
I4 −1.6 −0.8 −4.5 −3.4 −11.6 −5.4 −2.4 −2.6 −6.0 −5.2 −15.0 −7.4 0.1 0.2 −0.3 −0.3 1.8 −0.2
I5 −0.4 −2.5 −1.7 −1.5 −9.0 −4.7 −0.5 −3.9 −0.7 −1.9 −10.6 −4.7 0.3 0.3 −16.5 −16.4 −14.5 −16.1
I6 0.1 0.2 2.8 −0.5 −12.1 0.2 −0.2 −1.3 0.5 −4.4 −17.8 −2.7 0.3 0.3 −23.0 −23.6 −20.7 −25.2
I7 −0.5 0.7 1.2 1.9 −7.6 −0.5 −1.5 −2.7 2.2 −5.5 −11.4 1.1 0.3 0.4 −0.3 0.0 1.7 −0.5
I8 −1.5 −1.0 −1.2 −0.7 −9.1 −2.0 −4.2 −3.4 −2.9 −3.9 −17.7 −2.8 0.2 0.2 1.3 1.4 3.6 1.5
I9 −0.3 −0.5 −1.9 −1.8 −5.7 −2.0 6.3 −7.0 −1.2 −8.5 −13.2 −1.4 0.3 0.3 0.6 0.8 −4.1 0.9
I10 −1.0 1.3 −1.2 −3.2 −5.0 −4.1 1.1 −9.6 −0.5 −6.9 −12.2 −1.3 0.3 0.2 −2.4 −2.2 −4.2 −2.2
25
I1 −0.7 −0.8 −0.3 −1.3 −2.5 −0.4 −1.8 −2.4 −0.1 −2.5 −6.0 −0.0 0.3 −0.2 −1.6 −2.6 −5.2 −1.8
I2 −0.7 −0.9 0.7 4.0 −4.7 −1.4 −1.1 −2.2 0.0 −0.6 −7.1 −1.7 0.4 0.1 1.2 0.4 2.9 1.0
I3 −1.8 −0.0 2.4 2.0 −0.0 1.7 −1.2 −10.1 −1.3 −11.9 −12.0 1.0 0.2 0.0 −2.8 −3.6 −0.2 −3.2
I4 −0.7 −1.9 −1.7 −2.4 −6.1 −2.7 −1.7 −3.8 −3.1 −4.8 −9.6 −4.6 0.4 0.4 −0.2 −1.0 2.0 0.2
I5 −0.3 −0.9 0.0 −0.1 −5.0 −2.0 −0.9 −3.5 1.1 −0.7 −8.4 −2.9 0.3 0.0 −18.9 −19.1 −16.7 −18.5
I6 −0.7 −0.8 0.4 1.7 −5.7 2.3 −1.1 −1.9 −2.4 −0.8 −13.3 3.3 0.3 0.1 −26.9 −29.0 −23.1 −29.7
I7 −0.7 −1.0 0.6 0.3 −3.5 −0.5 −1.3 −5.2 1.2 −4.1 −8.7 −1.1 0.4 −0.1 −0.3 −0.5 −5.2 −0.4
I8 −0.5 −0.6 −0.9 −0.6 −3.9 −1.7 −0.7 −2.6 −0.6 −2.4 −10.0 −2.6 0.5 0.0 1.7 1.1 −4.6 1.7
I9 −0.7 −0.7 −0.5 −1.6 −3.8 −1.6 −1.0 −5.3 1.4 −6.7 −11.1 −1.8 0.5 0.0 0.6 0.9 −5.0 1.2
I10 −0.6 −0.1 0.6 0.4 −1.7 0.8 −1.4 −4.0 1.4 −4.8 −9.6 1.2 0.3 0.1 −2.2 −3.0 −4.7 −2.3
T in first column is the threshold level. I is the image name.
Mathematics 2021,9, 2363 23 of 25
5.3. Statistical Results
In this section, we apply the Friedman test to evaluate the robustness of the algo-
rithms based on all measures. This test statistically ranks the methods, where the highest
Friedman’s value is the best.
From Table 5, it can be seen that the AEODE achieved the highest mean rank compared
to all methods in both SSIM and PSNR measures, followed by AEO, MPA, CS, and GWO,
respectively, where the GOA ranked last in the PSNR measure. In the SSIM measure, the
AEO, CS, and SSO ranked second, third, and fourth, respectively, followed by MPA and
GWO. The GOA also came in last. However, the AEODE obtained the best values in the
PSNR and SSIM measure, and it achieved the fourth rank after AEO, MPA, and GOA in
the fitness function measure. This can be due to the fact that the fitness function did not
measure the quality of the image like the other two measures. Therefore, we can conclude
that the AEODE can effectively segment images better than the compared methods.
Table 5. Results of the Friedman test for all measures (bold means the best value).
AEODE AEO MPA CS GWO SSA GOA SSO
SSIM 6.617 5.617 5.267 5.017 4.742 2.083 2.183 4.475
PSNR 5.750 4.700 3.350 4.783 3.300 1.100 1.483 4.633
Fitness 4.750 6.100 6.267 4.233 4.633 1.000 5.217 3.800
The proposed AEODE achieved higher performance in both measures, SSIM and
PSNR, for most threshold levels, whereas it obtained good results in terms of fitness values.
It outperformed the basic version of AEO, as it combines and reserves the best features
for each of AEO and DE. It also shows advantages in SSIM, PSNR, and fitness values,
compared to other optimization algorithms, such as MPA, CS, GWO, SSO, and GOA,
which means high efficiency of exploring problem space. The proposed method achieves
two main tasks; avoiding stacking at a local point (and consequently being trapped) and
increasing the convergence ability. We believe that achieving acceptable performance with
a small (compared to other algorithms) number of parameters, which imply a relatively
simple implementation task, is considered a great advantage. The hybridization of swarm
algorithms is aligned with other literature, as they showed advantages toward solving
complex problems, such as determining an optimal threshold value for image segmentation.
6. Conclusions and Future Work
The segmentation process is the primary step in the image processing field, as it is
employed in different computer vision applications. Multilevel thresholding techniques
have confirmed their efficiency in solving image segmentation problems. This paper
presents a new multilevel thresholding method based on a modified artificial ecosystem-
based optimization (AEO) algorithm, using differential evolution (DE), called AEODE.
The AEO is a recently proposed optimization algorithm inspired by the chain of energy
transfer among living organisms, and it was successfully applied to address various
optimization problems. However, it suffers from some limitations, such as stacking at the
local optima. Therefore, in this paper, the DE was employed to overcome the drawbacks
of the AEO. It was applied as a local search for the AEO to improve the ecosystem of
the solutions. A set of images was applied to evaluate the performance of the AEODE
using three measures, namely structural similarity index (SSIM), peak signal-to-noise ratio
(PSNR), and fitness function values. The AEODE was compared with seven well-known
optimization algorithms, including the traditional AEO, gray wolf optimization (GWO),
marine predators algorithm (MPA), spherical search optimization (SSO), grasshopper
optimization algorithm (GOA), and cuckoo search (CS) algorithm. The evaluation outcomes
confirmed the competitive performance of the proposed AEODE, which outperformed
the traditional AEO and other compared algorithms on different tests. Furthermore, to
evaluate the robustness of the AEODE method and the compared algorithms, we applied
Mathematics 2021,9, 2363 24 of 25
a well-known statistical test, called the Friedman test. The AEODE obtained the highest
mean rank.
In future work, the proposed AEODE can be applied in different optimization applica-
tions, such as parameter estimation, feature selection, and data clustering. Moreover, other
recent optimizers can be applied to find alternative solutions to the multilevel thresholding
image segmentation problem, such as the arithmetic optimization algorithm (AOA).
Author Contributions:
A.A.E.: Conceptualization, supervision, methodology, formal analysis, re-
sources, data curation, and writing—original draft preparation. L.A.: Conceptualization, supervision,
methodology, formal analysis, resources, data curation, and writing—original draft preparation. D.Y.:
Writing—review and editing. A.T.S.: Writing—review and editing. M.A.A.A.-q.: Writing—review
and editing. S.A.: Writing—review and editing,supervision, project administration, and funding
acquisition. M.A.E.: supervision and writing—review and editing, methodology, formal analysis,
resources, data curation. All authors have read and agreed to the published version of the manuscript.
Funding:
This research was funded by the Deanship of Scientific Research at Princess Nourah Bint
Abdulrahman University through the Fast-track Research Funding Program.
Acknowledgments:
This research was funded by the Deanship of Scientific Research at Princess
Nourah bint Abdulrahman University through the Fast-track Research Funding Program.
Conflicts of Interest: The authors declare no conflict of interest.
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