![Results of Boxplots obtained all algorithms over CEC’2020 benchmark functions with Dim=20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Dim=20$$\end{document}.](https://www.researchgate.net/profile/Essam-Houssein/publication/371305117/figure/fig3/AS:11431281165442280@1686020539791/Results-of-Boxplots-obtained-all-algorithms-over-CEC2020-benchmark-functions-with_Q320.jpg)
![Results of convergence curves for the proposed algorithm with other competing algorithms over CEC’2020 benchmark functions with Dim=20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Dim=20$$\end{document}.](https://www.researchgate.net/profile/Essam-Houssein/publication/371305117/figure/fig4/AS:11431281165472209@1686020540306/Results-of-convergence-curves-for-the-proposed-algorithm-with-other-competing-algorithms_Q320.jpg)
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An ecient multilevel image
thresholding method based
on improved heap‑based optimizer
Essam H. Houssein
*, Gaber M. Mohamed , Ibrahim A. Ibrahim & Yaser M. Wazery
Image segmentation is the process of separating pixels of an image into multiple classes, enabling
the analysis of objects in the image. Multilevel thresholding (MTH) is a method used to perform
this task, and the problem is to obtain an optimal threshold that properly segments each image.
Methods such as the Kapur entropy or the Otsu method, which can be used as objective functions
to determine the optimal threshold, are ecient in determining the best threshold for bi‑level
thresholding; however, they are not eective for MTH due to their high computational cost. This
paper integrates an ecient method for MTH image segmentation called the heap‑based optimizer
(HBO) with opposition‑based learning termed improved heap‑based optimizer (IHBO) to solve the
problem of high computational cost for MTH and overcome the weaknesses of the original HBO. The
IHBO was proposed to improve the convergence rate and local search eciency of search agents of
the basic HBO, the IHBO is applied to solve the problem of MTH using the Otsu and Kapur methods as
objective functions. The performance of the IHBO‑based method was evaluated on the CEC’2020 test
suite and compared against seven well‑known metaheuristic algorithms including the basic HBO, salp
swarm algorithm, moth ame optimization, gray wolf optimization, sine cosine algorithm, harmony
search optimization, and electromagnetism optimization. The experimental results revealed that
the proposed IHBO algorithm outperformed the counterparts in terms of the tness values as well as
other performance indicators, such as the structural similarity index (SSIM), feature similarity index
(FSIM), peak signal‑to‑noise ratio. Therefore, the IHBO algorithm was found to be superior to other
segmentation methods for MTH image segmentation.
Segmentation has an important role in the eld of image processing1. Segmentation is the process of separating
an image into two or more homogeneous segments based on the characteristics of the pixels in the image. It is
utilized in various scopes, such as industry and medicine2, agriculture3, and surveillance4. resholding is one of
the most common image segmentation approaches. To dene the thresholds, most methods use the histogram of
the image5, which is vital for determining the probability distribution value of pixels in the image6. resholding
obtains the information of the histogram from an image and determines the best threshold ((th)) for categorizing
the pixels into various groups. Image thresholding approaches can be categorized into two types: multi-level and
bi-level thresholding. Bi-level thresholding techniques use one threshold to separate an image into two groups,
whereas multi-level thresholding (MTH) uses two or more thresholds to separate an image into many groups1.
To obtain the best threshold values in MTH segmentation, thresholding techniques can be classied into two
approaches: non-parametric and parametric. In parametric techniques, each group of grayscale range should be
consistent with a Gaussian distribution. Parametric approaches are dependent on the evaluation of the histogram
using mathematical operations. e Gaussian mixture is widespread, where used to dene the set of operations
that convergent the histogram, and the best thresholds are then selected. Non-parametric approaches employ
distinct methods to separate the pixels into homogeneous areas; then, the best threshold is dened using sta-
tistical information, such as entropy or variance. e Kapur method7 and Otsu method8 are used in this study.
e Otsu method selects the best thresholds by the maximization of the variance among groups. In the Kapur
method, the threshold value is dened by minimizing the cross entropy between a segmented image and the
original image. ese methods are ecient for one or two th values of thresholds. However, they have several
restrictions; for example, they are very costly in computation, mostly when the number of thresholds increases.
Non-parametric techniques have several advantages. Specically, in terms of computation, these methods are
computationally faster than parametric methods, especially when used in optimization problems. Metaheuristic
OPEN
Faculty of Computers and Information, Minia University, Minia, Egypt. *email: essam.halim@mu.edu.eg
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algorithms (MAs) can be used in the search process. Generally, these algorithms provide better results than
techniques dependent on thresholding methods9,10.
Metaheuristic algorithms are used to solve challenging real-world problems. In the past several decades,
researchers have extensively demonstrated the ability of MAs to solve several types of dicult optimization
problems in various areas, such as optimization11, communications12, bioinformatics13, drug design14, Image
segmentation15,16 and feature selection17, mainly due to the fact that these algorithms are general-purpose and
easy to implement18. MAs are commonly inspired by nature and can be classied into four main categories:
Evolutionary-based, swarm-based, physics-based, and human-based algorithms. Evolutionary-based algorithms
(use mechanisms inspired by biological evolution, such as recombination, crossover, mutation, and the herit-
age of features in ospring19. Candidate solutions to optimization problems are represented as individuals in a
population, and the quality of the solutions is determined by the tness function. Two main Evolutionary-based
algorithms are dierential evolution (DE)20 and the genetic algorithm (GA)21, which are inspired by biological
evolution, while swarm-based algorithms mimic the mass behavior of living creatures. Living creatures interact
with each other in nature to achieve optimal mass behavior22. An oshoot is particle swarm optimization (PSO)23,
which mimics the hunting behavior of groups of sh and birds. Physics-based algorithms are generally inspired
by physics to generate factors that enable search for the optimal solution in the search scope24,25. Some of the
most common categories in this branch are the gravitational search algorithm (GSA)26 and electromagnetism
optimization (EMO)27. Human-based algorithms are inspired by human gregarious demeanor. e common
and recent used algorithms in this category are teaching–learning-based optimization (TLBO)28, and the heap-
based optimizer (HBO)29.
With respect to MTH in image processing, it is possible to use thresholding approaches such as the Otsu
or Kapur method30 as the objective function. e problem is not only concerned with the increased number of
thresholds, but is also related to the image; for this reason, each image is an autonomous problem concerned
with the levels of thresholding used for segmentation31. e optimal segmentation threshold values must be
highly accurate in most processes. erefore, the use of MAs has been expanded in this eld. e moth swarm
algorithm discussed in32 was used to obtain the best threshold values with the Kapur method based on previ-
ous literature. In addition, a modied rey algorithm was proposed in33 for image processing, and used the
Kapur and Otsu methods as objective functions. In34, ant colony optimization was used in image segmentation
based on a multi-threshold image segmentation method with Kapur entropy and a non-local two-dimensional
histogram. In35, the researchers used a novel concept called a hyper-heuristic with MTH image segmentation, in
which each iteration determined the optimal execution sequence of MAs to determine the best threshold values.
In10, the black widow optimization algorithm10 was proposed to determine the optimal threshold using the
Kapur or Otsu method as an objective function with a multi-level threshold. In36, the crow search algorithm
was utilized in conjunction with the Kapur approach and 30th values to obtain the optimal threshold. In37, the
authors proposed the ecient krill herd algorithm to determine the best thresholds at various levels for color
images, where the Tsallis entropy, Otsu method, and Kapur entropy were utilized as tness functions. Harris
hawks optimization (HHO) is a new algorithm, and its hybridization was achieved by adding another powerful
algorithm, the dierential evolution (DE) algorithm38. Specically, the entire population was split into two equal
subpopulations, which were assigned to the HHO and DE algorithms, respectively. is hybridization used the
Otsu and Kapur approaches as tness functions. In39, the authors combined the classical Otsu’s method with an
energy curve for applying the segmentation of colored images in multilevel thresholding. e water cycle algo-
rithm (WCA) is integrated with Masi entropy (Masi-WCA) and Tsallis in40 to segment the color image. the results
of the experiment proved the superiority of the WCA for multilevel thresholding with Masi entropy compared
to other competitive algorithms. e authors in41 used a multi-verse optimizer (MVO) algorithm based on the
Energy-MCE thresholding approach for searching the accurate and near-optimal thresholds for segmentation.
In the same context, Elaziz etal.42 proposed DE as a technique to select the best MAs to determine the optimal
threshold for the Otsu method. Opposition-based learning (OBL)is one of the important eective methods to
improve search eciency of meta-heuristic algorithms43. e hyper-heuristic method based on a genetic algo-
rithm was presented in44 and estimates various MAs for determining the optimal threshold for each image using
a predetermined value of th using the Otsu method. In45, new ecient version of the recent chimp optimization
algorithm (ChOA) was proposed to overcome the weaknesses of the original ChOA and called opposition-based
Lévy Flight chimp optimizer (IChOA). e IChOA is applied to solve the problem of MTH using the Otsu and
Kapur methods as objective functions. In this paper, several MAs, including SCA, MFO, SSA, and EMO, were
combined with Otsu. As mentioned, the utilization of MAs in MTH is growing rapidly, and a summary of vari-
ous approaches can be found in46.
According to the No Free Lunch theorem, this signies that there is no ideal algorithm for a particular
problem47. For this reason, any algorithm must be evaluated for a real problem to demonstrate its performance.
MTH based on OBL are frequently used to solve a diversity of other optimization problems. erefore, this paper
seeks to further the research in the image segmentation eld by utilizing the recent heap-based optimizer (HBO).
e HBO was introduced in29 for optimization. is algorithm mimics the job responsibilities and descriptions
of employees. e sta are coordinated in a hierarchy, and a nonlinear tree-shaped data structure is used to rep-
resent the heap. e benet of these algorithms is that types with unsuitable tness are deleted from the circle,
leading to improved convergence speed. Based on the advantages of the HBO and the No Free Lunch theorem,
this paper aims to present an alternative version from HBO called IHBO algorithm to discover the optimal solu-
tion of complex MTH problems and overcoming the weaknesses of the original HBO.
e proposed method for MTH based on the HBO is called IHBO, and applies the Kapur and Otsu methods
individually to obtain the optimal threshold from benchmark images. IHBO explores the search area determined
by a histogram technique to provide the best threshold values using a set of factors inspired by humans’ career
hierarchy. e performance of IHBO is evaluated through various tests in which benchmark images are utilized
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with many levels of complexity. e segmentation results are estimated using various assessments, such as the
structural similarity (SSIM) index48, feature similarity (FSIM) index49 and peak signal-to-noise ratio (PSNR)50.
Furthermore, IHBO algorithm was evaluated on the CEC’2020 test suite and compared against seven well-
known metaheuristic algorithms including the basic HBO29, SSA51, MFO52, GWO53, SCA54, HS55, and EMO27.
e evaluations are executed through various non-parametric and statistical techniques to determine whether
the optimal solutions provided by the IHBO are superior.
e main contributions of this paper can be summarized as follows:
• An ecient HBO based on OBL called IHBO to overcome the weaknesses of the original HBO is presented.
• Evaluating the eectiveness of IHBO on the CEC’2020 test suite.
• IHBO is proposed to solve the problem of high computational cost for MTH .
• Proving the ability of the IHBO to solve the image segmentation problems using the Kapur’s entropy and
Otsu’s method as tness function.
• Verify the image quality using set of metrics FSIM, PSNR and SSIM to obtain optimal solutions.
• Evaluating the performance of the provided method based on the various segmentation degrees to estimate
stability of the optimizer and evaluate quality of the segmentation.
e remainder of this paper is organized as follows. “Preliminaries” section describes the materials and methods
used in this study, while “e proposed IHBO algorithm” section presents the proposed algorithm. “Environ-
mental and experimental requirements” section illustrates the environmental and experimental requirements,
while “Experimental results and discussion” section presents the performance evaluation and experimental
results. Finally, conclusions and proposals for future work are provided in “Conclusions and future works”
section.
Preliminaries
is section introduces the materials required to implement the proposed segmentation method, as well as the
approaches implemented based on the above-mentioned approaches.
Objective functions formulation. e entropy criterion of the Kapur7 approach and between-class vari-
ance of the Otsu8 approach are widely utilized to determine the optimal threshold value th in image segmenta-
tion. Both algorithms were developed for bi-level thresholding techniques. An approach can be readily extended
for solving MTH problems.
Otsu method for segmentation. e Otsu method is an automatic and non-parametric technique used to deter-
mine the optimal thresholds of an image8. is method is based on the maximum variance of the various classes
as a criterion to segment the image. e intensity levels L are taken from a grayscale image, and the equation
below is used to calculate the probability distribution of the intensity value:
where i is a specic intensity level in the range
0≤i≤L−1
and
ni
is the number of gray level i appearing in the
image. e number of pixels in the image is nk and
Phi
is the probability distribution of the intensity levels. For
the simplest segmentation (bi-level), two classes are represented as
e probability distribution for
C1
and
C2
are
ω0(th)
and
ω1(th)
, respectively, as illustrated in (3).
It is necessary to calculate the mean levels
µ0
and
µ1
that dene the classes using (4). Once these values are
calculated, the Otsu based between classes
σ2
B
is calculated using (5) as follows:
Moreover,
σ1
and
σ2
in (5) indicate the variance of regions
C1
and
C2
, and are calculated as
where
µT=ω0µ0+ω1µ1
and
ω0+ω1=1
. Based on the values
σ1
and
σ2
, (7) provides the tness function.
Subsequently, the optimization problem is reduced to determine the intensity level that maximizes (7):
(1)
Ph
i=
ni
nk ,Phi≥0,
L
i=1
Phi=
1,
(2)
C
1=
Ph1
ω0(th)
,...,Phth
ω0(th)
and C2=
Phc
th+1
ω1(th)
,...,PhL
ω1(th).
(3)
ω
0(th)=
th
i=1
Phiand ω1(th)=
L
th+1
Phi
.
(4)
µ
0=
th
i=1
iPhi
ω0(th)
and µ1=
L
i=th+1
iPhi
ω1(th
)
(5)
σ2
B
=
σ1
+
σ2
(6)
σ1=ω0(µ0+µT)2and σ2=ω1(µ1+µT)2,
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where
σ2
B(th)
is the Otsu method variance for a given th value. EBO methods are used determine the intensity
level th for maximizing the tness function according to (7). e tness or objective function
FOtsu(th)
can be
modulated for MTH as follows:
where
TH =[th1,th2,...thn−1]
represents a vector including MTH, and the variance calculations are as illus-
trated in (9).
Here i represents a class, and
ωi
is the occurrence probability, and
µj
is the mean of a class. For MTH, these
values are obtained as
and
Kapur entropy. Another non-parametric method used to determine the best threshold value of an image was
proposed by Kapur in7. e approach determines the best (th) implying the overall entropy to be maximized. For
a bi-level scenario, the Kapur target capacity can be determined as
where the entropies
H1
and
H2
are computed as follows:
In (13),
Phi
is the probability distribution of the intensity levels, which is computed by (1), and
ω0(th)
and
ω1(th)
are the probability distributions of classes
C1
and
C2
, respectively. ln(.) represents the natural logarithm. Like
the Otsu method, the entropy-based method can be modulated for MTH values. In this case, it is necessary to
separate an image into n groups using a similar number of thresholds. e equation below can dene the new
objective function:
where
TH =[th1,th2,...thn−1]
is the vector including MTH. Each entropy is computed separately with its
respective th values; thus, (14) is expanded for n entropies as follows:
erefore, the values of probability occurrence
(ωc
0,ω1,...,ωn−1)
of n classes can be determined using (10) and
the probability distribution
Phi
in (1).
Heap‑based optimizer (HBO). e HBO mimics the job responsibilities and descriptions of the employ-
ees within a company29. Although the job title dier from company to another and from business to another,
they are organized in a hierarchy and many of titles are given like corporate hierarchy structure, organizational
chart tree, or corporate rank hierarchy (CRH), etc. e collection of methods that outlines how particular activi-
ties are directed to realize the goals of an organization and also denes how information ows among levels
within the company56 is called an organizational structure. In this section, we explain the mathematical model
of the Heap-based optimizer.
Mathematical modeling of the interaction with immediate boss. e upper levels set the rules and laws for
employees within the centralized structure and subordinates follow their immediate boss. By the assumption
that each immediate boss is a parent node of its children, thus we can model this behaviour by upgrading the
location of each search agent
xi
with reference to its original node B by using the below equation:
(7)
FOtsu
(
th
)=
Max
(σ
2
B
(
th
))
where 0
≤
th
≤
L
−
1,
(8)
FOtsu(TH )
=
Max(σ 2
B(th)) where 0
≤
th
≤
L
−
1 and i
=[
1, 2, 3, ...,n
]
,
(9)
σ
2
B=
n
i=1
σi=
n
i=1
ω1(µ1−µT)2
.
(10)
ω
n−1(th)=
L
i=thn+1
Ph
i
(11)
µ
n−1=
L
i=thn+1
iPhi
ω1(thn)
.
(12)
Fkapur (th)=h1+h2,
(13)
h
1=
th
i=1
Phi
ω0
ln
Phi
ω0
and h2=
L
i=th+1
Phi
ω1
ln
Phi
ω1
.
(14)
F
kapur (TH)=
n
i=1
hi
,
(15)
h
c
n=
L
i=thn+1
Phi
ωn−1
ln
Phi
ωn−1
.
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where t is the current iteration, and | | calculates the absolute value.
k
is the
kth
component of vector
, and it is
generated random as following
where r is a random number in range
[0, 1]
. In Eq. (16), the designed parameter is
γ
, this parameter is computed
by the following rule:
e current iteration is t, T is the maximum iteration’s number, and C is a user dened parameter. while execut-
ing the iterations,
γ
decrease linearly from 2 to 0 and when reach to 0, it will increase again to 2 with iterations.
Modeling the interaction between colleagues mathematically. e employees having the same position are con-
sidered to be colleagues. Each employee interact with others to achieve the goals of an organization. By assuming
that the nodes at the same level in heap are colleagues and others are search agents
xi
and they update their posi-
tion based on the position of others selected colleagues
Sr
, the position of a search agent is calculated as follows:
where f is the objective function and calculates the tness of each search agent. Equation (19) enables the search
agents to explore the search space
Sk
r
if
(
Sr)<f(xi(t))
and allows to explore the search space
xk
i
otherwise.
Self contribution of an employee. is stage explains the concept of employees self contribution. Modeling of
this behavior are executed by retaining the prior position of the employee in the next iteration, as illustrated in
below equation:
In Eq.(20), the search agent
xi
does not change its rank for it’s kth design parameter in the next iteration. We
used this behavior to organize the rate of change of each search agent in population.
Putting it all together. is phase explains how to combine the equations of position updating and modelling
in previous subsections in one equation. ere are three probabilities of selection that are used to determine
equation used in updating position of search agents, this probabilities of selection is used to switch between
exploration and exploitation phase. ese probabilities is divided into three proportions
p1
,
p2
, and
p3
. e
search agent updates its location using Eq.(20) according to the proportion
p1
. e below equation computes
the outlines of
p1
.
e current iteration t, T is the maximum number of iterations. e search agent updates its location using Eq.
(16) according to the selection of proportion
p2
. e below equation compute the outlines of
p2
.
Finally, the search agent updates its location using Eq. (19) according to the selection of
p3
. e below equation
computes the outlines of
p3
.
A general position updating mechanism of HBO is computed as follows:
where
p1
,
p2
and
p3
are random numbers inside range [0,1]. is subsection claries that the Eq. (20) improves
exploration phase, Eq. (16) improves exploitation phase and convergence, and Eq. (19) allows the search agent
to move from the exploration phase to exploitation phase. According to this observations,
p1
is higher initially
and decreases linearly over iterations, this decreases the exploration phase and improves exploitation phase
(16)
Xk
i(t
+
1)
=
Bk
+
γ
k
|
Bk
−
Xk
i(t)|
(17)
�
=2r−1
(18)
γ
=
2−
tmod T
c
T
4c
(19)
X
k
i(t+1)=
Sk
r+γk|Sk
r−xk
i(t)|,f(�
Sr)<f(�xi(t
))
xk
i
+γk|Sk
r
−xk
i
(t)|,f(�
Sr)≥f(�xi(t
))
(20)
xk
i
(
t
+
1
)=
xk
i
(
t)
(21)
p
1=1−
t
T
(22)
p2=p1+
1−p
1
2
(23)
p
3=p2+
1−p
1
2
=
1
(24)
x
k
i(t+1)=
x
k
i(t),p≤p1
Bk+γk�
�Bk−xk
i(t)�
�,p>p1and p ≤p2
Sk
r+γk
�
�
Sk
r−xk
i(t)
�
�
,p>p2and p ≤p3and f (�
Sr)<f(�xi(t
))
xk
i
+γk
�
�
Sk
r
−xk
i
(t)
�
�
,p>p
2
and p ≤p
3
and f (�
S
r
)≥f(�x
i
(t
))
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with iterations. Aer calculating
p1
, the remainder of the span is splitted into two equal portions, which makes
attraction towards the colleague and boss equally probable.
Steps of HBO. is section summarizes the HBO steps and claries details about their implementation-related
calculations.
• Parameters initialization and denition: At rst, all the search agents are randomly initialized in a potential
solution space. e minimum and maximum boundaries of the search space are dened by variables
(Li,Ui)
respectively. e number of the population is (N) and maximum number of iteration (T). e specic param-
eter C can be calculated from
C=⌊T/25⌋
.
• Population initialization: e random population P is generated from N search agents, each consisting of D
dimensions. e population’s representation P is shown as follows:
• Heap building: We utilize
3−ary
heap to execute CRH. Although heap is a tree shaped data structure, it can
be executed using an array. e below operations are
d−ary
heap based operations required for the HBO
execution.
1. parent (i): By the assumption that the heap is performed as an array, this method receives the node’s index
then retrieves its parent’s index. e formulation of parent’s index for a node i is calculated by below equa-
tion:
where
⌊⌋
indicates the oor function, which retrieves the highest integer less than or equal to a given input.
2. child (i;k): e node can own a maximum of 3 childrens in a
3−ary
heap. erefore we can say, the man-
ager may not manages more than 3 direct persons. e index of the kth child of a node i is returned by this
function. e below equation shows mathematical formulation of this function.
For example,index of the 3nd child of 3nd node is calculated as:
3. depth (i): Assuming the last level depth equals to 0, therefore we can calculate the depth of any node i in
constant time through below formula:
e ceil function is
⌈⌉
, which retrieves the smallest integer greater than or equal to the input. For example,
depth of a node indexed 27 in heap is calculated as:
depth
(27)=
log
3
(81 −27 +1)
−1=⌈2.6476⌉=
3
4. colleague (i): Assuming that nodes at the same level of node i are the colleagues of this node. e index of
any elected colleague of node i is returned by this step and the index can be calculated by generating any
random integer in the range
dddepth(i)−1)
−1
D−1
+1, dd
depth(i)−1)
−
1
D−1
.
5. Heapify_Up (i): searching upward in the heap then add node i at its correct place to save the heap property.
Algorithm1 show the pseudo code of this operation.
Finally, the algorithm to build the heap is shown in Algorithm2.
p
=
�
x
T
1
�xT
2
.
.
.
�
x
T
N
=
x1
1x2
1x3
1xD
1
x1
2x2
2x3
2xD
2
x1
Nx2
Nx3
NxD
N
(25)
parent
(i)=
i
+
1
D
(26)
child(i,k)=D×i−D+k+1
child(3, 3)=12 −4+3+1=12
(27)
depth
(i)=
log(D×i−i+1)
−
1
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6. Repeated applications of position updating mechanism: search agents’ position is repeatedly updated accord-
ing to previously explained equations trying to converge on the optimum global. e pseudo code of HBO
is shown in Algorithm3.
Opposition‑based learning (OBL). e idea of opposition-based learning (OBL) is applicable strategy
of search strategy to avoid stagnancy in candidate solutions. OBL is a novel concept inspired from the opposite
relationship between entities57. e concept of opposition was presented in 2005 as the rst time, which has
attracted a many of research eorts in the last decennium. Many of Met-heuristic algorithms use the concept of
OBL to develop their performance such as harmony search algorithm58, grasshopper optimization59, ant colony
optimization60, articial bee colony61 and etc. OBL improve the exploitation phase of a search mechanism. Mostly
in meta-heuristic algorithms, convergence occurs quickly when the initial solutions are closer to the optimal
location; moreover, late convergence is expected. So that, OBL method produce novel solutions by considering
opposite search areas which may prove to be nearer to the best solution. OBL is regraded not only the candidate
solutions obtained by a stochastic iteration scheme, but also their ’opposite solutions’ located in opposite parts
of the search space. e OBL method has been hybridized with many bio-inspired optimization gives shorter
expected distances to the best solution compared to randomly sampled solution pairs62 such as cuckoo optimi-
zation algorithm63, shued complex evolution algorithm64, particle swarm optimization65, harmony search66,
chaotic dierential evolution algorithm67, and shued frog algorithm68. In optimization problems, the strategy
of simultaneously examining a candidate and its opposite solution has the purpose of accelerating the conver-
gence rate towards a globally best solution. According to previous related works, in initialization phase utilize
OBL only to improve the convergence and prevent stuck in local optima of HBO, then IHBO is utilized to solve
problem of multi-thresholding for image segmentation by use two objective functions called Kapur and Otsu.
The proposed IHBO algorithm
In this paper, the HBO algorithm is enhanced based on the OBL as local search strategy called IHBO to evade
the drawbacks of the random population and improve the rate of convergence of the algorithm by developing
the variety of its solutions. IHBO uses OBL strategy in the initialization phase to improve the search process as
following:
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where
Qi
is a vector-maintaining solution resulting from the use of OBL, and
UBj
and
LBj
are the upper and
lower bounds of the
jth
component of a vector X. e phases of the proposed image thresholding model are
described in depth below.
Initialization phase. In this phase, the algorithm starts by reading the image, converting it to grayscale,
computing the histogram of the selected benchmark images, and then computing the probability distribution
by (1). e algorithm initializes the IHBO parameters, which are the population size (N), maximum iteration
number (T), boundaries of the search space (
LI
,
UI
), and number of iterations per cycle (t). ereaer, the OBL
strategy is utilized to calculate the
Qi
vector-maintaining solution by (28).
Updating phase. is phase provides the best threshold values by evaluating the tness value of
Xi
and
Qi
populations. To update the search agents’ positions (X), we use the tness value of the optimal threshold of the
Otsu
FOtsu
method (8) or Kapur
Fkapur
method (14) as the objective function then comparing the tness value
of
Xi
and
Qi
and saving the global best solution with the highest tness. We dene the position of each agent
based on the tness value. In addition, we determine three probabilities of selection
P1
,
P2
, and
P3
using (21),
(22), and (23) sequentially, and then, based on the probabilities, we calculate the position of each agent within
the heap using (24). e agent’s position (X) is updated using important
D−ary
heap-based operations, such
as Heapify_Up(i), which is used to search for the superior node in the heap, and we insert the node at its correct
position to preserve the heap characteristics, as demonstrated in Algorithm1. en, each agent upgrades its
location frequently according to the best tness value, and seeks the global optimum, as depicted in Algorithm3.
Optimization scenarios of implementing the proposed IHBO algorithm illustrated in Figure1.
Segmentation phase. In this phase, we generate the segmented image with the optimal threshold value in
an image aer setting
xheap[1].value
as the threshold value of the image. e pseudo-code of the proposed IHBO
algorithm is illustrated in in Algorithm4.
(28)
Qi=LBj+UBj−Xi,i∈1, 2, ...,n
Figure1. Flowchart of the proposed algorithm.
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Computational complexity of the IHBO. is section discusses the computational complexity of IHBO
algorithm. e complexity of the population’s initialization can be represented as
O(N×D)
time complexity,
where D and N indicate the dimension of the problem and the size of the population, respectively. Additionally,
the IHBO calculates the complexity with the tness of each search agent as
O(N×D×Tmax)
, where the maxi-
mum number of iterations is
Tmax
. Besides, the IHBO requires
O(t)
time complexity for executing t number of
its main operations. erefore, the time complexity of the proposed IHBO is
O(N×D×t×Tmax)
. But, the
total amount of space occupied by the algorithm is called the space complexity. So, the space complexity of the
proposed IHBO can be represented by
O(N×D)
.
Performance evaluation of the proposed IHBO algorithm
Parameter settings. is section provides the estimation of the proposed IHBO algorithm. As we all know,
adjusting parameters will certainly aect the performance of an algorithm. However, according to the sugges-
tion of Arcuri etal.69, when comparing algorithm performance, all algorithm parameters should be kept at their
default values, which come from their original papers, to ensure they are in a relatively optimal state. Moreover,
we reduce the risk of better parametrization bias as each algorithm is set to its default values. erefore, in this
work, all algorithm parameters are kept at their default values.
us, the performance of the proposed IHBO algorithm is evaluated over the IEEE Congress on Evolution-
ary Computation (CEC’2020)70 as test problems. e CEC’2020 benchmark functions is utilized to test the
performance of IHBO algorithm. Initially, this benchmark functions contained 10 test functions referred to
as
f1
–
f10
. Consequently, function 1 is unimodal functions, functions 2–4 are multimodal functions, functions
5–7 are hybrid functions, and functions 8–10 are composition functions. Table1 illustrates the parameters set-
ting and mathematical formulation of the CEC’2020 benchmark functions; ’Fi*’ refers to the best global value.
Figure2 illustrates a 2D visualization of the CEC’2020 benchmark functions to understand the dierences and
the nature of each problem.
Statistical results analysis of CEC’2020 benchmark. is section illustrates CEC’2020 benchmark
test are utilized to estimate the performance of the proposed IHBO that contain qualitatively and quantitatively
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metrics. e standard deviation (STD) and mean of optimal solutions acquired by the proposed algorithm and
all another algorithms utilized in the comparison is calculated. Furthermore, the qualitative metrics consists of
average tness history, convergence curve, and search history is used to evaluate the performance of the IHBO
on the CEC’2020 test suite against seven well-known metaheuristic algorithms including the original HBO algo-
rithm, SSA, MFO, GWO, SCA, HS, and EMO. Table2 shows the STD and mean of the optimal value obtained
from the proposed algorithm and the other competing algorithms for each CEC’2020 benchmark functions with
20 dimensional, and the optimal results of the STD and mean is minimum values in results. e results of the
mean and STD of the proposed algorithm are proved superiority in solving seven CEC’2020 benchmark func-
tions against to other competing algorithms.
Table 1. Parameter settings of CEC’2020 benchmark test.
No. Function description Fi*
Unimodal function
F1 Shied and Rotated Bent Cigar Function 100
Multimodal shied and rotated functions
F2 Shied and Rotated Schwefel’s Function 1100
F3 Shied and Rotated Lunacek bi-Rastrigin Function 700
F4 Expanded Rosenbrock’s plus Griewangk’s Function 1900
Hybrid functions
F5 Hybrid Function 1 (
N=3
) 1700
F6 Hybrid Function 2 (
N=4
) 1600
F7 Hybrid Function 3 (
N=5
) 2100
Composition functions
F8 Composition Function 1 (
N=3
) 2200
F9 Composition Function 2 (
N=4
) 2400
F10 Composition Function 3 (
N=5
) 2500
Figure2. CEC’2020 benchmark functions in 2D view.
Table 2. Mean and STD values of the optimal tness obtained with competing algorithms on the CEC’2020
functions with
Dim =20
. Signicant values are in bold.
Functions
HS SCA MFO GWO SSA HBO IHBO
Mean Std Mean Std Mean Std Mean Std Me an Std Mean Std Mean Std
F1 2.7133E+10 1.4183E+05 8.6737E+09 3.2590E+04 2.4237E+07 1.3700E+09 4.8253E+10 4.8874E+03 3.5695E+09 3.2884E+03 1.4994E+10 3.7401E+01 4.1417E+11 1.5482E+09
F2 3.1859E+09 3.1203E+10 5.1390E+09 4.2318E+09 4.4332E+09 6.6776E+09 4.9220E+10 6.6198E+07 4.2912E+09 4.4798E+08 3.3996E+09 2.1044E+08 2.6221E+09 2.7699E+07
F3 9.6206E+08 6.8217E+06 9.6568E+09 3.7090E+06 7.7019E+07 2.7089E+06 8.9464E+08 2.7347E+08 9.0421E+07 6.3036E+08 7.9531E+07 3.8975E+10 1.1088E+07 7.0401E+07
F4 2.0157E+08 1.3668E+08 3.4429E+08 5.3612E+05 1.9991E+07 9.7905E+08 6.6635E+09 2.4878E+11 2.5966E+08 1.5150E+05 2.0029E+07 2.3428E+06 6.7985E+07 5.9369E+10
F5 6.2557E+08 6.4993E+08 1.1866E+10 7.4304E+08 1.4980E+09 6.7269E+05 8.5775E+11 3.9047E+06 1.5708E+10 1.7314E+06 5.9495E+09 6.7486E+07 1.4908E+08 7.8877E+07
F6 3.3488E+04 7.5211E+00 2.7999E+05 2.6069E+04 9.4696E+04 3.7214E+03 2.2082E+05 0.0000E+00 1.7973E+04 6.2490E+02 4.0739E+03 0.0000E+00 1.0872E+03 3.7049E+01
F7 3.3488E+07 7.5211E+03 2.7999E+07 2.6069E+07 9.4696E+06 3.7214E+06 2.2082E+08 0.0000E+00 1.7973E+07 6.2490E+05 4.0739E+05 0.0000E+00 1.0872E+05 3.7049E+04
F8 2.3744E+08 5.3560E+05 3.2600E+08 1.4781E+08 2.6752E+08 7.2306E+08 5.1174E+09 6.4210E+06 5.1713E+08 2.7202E+10 2.3802E+08 6.4911E+11 6.6854E+08 0.0000E+00
F9 3.3557E+10 4.2292E+10 3.0644E+10 6.3500E+08 2.9123E+09 5.6238E+08 3.6391E+10 1.6099E+12 3.0745E+09 6.5302E+10 2.9969E+09 3.5113E+08 1.0346E+09 3.6709E+09
F10 3.1568E+08 6.0725E+08 3.3829E+08 3.1844E+08 3.0797E+09 3.8147E+05 3.4776E+10 2.1653E+07 3.3225E+08 6.2899E+07 3.0599E+08 1.6946E+10 2.2474E+08 5.9369E+09
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Boxplot analysis. Boxplot analysis is a graphical technique used to display data distribution characteris-
tics. e boxplot technique is designed to report data that follow a normal distribution and have homogeneous
variances, the results of boxplot for all algorithms and them functions are illustrated in Fig.3. Boxplot is most
important plots to describe data distributions into quartiles. is quartiles are the median of the rst half of
the data is rst quartile, the second quartile is the median, the third quartile is median of the second half of the
data, and the largest observation. e region among the rst and third quartile is called the interquartile range
and used to give an indication of spread in the data. e ends of the rectangles determine the lower and upper
quartiles and a narrow boxplot refers to highest agreement among data. Figure3 shows the boxplots of the pro-
posed IHBO algorithm and illustrates the results of ten functions boxplot for Dim = 20. In reality, the results of
proposed algorithm are proved superiority than all other competing algorithms on most of the test functions,
but the performance of proposed algorithm is limited on F2, and F7.
Convergence curves analysis. is subsection explains the convergence plots of the proposed algorithm
with other competitor algorithms. Figure4 illustrates the convergence plots of IHBO, HBO, SSA, GWO, MFO,
HS, and SCA for the CEC’2020 benchmark functions. Furthermore, the proposed algorithm achieved optimal
solutions and reached a stable point for most functions. us, IHBO can solve problems that require fast com-
putation, such as online optimization problems. e proposed algorithm exhibited stable behavior, and its solu-
tions converged easily in most of the problems it was tested on. Due to space limitations.
Figure3. Results of Boxplots obtained all algorithms over CEC’2020 benchmark functions with
Dim =20
.
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Qualitative metrics analysis. Even though the earlier outcome analyses assure the high performance of
the proposed IHBO algorithm, the performance of more experiments and analyses would help us to draw more
clear conclusions about the algorithm performance in real problem solving. Figure5 illustrates the qualitative
analysis of the proposed IHBO algorithm. e rst column illustrates a set of the CEC’2020 benchmark func-
tions as plots in two-dimensional space. e second column illustrates the search history of search agents, from
the rst to the last iteration and display their exploitation behavior to realize the desired outcomes. e third col-
umn shows the average tness history over 350 iterations, explaining the general behavior of the agents and the
role that they play in the search of the best solution. According to average tness history, all the history curves are
decreasing, which means that the population improves at each iteration. e fourth column shows convergence
curve and optimization history revealed the progress of tness over a number of iterations. Optimization history
is decreasing indicates that the solutions are optimized during iterations until reach the best solution.
Environmental and experimental requirements
is section presents the test images used for the experiments, then describes the empirical setup, and analyzes
the results.
Figure4. Results of convergence curves for the proposed algorithm with other competing algorithms over
CEC’2020 benchmark functions with
Dim =20
.
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Benchmark images. To evaluate the proposed method, ten images of common benchmark were used. e
selected benchmark images due to the various levels of complexity and included the following images: Baboon,
Lena, Buttery, Pirate, Cameraman, Peppers, Bridge, Living Room, Barbara, and Jetplane71,72. Most images had
the same dimensions (512
×
512 pixels); however, two images (Cameraman and Lena) were 256
×
256 pixels.
Table3 displays the set of test images used.
Environmental setup. In this study, the proposed IHBO is compared with seven well-known metaheuris-
tic algorithms including the original HBO, SSA, MFO, GWO, SCA, HS, and EMO. All competitor algorithms
were applied and executed in Matlab 2015, and implemented on PC with 6G RAM running in a Windows 8.1
64-bit environment with an Intel Core I5 processor. e counterparts were executed 30 times per test image,
number of iterations was set to 350, and population size is 50. e parameters settings of each algorithm were
determined according to standard criteria and information found in previous literature (default values). e
number of thresholds used was
th2,th3,th4,
and
th5
according to related literature73. e parameters settings of
the IHBO and their values are presented in Table4.
Evaluation metrics. Two metrics were utilized to estimate the performance of the IHBO algorithm. e
rst metric was used to evaluate the quality of the image, while the second metric was used to compare the edges
of the segmented image. ese metrics are important for evaluating the performance of the IHBO approach
based on the Otsu and Kapur methods as objective functions. Statistical tests, such as the standard deviation
(STD), Wilcoxon rank test, and average, were used to analyze the tness of the proposed algorithm. We used the
SSIM48, FSIM74, and PSNR75 to measure the quality and stability of the image.
Structural similarity index (SSIM). e SSIM48 index is a metric that is used to analyze the internal structures
in a segmented image. A higher SSIM value indicates better segmentation of the original image due to the fact
that structures in the two images match. e equation below describes the SSIM:
e mean of the intensities of the original image I and segmented image Seg are
µI
and
µSeg
, respectively, and
σI
and
σSeg
are the standard deviations of the original image I and segmented image Seg, respectively.
σI,Seg
is the
covariance of the original image I and segmented image Seg, and
c1
and
c2
are two constants.
(29)
SSIM
(I,Seg)=
(2µ
1
µ
Seg
+c
1
)(2σ
1,Seg
+c
2
)
(µ2
I
+µ2
Seg
+c1)(σ 1
I
+σ2
Seg
+c2
)
Figure5. Qualitative metrics on F2, F4, F8, F9, and F10 in 2D view of the functions.
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Feature similarity index (FSIM). e FSIM74 index is a metric that is used to compute the similarity between
the segmented image and original image based on their internal features. A higher FSIM value indicates better
segmentation by the thresholding method. e FSIM can be described in the following steps:
e entire domain of the image is
ω
:
(30)
FSIM
=
vǫ� SL(v)PCm(v
)
vǫ� PCm(v)
(31)
SL(v)=SPC (v)SG(v)
(32)
S
PC(v)=
2PC
1
(v)PC
2
(v)+T
1
PC
2
1
(
v
)+
PC
2
2
(
v
)+
T1
Table 3. Set of test images.
Table 4. e parameters of IHBO and it’s value.
Parameter Val u e
Maximum number of iterations 30
Number of local iterations 350
Size of population 50
Dimension of problem (Dim) 20
Mutation ratio 0.5
Values [C,
p1
,
p2
] from corresponding equations
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and G is the image’s gradient magnitude and can be computed as follows:
e vector’s magnitude in v on n is E(v), and the local amplitude of scale n is
An(v)
. e small positive number
is
ǫ
and
PCm(v)=max(PC1(v),PC2(v))
.
Peak signal‑to‑noise ratio (PSNR). e PSNR75 is another metric used to evaluate the quality of segmentation
by determining the dierence between the quality of the original image and that of the segmented image. e
PSNR is used to compare the original and segmented image using the root mean square error (RMSE) of each
pixel, as expressed in (37). e PSNR can be dened as follows:
where
In (37), I and Seg are the segmented and original images of size
M×N
, respectively. A higher PSNR value indi-
cates that there is higher similarity between the segmented and original images, which reects a more eective
segmentation process.
Experimental results and discussion
e experimental results are discussed in this section to evaluate the eciency of the proposed algorithm.
Otsu results analysis. is subsection analyzes the outcomes of the IHBO based on the between-class
variance as the tness function, as proposed by Otsu. Table7 illustrates the best threshold values obtained by
applying the IHBO with the Otsu entropy as the objective function (8). Tables5 and 6 present a graphical analy-
sis of the thresholds, illustrating the resulting images of the IHBO with a dierent number of thresholds. Table8
shows the computational time values of comparison algorithms obtained by Otsu’s method. e IHBO proved
its superiority in computational time compared to other competitive algorithms with 23 cases in 40 experi-
ments and came in the rst place. GWO came in second place with 10 experiments, while HBO come in third
place with nine experiments, followed by EMO with two experiments. Finally, the MFO came in h place with
only one experiment, and the remaining algorithms could not obtain the best computational time in any of the
experiments. Table9 illustrates the Otsu STD and average of the tness results for the benchmark images. e
IHBO demonstrated superiority in MTH by obtaining an optimal tness values for 23 cases in 40 experiments.
e HBO algorithm obtained the best tness value in eight experiments, while the SCA obtained the optimal
tness value in ve experiments and SSA come in fourth place with four experiments followed by MFO with
three experiments. Finally, HS obtained the optimal tness value in only one experiments and the remaining
algorithms could not obtain the optimal tness value in any of the experiments. Table9 illustrates the STD values
calculated for the 40 independent outcomes for each tested image with various thresholds. A lower STD value
indicates that the algorithm is more stable.
Table10 presents the STD and mean PSNR for the benchmark images using the eight MAs. e IHBO was in
rst place in terms of the mean values of PSNR in 22 experiments. e SSA was in second place in seven experi-
ments, while HBO was in third place, as it was superior in only six experiments. In fourth place was SCA with the
best PSNR in only ve experiments followed by MFO and HS with three experiments. Finally, the worst results
were obtained by EMO which did not obtain the optimal values of the PSNR in any of the experiments. With
respect to the STD, the IHBO was not the best alternative for lower dimensions (2 or 3 th). is is because the
STD value was higher, which represents higher instability of the algorithm. However, MFO was a more unstable
algorithm in terms of the PSNR. For the remaining approaches, the STD values followed the same tendency:
lower for small dimensions and higher for four thresholds. However, the SSA was the least unstable algorithm,
while the SCA was in second place. HBO was in the third place, HS was in fourth place, and the IHBO was in
h place. Furthermore, GWO was in sixth place, and EMO was in seventh place.
Table11 illustrates the STD and mean of the FSIM obtained from 40 experiments. e results of the FSIM
indicate that the IHBO obtained the highest FSIM in 22 experiments and was in rst place, while the HBO was
in second place in ten experiments. However, the SCA was in third place in eight experiments. SSA was in fourth
place in two experiments, followed by HS and EMO, which appeared in h place in only one experiment.
Finally, GWO and MFO came in last place in the experiments. e SCA was thus the best approach in terms of
the STD because its values were lower in most experiments. e SSA came in second place, followed by EMO
(33)
S
G(v)=
2G
1
(v)G
2
(v)+T
1
G
2
1
(
v
)+
G
2
2
(
v
)+
T1
,
(34)
G
=
G2
x+G
2
y
(35)
PC
(v)=
E(v)
ǫ+
n
A
n
(v)
(36)
PSNR
=20log10
255
RMSE ,
(37)
RMSE
=
M
i=1
N
j=1((I(i,j)−Seg(i,j))2)
MxN .
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in third place. en, the IHBO appeared in fourth place. GWO was in h place, followed by HBO. Finally, the
least stable approaches were MFO and HS due to their high STD values in most cases.
Table12 presents the results of the STD and mean SSIM obtained in 40 experiments. e IHBO came in
rst place in terms of mean PSNR with the best SSIM in 22 experiments, while the SCA, HBO, and SSA came
in second place in six experiments with higher SSIM values. EMO came in the third place in two experiments,
followed by HS and MFO, which came in fourth place with only one experiment. Finally, GWO came in last place
in the experiments. Because it provided the largest number of minimum values of the STD of all algorithms,
Table 5. Implementation results of IHBO-Otsu over the set of benchmark images.
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SCA was the best method. In second place was the HBO, followed by EMO, which was in third place. e IHBO
was in fourth place, while GWO was in h. Finally, MFO, HS, and SSA had no minimum STD values in the
experiments.
Table7 illustrates the thresholds that were applied on the selected benchmark images. In Tables5 and6, the
histograms are illustrated with the respective threshold values and the segmented images of the selected images
using 2, 3, 4, and 5 thresholds. ese results indicate that for some images, there was improvement in the qual-
ity of their contrast as the number of thresholds increased, particularly for the images Buttery, Living Room,
Table 6. Implementation results of IHBO-Otsu over the set of benchmark images.
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Jetplane, Lena, Pirate, Cameraman, Lake, and Bridge, presenting a higher amount of information in the image
with the largest number of thresholds when compared with an image with only two thresholds. e most dif-
cult histograms to segment were for Test 6, 9, and 10, relating to Bridge, Buttery, and Barbara, respectively.
e complexity was due to dierent numbers of pixels in the images, which could produce several classes or even
make it impossible to select the optimal thresholds.
Table13 presents the p-values resulting from the Wilcoxon test for tness using the Otsu tness function.
is table presents the dierence between the proposed algorithm and the compared algorithms (HBO, SSA,
MFO, GWO, SCA, HS, and EMO).
A dierence between the SCA and MFO in comparison to the IHBO can be observed, which indicates that
the proposed algorithm has a signicant development. However, for the number of thresholds (n) = 5, the
dierences between the IHBO and most of the competing algorithms are clear by performing the comparison
over 30 runs in each experiment. In the results, NaN indicates that the dataset to be compared is the same. is
signies that the algorithms obtained the same solution; thus, their results from the Wilcoxon test reveal that
they are similar and that there are no dierences between the methods.
Table 7. e best thresholds values obtained by Otsu’s method.
Test Image n IHBO HBO SSA MFO GWO SCA HS EMO
Test 1
2 81 124 80 125 79 125 79 125 79 125 79 125 79 125 79 125
3 72 114 144 71 112 143 75 115 139 71 111 142 74 119 144 70 116 144 70 116 140 71 106 140
4 72 108 135 159 71 108 131 155 70 102 124 149 71 105 130 153 73 105 139 159 70 106 134 157 73 102 133 148 70 100 121 144
5 62 88 114 139 161 62 88 125 139 167 36 82 102 139 163 62 87 125 140 169 52 77 117 141 163 58 51 116 138 162 63 81 118 136 160 62 88 122 136 165
Test 2
2 65 122 65 122 65 122 63 122 65 122 65 121 65 122 65 122
3 50 88 128 65 122 129 55 89 126 58 88 124 59 92 129 53 85 124 59 91 127 50 86 123
4 48 85 118 150 48 85 118 150 50 88 118 151 48 82 115 149 46 81 117 144 42 87 110 141 45 88 115 150 39 80 112 140
5 48 81 107 133 160 48 85 102 145 159 48 85 108 140 159 48 88 101 143 160 48 83 106 138 157 44 83 101 138 155 48 85 103 145 150 48 79 102 141 160
Test 3
2 98 152 97 149 97 149 97 150 97 149 97 149 97 149 97 149
3 72 116 137 80 120 131 85 121 131 70 118 136 77 110 134 82 110 135 73 119 132 85 116 130
4 66 98 124 149 58 102 122 160 52 106 127 158 58 92 120 145 51 79 124 151 63 84 125 153 51 104 127 155 51 108 125 155
5 58 85 110 133 157 67 91 120 136 154 54 83 109 134 151 66 90 122 134 152 54 86 105 135 151 59 86 110 134 160 69 91 125 133 153 67 88 124 130 151
Test 4
2 59 119 59 119 59 119 59 119 59 119 59 119 59 119 59 119
3 42 95 140 59 112 139 38 92 138 60 110 140 52 115 140 48 110 137 38 93 139 34 90 136
4 36 83 123 150 31 86 140 152 38 92 138 159 30 85 142 151 38 91 137 149 37 83 139 152 37 92 137 149 29 75 130 149
5 36 81 120 147 170 36 82 122 149 172 35 81 118 147 162 35 82 121 148 171 35 81 121 144 160 35 82 120 147 173 36 82 120 148 171 35 82 121 145 170
Test 5
2 93 145 93 145 93 145 92 145 93 145 93 145 93 145 92 145
3 84 129 172 83 125 171 79 121 168 83 124 170 82 126 164 83 125 171 79 120 168 83 123 171
4 67 105 142 180 63 109 152 173 84 129 152 182 67 103 141 178 68 107 140 178 68 107 143 175 67 103 140 176 84 128 154 180
566 101 135 166
191 68 106 142 169
190 68 106 132 159
184 68 104 140 169
193 22 91 115 147 187 22 91 115 147 187 68 106 140 170
191 68 108 131 160
182
Test 6
2 75 126 75 124 75 124 75 124 75 124 75 124 75 124 75 124
3 63 103 145 65 104 140 65 102 139 65 102 136 66 107 136 62 107 141 65 100 134 65 102 137
4 55 88 120 156 60 103 125 190 50 88 128 171 60 106 122 175 56 77 124 177 59 69 122 172 50 89 125 170 52 88 129 170
5 46 76 103 134 164 46 81 104 139 161 48 80 103 132 165 44 83 106 139 160 39 69 101 138 168 41 87 109 138 161 46 81 104 139 161 46 81 111 134 160
Test 7
2 76 123 75 123 75 123 75 123 71 120 75 123 75 123 75 123
3 55 97 132 56 103 133 56 103 133 49 105 132 45 108 128 56 103 133 56 105 131 44 108 131
4 49 88 121 147 46 87 122 148 46 86 124 148 46 87 122 148 47 88 123 147 46 84 123 148 46 85 123 149 46 86 120 145
5 41 76 105 129 153 43 88 102 126 158 41 88 108 129 159 43 88 100 120 157 41 61 111 121 161 38 81 110 123 156 43 88 104 125 155 43 89 110 121 159
Test 8
2 51 116 51 116 51 116 51 115 51 116 51 116 51 115 51 115
3 31 73 112 36 76 115 36 74 116 36 72 110 35 74 114 36 76 114 38 74 114 40 62 108
4 25 55 89 126 23 59 90 106 127 30 61 90 126 22 59 94 125 123 31 56 88 127 30 52 85 126 30 61 92 125 30 65 8790 125
5 22 52 83 110 136 22 53 86 112 132 22 53 88 111 132 20 52 80 109 134 22 55 78 111 130 21 53 99 125 140 22 53 86 114 131 22 51 82 115 130
Test 9
2 83 119 81 118 81 118 81 118 81 118 81 118 81 118 81 118
3 72 99 127 70 96 129 71 108 129 70 90 125 71 95 127 72 92 120 71 105 128 70 94 128
4 71 98 124 152 72 99 123 152 63 103 125 153 72 95 120 151 74 94 124 158 69 96 123 152 72 99 127 154 63 113 126 154
5 69 90 111 132 155 71 89 114 132 159 68 71 113 139 159 71 85 114 130 155 61 80 92 115 158 63 81 111 130 159 71 89 114 131 165 70 89 111 130 154
Test 10
2 55 114 52 112 52 112 52 112 52 112 52 112 52 112 52 112
3 52 101 138 49 102 138 53 102 138 50 101 136 54 100 129 47 102 132 51 102 137 52 109 135
4 50 95 125 150 50 101 127 147 49 100 126 146 50 101 127 147 42 97 124 143 48 92 120 142 49 98 121 140 49 98 124 148
5 46 91 117 139 160 48 90 115 140 159 47 88 103 139 162 46 90 117 137 158 41 86 103 141 159 40 79 119 143 161 48 92 114 142 157 50 91 112 141 153
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Kapur results analysis. e best results are illustrated in Table16, and were obtained by the IHBO using a
tness function such as Kapur entropy (14). Tables14, and15, present the histogram distribution of the bench-
mark images and segmented images with dierent numbers of thresholds produced by the IHBO. e results in
Table18 illustrate that the proposed algorithm with the Kapur entropy method proved outperform other algo-
rithms in terms of SSIM (Table18); in addition to, it outperformed other algorithms in terms of the mean FSIM
(Table20), PSNR (Table19), and mean tness.
e values of the computational time of comparison algorithms obtained by Otsu’s method are presented in
Table17. e IHBO came in rst place with 24 cases in 40 experiments and proved its superiority in computa-
tional time compared to other competitive algorithms. HBO came in second place with 13 experiments, while
GWO came in third place with ten experiments, followed by SSA with four experiments. Finally, the SCA came
in h place with only one experiment, and the remaining algorithms could not obtain the best computational
time in any of the experiments.
Table18 presents the STD and average tness results of the Kapur method on the benchmark images. e
IHBO was in rst place by obtaining optimal tness values with 24 cases in 40 experiments. e SCA was in
second place in seven experiments, while the HBO was in third place in ve experiments. SSA was in fourth
place in three experiments, and HS was in h place in two experiments followed by GWO in sixth place in one
experiments. Finally, EMO and MFO could not produce optimal tness values. Table18 also presents the STD
values to demonstrate the stability of the algorithm according to the repetition of the values.
Table19 illustrates the STD and mean PSNR. e IHBO came in rst place in 23 experiments with optimal
PSNR values, while the SCA came in second place in ten experiments only. HBO came in third place in four
experiments, while HS and GWO came in fourth place in two experiments. Finally, SSA, MFO, and EMO came
in last place with no experiments. According to the STD values, EMO came in rst place with the maximum
Table 8. e computational time values of comparison algorithms obtained by Otsu’s method. Signicant
values are in bold.
Test Image n
IHBO HBO SSA MFO GWO SCA HS EMO
Mean STD Mean STD Mean STD Mea n STD Mean STD Mean STD Mean STD Mea n STD
Test 1
2 0.5652 3.1416E-01 0.6926 7.0576E-01 0.7961 4.9555E-01 0.5781 3.1218E-01 0.5043 2.8886E-01 0.6464 6.3756E-01 0.6359 2.6455E-01 0.6463 3.1111E-01
30.4634 3.8720E-02 0.5323 1.0758E-01 0.6453 2.0229E-01 0.5630 1.2397E-01 0.5793 1.0142E-01 0.6819 3.6190E-01 0.6193 1.1858E-01 0.6819 1.3945E-01
40.4337 4.5210E-02 0.9825 2.7709E-01 0.6215 1.6423E-01 0.5252 5.0820E-02 0.4713 2.6840E-02 0.7821 9.3830E-02 0.5777 3.8720E-02 0.7820 4.5535E-02
50.4352 4.7080E-02 0.9205 1.0670E-01 0.6137 1.8678E-01 0.5482 1.3783E-01 0.4926 3.7180E-02 0.7857 7.9750E-02 0.6030 5.0710E-02 0.7856 5.9635E-02
Test 2
2 0.6488 3.3770E-02 0.5741 3.0052E-01 0.7236 2.3441E-01 0.7423 9.0090E-02 0.5295 1.2100E-02 0.9135 2.4068E-01 0.8165 1.1792E-01 0.9134 1.3867E-01
3 0.9051 1.3200E+01 0.5305 1.2859E-01 0.9649 3.6256E-01 0.8414 1.9767E-01 0.5305 1.0120E-02 0.8508 2.3100E-01 0.9256 4.0733E-01 0.8507 4.7902E-01
4 0.7256 1.4960E-02 0.8701 1.8095E-01 0.9772 7.5636E-01 0.8050 1.2320E-01 0.6547 1.1990E-02 0.8758 1.8535E-01 0.8855 1.5367E-01 0.8757 1.7303E-01
50.6570 1.1110E-02 0.8759 9.5480E-02 0.9779 7.9266E-01 0.8343 9.4820E-02 0.6749 1.4628E-02 0.8893 1.5235E-01 0.9177 4.2570E-01 0.8892 4.7934E-01
Test 3
20.5602 1.0670E-02 0.8046 1.4916E-01 0.8584 5.4142E-01 0.7366 6.5120E-02 0.6113 2.4100E-02 0.8544 2.7258E-01 0.8103 2.7026E-01 0.843 3.0431E-01
3 0.5259 9.1300E-03 0.4801 9.8010E-02 0.8131 4.8246E-01 0.8016 1.0615E-01 0.6253 1.7146E-02 0.8406 2.2460E-01 0.8818 1.4435E-01 0.8405 1.6253E-01
40.4662 9.1300E-03 0.4900 3.0569E-01 0.8405 4.3813E-01 0.9729 2.8446E-01 0.6547 4.8823E-01 0.9092 1.5528E-01 1.0701 9.9876E-02 0.9091 1.1246E-01
50.4178 2.5892E-02 0.4718 2.2671E-01 0.9731 9.1850E-01 0.7714 1.0098E-01 0.6940 3.6330E-02 0.8769 1.4981E-01 0.8485 1.5409E-01 0.8768 1.7351E-01
Test 4
2 0.4121 6.9300E-02 0.4039 1.9448E-01 0.8385 4.8444E-01 0.6912 4.0810E-02 0.4285 5.7672E-02 0.6760 2.7763E-01 0.7603 2.5801E-02 0.4039 2.9052E-02
3 0.4333 3.4948E-02 0.4682 1.2121E-01 0.8624 4.7234E-01 0.7288 6.7100E-02 0.4451 3.1893E-02 0.6149 1.4125E-01 0.8017 3.5551E-02 0.4149 4.0030E-02
4 0.4585 3.5388E-02 0.4436 1.1528E-01 0.9959 6.2370E-01 0.7387 3.7400E-02 0.4647 1.1870E-02 0.6912 2.5730E-01 0.8126 2.9131E-02 0.4912 3.2802E-02
50.4473 8.8689E-02 0.4604 2.3199E-01 1.2890 1.3500E+00 0.7356 5.1590E-02 0.4898 2.3980E-02 0.6012 1.9012E-01 0.8092 1.0297E-01 0.5012 1.1594E-01
Test 5
2 0.6028 3.5168E-02 0.5735 7.2930E-02 0.8597 4.9159E-01 0.6680 4.1030E-02 0.7032 1.3549E-02 0.6204 2.9749E-01 0.7348 7.0983E-02 0.6142 7.9927E-02
30.5067 2.1760E-02 0.5765 2.0988E-01 0.8965 7.4514E-01 0.7108 5.0118E-02 0.7249 9.7120E-03 0.8450 4.3708E-01 0.7818 6.0877E-02 0.8365 6.8548E-02
40.5189 5.7150E-03 0.6712 6.0280E-02 1.0702 8.2478E-01 0.7183 6.6784E-02 0.7514 1.4028E-02 0.9586 3.8987E-01 0.7902 1.8810E-01 0.9490 2.1180E-01
50.5004 5.1763E-02 0.7004 7.6230E-02 1.1645 7.8078E-01 0.7389 6.5945E-02 0.8056 5.5394E-02 0.9699 1.5457E-01 0.8128 2.6527E-01 0.9601 2.9869E-01
Test 6
20.3997 2.4837E-02 0.7587 3.5472E-02 0.9604 1.8348E-01 1.0563 6.9926E-01 0.3997 1.3189E-02 0.8271 3.0866E-01 1.1620 5.3753E-02 0.8187 5.9451E-02
30.4492 5.4730E-02 0.6675 7.1221E-02 0.8845 6.4251E-01 0.9728 3.1318E-01 0.4573 1.9784E-02 0.8071 2.5849E-01 1.0701 1.7153E-01 0.7989 1.8971E-01
4 0.4549 2.0771E-02 0.7090 8.5269E-02 0.6991 3.0492E-01 0.7689 1.9939E-01 0.4539 1.3549E-02 0.8437 3.1925E-01 0.8458 6.2836E-02 0.8352 6.9497E-02
50.4692 2.4177E-02 0.7047 1.0302E-01 0.7083 1.2606E-01 0.7791 7.2899E-02 0.4900 4.9399E-02 0.8053 9.9400E-02 0.8570 1.2805E-01 0.7972 1.4163E-01
Test 7
2 0.4509 3.3190E-02 0.6757 1.2360E-01 0.4790 3.6707E-02 0.5268 1.3225E-01 0.3931 1.5107E-02 0.7903 1.9452E-01 0.5742 4.5502E-02 0.7824 5.0325E-02
30.4036 2.5387E-02 0.6619 1.0912E-01 0.5345 3.3959E-02 0.5879 1.6738E-01 0.4636 1.1414E-01 0.8713 2.9999E-01 0.6408 1.1446E-01 0.8625 1.2660E-01
40.4834 1.3562E-01 0.6949 1.8872E-01 0.7648 1.6870E-01 0.8412 9.4601E-02 0.6201 1.6906E-02 0.8249 1.0618E-01 0.9169 3.9701E-02 0.8166 4.3909E-02
50.4511 1.6661E-01 0.6434 4.0184E-02 1.0059 1.0034E-01 1.1064 6.2228E-02 0.6565 4.3670E-01 0.8755 2.2056E-01 1.2060 1.3547E-01 0.8667 1.4983E-01
Test 8
20.4889 1.2188E-01 0.6315 6.1202E-02 0.7423 7.6161E-02 0.8165 2.0503E-01 0.5219 2.7589E-02 0.8219 7.6893E-01 0.8900 4.2408E-02 0.8136 4.6903E-02
30.4311 6.6160E-02 0.6312 6.3380E-02 0.8094 5.5390E-02 0.8903 1.1814E-01 0.5363 2.1340E-02 0.9446 4.5270E-01 0.9704 7.9402E-02 0.9350 9.0201E-02
4 0.5785 2.6365E-01 0.5548 3.8006E-02 0.8652 7.8029E-02 0.9516 1.0276E-01 0.5548 1.0729E-02 0.9682 7.9573E-01 1.0372 7.5407E-02 0.9584 8.5662E-02
50.4404 1.0429E-01 0.7974 2.2139E-01 1.0382 9.3762E-02 1.1420 2.4321E-01 0.6679 1.0139E-02 0.9837 6.0668E-01 1.2447 5.9423E-02 0.9738 6.7505E-02
Test 9
2 0.5814 4.0773E-02 0.5747 3.4794E-01 0.4466 1.0239E-01 0.4913 9.3195E-02 0.6124 1.6911E-02 0.7452 5.8153E-02 0.5355 3.1984E-02 0.7376 3.3135E-02
30.5994 2.8904E-02 0.6029 9.4721E-02 0.5045 1.9628E-01 0.5549 1.2631E-01 0.6341 1.7031E-02 0.7472 9.1132E-02 0.6048 3.6978E-02 0.7397 3.8309E-02
4 0.5188 5.6489E-02 0.4566 4.1557E-01 0.6027 2.6702E-01 0.4566 1.6636E-01 0.6585 3.2251E-02 0.7911 1.1743E-01 0.7226 2.9131E-02 0.7831 3.4479E-02
5 0.5332 1.9848E-02 0.4842 5.1341E-01 0.6898 2.2361E-01 0.7587 1.2510E-01 0.6787 2.8386E-02 0.8748 4.9614E-01 0.8270 3.1434E-02 0.8660 3.7205E-02
Test 10
20.3844 1.7914E-02 0.9624 3.1042E-01 0.7397 3.5095E-02 0.8136 1.3587E-01 0.4082 1.5582E-02 0.8474 1.3220E+00 0.8869 4.1000E-02 0.8388 4.8528E-02
3 0.3418 3.0178E-02 0.9286 4.7516E-01 0.7943 2.1144E-02 0.8736 2.1161E-01 0.3513 5.5684E-02 0.9564 1.2279E+00 0.9523 2.7447E-02 0.9467 3.2486E-02
4 0.4190 8.2430E-03 0.9657 8.4410E-02 0.8262 2.1362E-02 0.9088 3.8729E-01 0.3815 6.2569E-02 0.9658 6.8974E-01 0.9905 4.3848E-02 0.9560 5.0660E-02
50.4238 5.8797E-02 0.9911 1.7805E-01 0.9468 8.3050E-01 1.0414 1.4083E-01 0.5098 7.4769E-02 0.9882 4.0178E-01 1.1352 6.8562E-02 0.9782 7.9214E-02
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number of minimum STD cases, followed by IHBO in second place, HS in third place, MFO and the HBO in
fourth place, SSA in h place, and SCA in sixth place. GWO had no optimal STD values.
Table20 provides the STD and mean FSIM. It can be observed that the IHBO was in rst place in 17 experi-
ments, while SCA was in second place in nine experiments. HS came in third place with six experiments, while
MFO, HBO and GWO came in fourth place with three experiments, followed by SSA in h place with one
experiment. Finally, EMO came in last place with no experiments. In terms of the STD, MFO came in rst place
with the maximum number of minimum STD cases, followed by SSA in second place, EMO and the IHBO in
third place, HS in fourth place, HBO in h place, and SCA in sixth place. GWO had no optimal STD values.
e mean and STD of SSIM are presented in Table21. e results indicate that IHBO was in rst place in
19 experiments in terms of the SSIM, followed by SCA, which were in second place with seven experiments.
e GWO and MFO were in third place in six experiments, while HBO and HS were in fourth place with three
experiments, followed by SSA in h place with only one experiment. Lastly, EMO had no optimal experiments
in terms of SSIM. According to the STD values, MFO came in rst place with the maximum number of minimum
STD cases. GWO, EMO, and SCA were in second place, while the IHBO was in third place. SSA and HBO were
in fourth place, while HS was in h place.
Finally, Table22 presents the p-values resulting from the Wilcoxon test for tness using the Kapur tness
function. is table presents the dierence between the proposed algorithm and the compared algorithms (HBO,
SSA, MFO, GWO, SCA, HS, and EMO). e results in Table22 indicate that the IHBO was dierent from the
SCA and EMO but similar to the remaining algorithms. e exceptions occurred for nTH = 5, where in some
cases the values exhibited dierences as well as similarities (NaN values).
Human participants or animals. is article does not contain any studies with human participants or
animals performed by any of the authors.
Table 9. Mean and STD values of the optimal tness obtained by Otsu’s method. Signicant values are in bold.
Test Image n
IHBO HBO SSA MFO GWO SCA HS EMO
Mean STD Mean STD Mean STD Mea n STD Mean STD Mean STD Mean STD Mea n STD
Test 1
22194.885 5.7134E-13 2131.370 5.2000E-12 1958.607 3.2500E-12 1964.426 9.2000E-13 1604.246 8.0200E-02 1964.402 8.2900E-13 1963.413 9.2300E-13 1964.417 1.6600E-02
32260.382 4.0653E-06 2194.972 3.7000E-05 2123.712 1.4600E-02 2131.265 5.7800E-03 1604.246 5.6000E-01 2129.935 4.6100E-13 2111.747 0.0000E+00 2131.372 4.1600E-02
42286.649 2.7735E-05 2220.479 2.8100E-02 2184.805 4.0400E-02 2194.973 3.4900E-02 1601.951 1.8900E+01 2437.504 2.3100E-12 2179.246 9.3000E-03 2194.947 2.0500E-01
52307.986 7.8450E-03 2241.198 7.1400E-02 2208.524 4.6100E-02 2220.834 9.8000E-03 1600.293 1.4400E+01 2587.055 1.3500E+00 2215.602 1.2300E-02 2219.391 4.3100E-01
Test 2
2 2221.765 7.6033E-14 2157.472 6.9200E-13 2430.503 6.9200E-13 2537.504 6.9200E-13 2034.527 8.2500E-02 2437.504 6.9200E-13 2437.871 6.9200E-13 2437.504 6.9200E-13
3 2256.541 1.0141E-13 2191.242 9.2300E-13 2580.323 2.5000E-02 2588.330 9.2300E-13 2039.244 8.2000E-01 2587.583 9.2300E-13 2582.915 4.6100E-13 2588.300 1.3700E-01
42777.255 2.1865E-03 2696.888 1.9900E-02 2646.375 4.1400E-02 2657.473 1.5000E+01 2044.527 2.9300E+01 2653.467 4.6100E-13 2638.716 1.3800E-02 2657.472 2.1700E-01
52796.418 6.6584E-01 2715.496 6.0600E+00 2686.608 6.8200E-01 2696.804 7.3500E+00 2058.534 2.0500E+01 2653.351 5.7800E-01 2696.471 7.6900E-01 2696.889 1.4400E+00
Test 3
21692.447 0.0000E+00 1643.472 0.0000E+00 1546.032 0.0000E+00 1552.457 0.0000E+00 1222.792 1.2000E-01 1552.457 0.0000E+00 1543.807 0.0000E+00 1552.457 0.0000E+00
3 1747.897 4.3950E-04 1757.917 4.0000E-03 1637.029 5.3100E-02 1643.472 6.9200E-13 1242.797 2.8900E+01 1697.317 6.9200E-13 1628.171 2.9000E-03 1643.472 3.8400E-02
4 1744.546 1.5798E-05 1755.063 1.6000E-02 1654.125 6.2200E-02 1697.017 1.7200E-02 1649.755 2.3900E+01 1693.695 2.8900E-04 1679.165 9.1000E-03 1697.317 1.3000E+00
5 1792.750 5.3320E-05 1740.872 5.4000E-02 1712.485 5.3300E-01 1803.288 4.3300E+00 1685.315 2.0200E+01 1692.815 2.3000E-03 1711.761 6.0000E-02 1723.199 1.0500E+00
Test 4
23838.491 1.5163E-14 3727.414 1.3800E-12 3641.278 1.3800E-12 3651.867 1.3800E-12 3067.814 4.2000E-02 3651.613 1.3800E-12 3830.909 9.2300E-13 3651.867 1.2000E-02
3 3895.112 1.4284E-05 3782.397 1.3000E-03 3716.225 2.3700E-02 3727.414 2.3100E-12 3208.087 5.7600E-01 3727.232 2.3100E-12 3895.142 1.3800E-12 3727.371 5.1300E+00
43925.344 4.6037E-04 3911.754 4.1900E-02 3769.229 6.8300E-02 3782.398 1.5100E-02 3258.632 1.5700E+00 3778.105 9.1000E-03 3774.971 7.4200E-02 3782.398 1.2200E+00
5 3985.776 9.2184E-04 3987.697 8.3900E-02 3800.264 1.1600E-01 3813.742 1.3300E-02 3295.114 1.9100E+01 3806.778 6.0900E-04 3833.697 3.5100E-02 3805.724 5.0000E+00
Test 5
2 2000.114 1.0141E-14 2025.432 9.2300E-13 1942.973 8.0000E-03 1949.294 9.2300E-13 1974.575 3.6900E-02 1949.234 9.2300E-13 1947.463 4.6100E-13 1949.293 9.2300E-13
3 2045.033 8.7899E-06 2070.919 8.0000E-04 2118.918 4.2000E-02 2025.433 0.0000E+00 2002.571 5.6400E-01 2118.918 0.0000E+00 2012.874 8.4900E-04 2025.433 6.8500E-02
4 2134.909 1.3734E-04 2196.931 1.2500E-02 2061.398 7.4600E-02 2070.527 5.0000E-03 2059.734 1.8400E+01 2125.228 4.6100E-13 2067.908 7.4000E-03 2070.926 1.5100E+00
52150.180 4.8345E-02 2111.259 4.4000E+00 2086.816 8.0000E-02 2096.967 6.1300E+00 2098.914 1.4700E+01 2150.180 3.7000E-03 2050.263 1.6900E-02 2096.935 6.7300E-01
Test 6
2 2690.368 0.0000E+00 2724.423 0.0000E+00 2527.085 0.0000E+00 2534.441 3.6400E-02 2236.247 9.4000E-02 2534.441 0.0000E+00 2534.492 0.0000E+00 2534.441 2.8500E-02
32891.223 0.0000E+00 2824.839 0.0000E+00 2714.836 4.0300E-04 2724.423 1.7000E-03 2448.361 2.4400E+01 2723.995 0.0000E+00 2696.862 0.0000E+00 2723.047 1.7900E-02
42943.991 1.0987E-04 2876.396 1.0000E-02 2811.446 4.6000E-03 2824.843 4.5600E-02 2625.646 2.9000E+01 2823.443 9.9800E-04 2824.348 1.3800E-12 2824.208 2.3800E-02
52975.808 7.6622E-05 2907.482 7.7600E-02 2863.994 1.7500E-01 2876.308 4.4600E-02 2650.884 2.4200E+01 2828.659 8.3000E-03 2875.682 3.2900E-02 2876.308 1.3800E-01
Test 7
21807.663 1.7690E-14 1760.103 1.6100E-12 1623.129 1.6100E-12 1627.909 1.6100E-12 1606.711 8.9100E-02 1807.663 1.6100E-12 1627.294 1.6100E-12 1627.909 5.2600E-02
31871.842 1.0141E-14 1828.864 9.2300E-13 1753.192 3.7900E-04 1760.103 1.9000E-03 1740.822 2.1600E+01 1857.629 9.2300E-13 1760.981 1.3800E-12 1760.103 3.0800E-01
4 1879.158 1.7360E-04 1871.971 1.5800E-02 1813.172 5.3300E-02 1828.792 5.6600E-02 1801.213 2.7600E+01 1897.867 1.1500E-12 1827.486 9.2300E-13 1828.307 1.7400E-01
5 1901.635 4.7691E-05 1907.029 4.8300E-02 1859.064 4.2300E-01 1871.924 5.3400E-02 1846.239 2.3700E+01 1824.237 4.5000E-03 1865.092 2.8700E-02 1871.971 7.3800E-02
Test 8
23288.010 6.8328E-16 3212.516 6.9200E-13 3053.809 6.9200E-13 3063.526 6.9200E-13 2501.315 6.1000E-02 3263.475 6.9200E-13 3063.343 6.9200E-13 3063.475 6.9200E-13
3 3345.355 1.5163E-13 3268.544 1.3800E-12 3200.527 1.3800E-12 3212.517 1.4100E-01 2650.271 9.6000E+00 3350.259 1.3800E-12 3212.577 6.9200E-13 3212.517 8.3000E-02
43383.481 4.7685E-01 3305.795 4.3400E+00 3253.889 2.0200E-02 3268.741 6.3200E-02 2674.957 1.6600E+01 3367.113 1.2700E-02 3268.748 1.1500E-12 3268.497 1.4700E-01
53403.903 4.1422E-01 3325.748 3.7700E+00 3291.844 1.9600E-01 3307.331 3.7800E+00 2690.816 1.1400E+01 3400.386 2.5000E-03 3285.643 9.4000E-03 3307.331 2.6900E-01
Test 9
2 1650.510 0.0000E+00 1671.403 0.0000E+00 1551.144 0.0000E+00 1555.731 0.0000E+00 1530.216 5.0400E-02 1555.691 0.0000E+00 1552.766 0.0000E+00 1555.691 0.0000E+00
31753.172 2.5381E-13 1712.918 2.3100E-12 1655.822 6.0300E-02 1671.403 1.1900E-04 1610.256 2.8700E+01 1669.475 2.3100E-12 1657.173 1.3800E-12 1671.403 4.1300E-02
41779.966 3.8456E-03 1739.097 3.5000E-02 1700.282 2.5600E-01 1713.252 6.7100E-05 1706.167 3.1900E+01 1669.737 1.3800E-12 1701.348 3.9600E-06 1713.127 2.4500E-01
51797.314 4.1972E-03 1756.047 3.8200E-02 1727.451 6.4500E-01 1736.528 4.1400E-01 1733.115 1.9400E+01 1706.363 4.5400E-02 1710.863 3.9700E-01 1759.022 2.7300E-01
Test 10
2 1679.010 7.6033E-14 1640.459 6.9200E-13 1679.369 9.4000E-03 1542.907 6.9200E-13 1454.615 2.0700E-01 1542.899 6.9200E-13 1540.462 1.1500E-12 1642.899 1.1300E-02
31742.790 7.6033E-14 1702.775 6.9200E-13 1740.314 6.8000E-03 1640.459 2.6000E-03 1457.515 2.7000E+01 1639.327 6.9200E-13 1636.417 1.9600E+01 1640.459 2.0300E-02
41787.985 1.1553E-05 1731.011 1.1700E-02 1787.985 1.8600E-02 1702.775 9.8600E+00 1587.387 2.9600E+01 1702.246 9.2300E-13 1677.415 3.4000E-03 1742.775 9.5100E-01
5 1787.292 1.2979E-02 1746.255 3.3400E+01 1789.766 1.5900E+01 1731.011 2.0700E+01 1721.601 1.5200E+01 1722.038 2.0100E-01 1712.117 1.6800E-03 1757.986 6.2000E-12
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Conclusions and future works
Image segmentation is the most substantial pivotal phase that should be performed for image analysis and under-
standing. To handle this growing challenge, dierent methods using MTH, including feature-based, threshold-
based, and region-based segmentation, have been implemented. e most common technique used to perform
and analyze image segmentation is threshold-based segmentation. is paper presented an improved variant
of the Heap-based optimizer (HBO) called IHBO. e eectiveness of the proposed IHBO was estimated using
the functions in the CEC’2020 benchmark functions, however, the proposed algorithm superiority on the com-
peting algorithms regarding various statistical metrics. In addition, IHBO was applied to image segmentation
using objective functions such as the Otsu and Kapur methods. e main target of IHBO is to determine the
best thresholds that maximize the Otsu and Kapur methods. e IHBO was implemented on a set of test images
with dierent characteristics, and the results were compared against seven well-known metaheuristic algorithms
including the original HBO algorithm, SSA, MFO, GWO, SCA, HS, and EMO. e experimental results revealed
that the IHBO algorithm outperformed all counterparts in terms of FSIM, SSIM, and PSNR. It should be noted
that the IHBO results using the Otsu method provided better class variance in most metrics. However, when
applying the Kapur method, the IHBO produced SSIM, FSIM, PSNR, and tness values were better than those
of all counterparts. e IHBO produced promising results because it preserved an eective balance between
exploration and exploitation, and had the ability to avoid being trapped in local optima.
For future work, there are many research directions in this eld, such as studying the performance of the
IHBO algorithm on dierent datasets, and other real-world complex problems. In addition, future work can study
the hybridization of the original HBO with other metaheuristic or machine learning algorithms to automate the
search process for the optimal number of thresholds in a specic image.
Table 10. Mean and STD values of PSNR results obtained by Otsu’s method. Signicant values are in bold.
Test Image n
IHBO HBO SSA MFO GWO SCA HS EMO
Mean STD Mean STD Mean STD Mea n STD Mean STD Mean STD Mean STD Mea n STD
Test 1
219.0524 2.2436E-03 17.4279 2.0400E-02 15.4016 1.8500E-12 15.4016 2.4800E-13 15.7527 6.5800E-02 15.2431 2.5800E-13 15.0441 1.1400E-12 14.2494 3.1400E-02
320.5241 9.2384E-04 18.7742 8.4000E-03 17.4279 1.4500E-02 18.7742 1.2700E-02 15.7523 1.9980E-01 17.2103 2.3500E-02 17.2318 8.7000E-03 15.9879 4.1500E-02
4 18.1574 4.2673E-03 19.4801 3.8800E-02 18.7742 3.9200E-02 18.7742 4.1500E-02 16.7526 6.0230E-01 20.0341 7.2100E-15 18.1213 1.5200E-02 17.9861 1.1150E-01
5 19.2599 4.2233E-04 20.6629 3.8400E-02 20.6851 2.0500E-02 19.4028 3.0200E-02 18.7529 8.8760E-01 19.2071 1.5260E-01 20.2814 2.1500E-02 20.2616 1.1380E-01
Test 2
2 18.0705 7.9296E-04 18.3592 7.2100E-02 16.2997 7.2100E-15 16.2997 7.2100E-15 16.4912 3.1600E-02 15.4016 7.2100E-15 15.4016 7.2100E-15 15.4085 1.6200E-02
321.6706 1.1878E-04 20.7376 1.0800E-02 18.3592 1.6500E-02 18.3592 1.0800E-14 17.2917 6.4410E-01 17.4279 1.0800E-14 17.4279 1.0800E-14 17.4333 4.4300E-02
422.3900 4.2893E-05 22.3104 3.9000E-03 20.7376 5.8000E-03 20.7376 2.3680E-01 18.2907 6.2630E-01 18.7742 0.0000E+00 18.7752 2.5000E-03 18.7771 4.2300E-02
523.3803 3.1103E-03 23.2163 2.8280E-01 23.3104 3.2150E-01 22.2854 2.0580E-01 19.2977 5.3970E-01 19.5116 2.6470E-01 19.7961 3.6460E-01 19.5624 3.1610E-01
Test 3
218.3590 1.3858E-03 17.7084 1.2600E-02 15.4217 1.2600E-14 15.4217 1.2600E-14 15.4087 4.9600E-02 17.7117 1.2000E-13 15.4217 1.2600E-14 15.3238 1.1100E-02
3 20.0802 3.9593E-01 20.1976 3.6000E-03 17.7084 6.3400E-02 17.7084 7.2100E-15 17.9785 7.4620E-01 20.2084 7.2100E-15 17.7089 2.6000E-03 17.7589 8.2900E-02
4 21.4252 3.9703E-03 21.4279 3.6100E-02 20.1976 5.0100E-02 21.4758 4.1400E-02 19.7216 1.4092E+00 20.2001 1.4800E-02 20.2119 3.2000E-02 20.2669 9.7700E-02
523.4331 8.3036E-02 23.2646 7.5500E-02 21.5676 1.9760E-01 21.7046 2.4100E-01 20.5034 3.2110E-01 21.5699 5.0000E-03 21.6749 1.0130E-01 21.6435 1.7910E-01
Test 4
2 20.0953 2.5626E-02 20.2114 2.3300E-02 20.2474 0.0000E+00 17.2474 2.6300E-02 18.8571 1.8300E-02 18.5476 3.1400E-02 18.9077 1.7300E-02 17.8087 6.7000E-03
321.6474 3.9813E-03 21.5328 3.6200E-02 21.2113 9.1000E-03 20.2114 1.0800E-14 20.7496 1.1220E-01 20.3452 1.0800E-14 20.3442 1.0800E-14 20.3376 2.3900E-02
4 23.0580 1.4078E-02 23.4783 1.2800E-02 21.5328 2.1700E-02 21.5328 8.0000E-03 21.9144 7.3410E-01 21.1737 5.0000E-04 21.1807 1.3100E-02 21.1741 2.8600E-02
5 23.3517 2.9035E-02 23.7773 2.6400E-02 23.7827 1.7400E-02 23.2827 1.6200E-02 22.4174 7.3460E-01 23.6934 3.6000E-03 23.6814 1.3100E-02 23.6585 6.0200E-02
Test 5
219.5379 1.3858E-03 18.7868 1.2600E-02 15.0294 1.2400E-02 15.0295 1.2600E-14 18.0546 6.5100E-02 15.0347 1.2600E-14 15.1295 1.2600E-14 15.2374 2.6000E-02
321.6678 2.8375E-03 20.7351 2.5800E-02 18.7868 6.1700E-02 18.7868 1.4400E-14 20.8075 1.3422E+00 18.7868 1.4400E-14 18.8143 2.7400E-02 18.8376 8.7900E-02
423.5257 3.7833E-03 23.1664 3.4400E-02 20.7351 5.5700E-02 20.8801 1.9300E-02 23.1682 1.0403E+00 20.0359 1.0800E-14 21.3378 1.0000E-02 20.7131 1.1300E-01
5 24.8769 3.9725E-02 24.5853 3.6120E-01 24.6603 3.8100E-02 24.6603 5.5720E-01 20.2984 9.2180E-01 22.1518 2.2900E-02 24.9666 3.0500E-02 22.0424 1.6140E-01
Test 6
217.1204 1.9577E-03 16.5754 1.7800E-02 13.9437 1.0800E-14 13.9437 1.4300E-04 15.4016 2.3400E-02 16.2997 1.0800E-14 16.2037 1.0800E-14 16.2878 1.1000E-03
319.6319 1.5837E-04 18.8728 1.4400E-02 16.5753 4.6000E-03 16.5753 5.4200E-03 17.2654 3.4510E-01 18.3502 1.4400E-14 18.0481 1.4400E-14 18.1785 2.4700E-02
4 20.6370 1.0976E-02 20.7069 9.9800E-02 18.8728 9.2000E-03 18.8728 2.4900E-02 19.5449 9.5150E-01 20.7371 2.6000E-03 20.7076 1.0800E-14 20.3278 4.0600E-02
523.3362 2.2986E-03 22.2612 2.0900E-02 20.5396 1.7000E-02 20.5396 1.2800E-02 20.7012 1.0633E+00 22.3144 1.5000E-03 22.2041 1.1100E-02 22.1665 4.6600E-02
Test 7
218.8937 5.9610E-01 18.1975 5.4200E-02 15.9994 5.4100E-15 15.9994 5.4100E-15 16.3202 2.4500E-02 15.7494 5.4100E-15 15.9351 5.4100E-15 16.0148 3.6000E-03
3 20.6004 7.9296E-04 20.6734 7.2100E-02 20.1974 9.4000E-03 18.1974 2.1800E-02 17.0135 6.8190E-01 18.1774 7.2100E-15 18.1684 7.2100E-15 18.1068 3.4000E-02
4 22.2613 2.1006E-04 22.1927 1.9100E-02 22.6734 1.1100E-02 20.6829 2.9500E-02 19.6592 1.1060E+00 20.6734 1.8000E-14 20.6734 1.8000E-14 20.6658 5.2000E-02
523.9678 3.7174E-03 23.7537 3.3800E-02 22.2254 2.6400E-02 22.2736 3.1300E-02 20.3591 9.3050E-01 22.3205 7.7000E-03 22.2565 5.2000E-03 22.1581 7.9100E-02
Test 8
220.5438 4.0253E-03 20.3472 3.6600E-02 20.3472 2.9800E-02 17.8874 0.0000E+00 14.9097 0.0000E+00 14.6065 0.0000E+00 14.6291 7.6000E-03 14.6071 0.0000E+00
322.3472 3.9593E-03 22.1504 3.6000E-02 22.3472 3.6000E-15 20.3472 1.3000E-01 17.0926 1.8104E+00 19.1571 3.6000E-15 19.1531 3.6000E-15 19.1178 1.6760E-01
423.6265 8.3586E+00 23.4415 7.6000E-02 23.5732 7.2400E-02 22.1732 1.4060E-01 17.8655 1.9656E+00 21.1671 5.0800E-02 21.1802 7.2100E-15 20.5465 2.5600E-01
5 24.2248 1.9753E-02 24.7531 1.7960E-01 24.6945 2.5100E-02 23.6945 3.8600E-01 20.2017 1.8785E+00 21.4112 2.2800E-02 22.2684 3.4700E-02 22.1879 1.7990E-01
Test 9
2 16.5384 1.4847E-03 16.9578 1.3500E-02 13.6937 1.3200E-02 16.9978 1.3500E-02 14.9607 2.6500E-02 13.9573 1.4200E-02 15.9581 1.4200E-02 13.9576 1.6400E-02
3 18.6995 1.1922E-02 18.9346 1.0840E-01 16.9578 1.6960E-01 19.0346 8.0000E-03 15.3148 1.0297E+00 16.5753 1.0800E-14 18.5753 1.0800E-14 16.7093 1.8240E-01
4 19.5683 2.3866E-03 19.7293 2.1700E-02 19.1112 1.2830E-01 19.1112 2.1500E-02 16.1401 1.0999E+00 18.8728 7.2100E-15 19.7293 6.4000E-03 18.9318 1.1340E-01
520.8438 6.3019E-02 20.7638 5.7300E-02 19.7293 1.6300E-01 20.1234 9.1200E-02 18.3409 8.7270E-01 20.5658 3.0300E-02 20.8438 1.3940E-01 20.5454 1.2450E-01
Test 10
2 19.9428 9.8323E-03 19.9572 8.9400E-02 19.6091 2.6800E-02 14.6091 0.0000E+00 19.1375 7.7500E-02 19.9572 0.0000E+00 13.6937 0.0000E+00 13.7163 4.5900E-02
3 21.1545 5.0591E-03 21.1803 4.6000E-02 21.2571 2.0000E-04 19.1571 4.0600E-02 21.3725 1.1890E+00 21.9578 0.0000E+00 16.8803 4.5820E-01 16.9389 7.6600E-02
422.9984 7.0168E-03 22.3993 6.3800E-02 21.3803 2.9700E-02 21.1803 5.1600E-01 22.8746 1.1509E+00 22.1112 1.0800E-14 19.1148 2.0100E-02 19.1497 1.1920E-01
523.6993 1.0206E-02 23.1291 9.2800E-02 23.6993 1.4700E-01 22.3993 7.5300E-02 23.3993 9.4460E-01 23.6993 2.2000E-02 19.7688 5.8100E-02 19.7809 1.3630E-01
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Test
Image n
IHBO HBO SSA MFO GWO SCA HS EMO
Mean STD Mean STD Mean STD Mean STD Mean STD Mean STD Mean STD Mean STD
Test 1
20.7771 1.0269E-
05 0.7536 1.0000E-
04 0.6984 3.3800E-
16 0.6984 3.4000E-
16 0.5984 1.2000E-
02 0.7611 3.3800E-
16 0.7011 3.0500E-
16 0.7111 4.2000E-
03
3 0.7936 1.6225E-
03 0.8036 1.5800E-
02 0.7536 2.0000E-
04 0.75647 2.3800E-
05 0.5531 5.9000E-
03 0.8157 2.2500E-
16 0.8107 2.3500E-
16 0.8044 2.1000E-
02
4 0.8230 9.2418E-
05 0.8333 9.0000E-
04 0.8036 6.0000E-
04 0.8036 7.6500E-
04 0.6301 1.4800E-
02 0.8316 5.6300E-
16 0.8275 3.0000E-
04 0.8062 2.4000E-
03
5 0.8857 5.1344E-
05 0.8589 5.0000E-
04 0.8322 2.0000E-
04 0.8329 4.0000E-
04 0.6541 1.8400E-
02 0.8459 2.1000E-
03 0.8862 2.0000E-
04 0.8436 1.7000E-
03
Test 2
20.8054 2.1359E-
03 0.7810 2.0800E-
02 0.7244 3.3800E-
16 0.7044 3.3800E-
16 0.6996 1.8000E-
03 0.6974 3.3800E-
16 0.6923 3.4300E-
16 0.5908 1.3000E-
03
30.8490 2.3105E-
03 0.8233 2.2500E-
02 0.7811 7.0000E-
04 0.7817 2.2500E-
16 0.7494 1.9000E-
02 0.7906 2.2500E-
16 0.7436 2.2500E-
16 0.7244 1.7000E-
03
40.8843 2.0537E-
04 0.8575 2.0000E-
04 0.8633 4.0000E-
04 0.8233 1.0900E-
02 0.7863 2.2500E-
02 0.8036 3.3800E-
16 0.8036 2.0000E-
04 0.8037 1.0000E-
03
5 0.8876 4.9290E-
03 0.8881 4.8000E-
03 0.8575 2.7000E-
03 0.8584 6.8000E-
03 0.8035 1.6200E-
02 0.8342 4.0000E-
04 0.8336 1.6000E-
03 0.8802 1.5000E-
03
Test 3
2 0.8856 1.5608E-
02 0.8882 1.5200E-
02 0.8452 1.0300E-
16 0.8452 1.1900E-
16 0.8448 1.6000E-
03 0.8882 1.0700E-
16 0.8482 1.5100E-
16 0.8458 3.6000E-
03
3 0.9223 4.1075E-
04 0.9235 4.0000E-
04 0.8879 9.0000E-
04 0.9235 7.8900E-
16 0.8834 1.2600E-
02 0.9279 7.8900E-
16 0.8878 3.0000E-
04 0.8876 1.0000E-
03
40.9499 5.1344E-
04 0.9406 5.0000E-
04 0.9235 7.0000E-
04 0.9426 6.0000E-
04 0.9055 2.1800E-
02 0.9290 2.0000E-
04 0.9237 4.0000E-
04 0.9243 1.6000E-
03
5 0.9547 1.9511E-
03 0.9549 1.9000E-
03 0.9398 1.0000E-
03 0.9395 2.7000E-
03 0.9309 1.7800E-
02 0.9398 1.3000E-
03 0.9404 2.1000E-
02 0.9609 2.1000E-
03
Test 4
2 0.8101 6.6849E-
02 0.8147 6.5100E-
02 0.7711 4.5100E-
16 0.7711 2.0100E-
16 0.7975 4.0000E-
04 0.7902 3.1500E-
16 0.7932 4.5100E-
16 0.7951 2.1000E-
03
3 0.8371 2.1872E-
02 0.8476 2.1300E-
02 0.8476 2.0000E-
04 0.8147 5.6300E-
16 0.8025 7.0000E-
04 0.8735 5.6300E-
16 0.8735 5.6300E-
16 0.8734 4.0000E-
04
4 0.9130 3.0806E-
05 0.8854 3.0000E-
04 0.8476 4.0000E-
04 0.8476 2.7300E-
04 0.8386 1.5900E-
02 0.9122 9.0000E-
05 0.9118 3.0000E-
04 0.9117 2.0000E-
03
5 0.9225 2.0537E-
06 0.8946 2.0000E-
04 0.8863 1.0000E-
04 0.8863 5.1800E-
05 0.8851 1.4700E-
02 0.9382 6.4700E-
05 0.9362 1.1000E-
03 0.9377 2.4000E-
03
Test 5
20.8500 2.5672E-
05 0.8500 2.5000E-
03 0.8109 3.0000E-
04 0.8129 2.2500E-
16 0.8109 1.2000E-
03 0.8171 2.2500E-
16 0.8119 2.2500E-
16 0.8102 7.0000E-
04
30.9120 9.2418E-
06 0.8844 9.0000E-
04 0.8589 2.1000E-
03 0.8586 1.1300E-
16 0.8477 1.5700E-
02 0.8501 1.1300E-
16 0.8505 1.0000E-
03 0.8514 1.9000E-
03
40.9465 1.0269E-
04 0.9179 1.0000E-
03 0.8843 1.5000E-
03 0.8847 5.4300E-
04 0.8735 1.6800E-
02 0.8853 3.3800E-
16 0.8844 4.0000E-
04 0.8856 2.1000E-
03
50.9700 4.3129E-
04 0.9407 4.2000E-
03 0.9178 1.9000E-
03 0.9178 7.6000E-
03 0.8785 1.4700E-
02 0.9171 6.0000E-
04 0.9191 2.6000E-
03 0.9155 2.0000E-
03
Test 6
20.8595 3.9021E-
04 0.8335 3.8000E-
03 0.7661 3.3800E-
16 0.7662 5.0200E-
05 0.7267 5.0000E-
04 0.8335 3.3700E-
16 0.7245 8.0000E-
04 0.7204 6.4400E-
07
3 0.8748 3.9021E-
04 0.8858 3.8000E-
03 0.8335 8.1000E-
05 0.8335 2.0900E-
04 0.7796 9.9000E-
03 0.8811 3.3800E-
16 0.7818 3.3800E-
16 0.7828 5.0000E-
04
4 0.9029 3.2860E-
04 0.9142 3.2000E-
03 0.8858 1.0000E-
04 0.8858 2.9100E-
04 0.8097 1.7300E-
02 0.9233 6.2200E-
05 0.8203 7.8900E-
16 0.8205 1.7000E-
03
5 0.9251 1.3349E-
04 0.9367 1.3000E-
03 0.9362 5.0000E-
04 0.9162 1.3500E-
04 0.8255 2.0400E-
02 0.8575 2.1900E-
05 0.8576 2.2000E-
03 0.8574 1.6000E-
03
Test 7
20.8355 4.6312E-
03 0.6163 4.5100E-
02 0.8355 4.5100E-
16 0.7572 4.5000E-
16 0.7579 2.5000E-
03 0.7502 4.0100E-
16 0.7512 4.1100E-
16 0.7522 1.1000E-
03
30.9108 2.5672E-
04 0.8832 2.5000E-
03 0.8288 1.4100E-
05 0.8288 1.3600E-
04 0.8274 2.2500E-
02 0.8278 2.2500E-
16 0.8284 2.2500E-
16 0.8241 5.0000E-
04
40.9401 3.1833E-
04 0.9117 3.1000E-
03 0.8832 4.0000E-
04 0.8837 6.3300E-
04 0.8624 2.4400E-
02 0.9130 3.3800E-
16 0.8802 3.3800E-
16 0.8825 7.0000E-
04
50.9627 1.4376E-
04 0.9336 1.4000E-
03 0.9109 9.0000E-
04 0.9129 1.3000E-
03 0.8893 2.2100E-
02 0.9209 1.0000E-
04 0.9116 7.0000E-
04 0.9429 1.3000E-
03
Test 8
20.9009 2.1154E-
17 0.8736 2.0600E-
16 0.7952 2.5100E-
16 0.7952 2.3200E-
16 0.7578 1.5000E-
03 0.7573 2.2500E-
16 0.7473 2.2500E-
16 0.7674 2.2000E-
03
30.9386 4.6312E-
16 0.9102 4.5100E-
16 0.8735 4.5100E-
16 0.8735 1.3000E-
03 0.8667 3.0100E-
02 0.8352 4.5100E-
16 0.8356 4.5100E-
16 0.8354 1.5000E-
03
40.9469 9.8580E-
04 0.9376 9.6000E-
03 0.9413 8.0000E-
04 0.9113 9.0000E-
04 0.8428 2.8200E-
02 0.8986 5.0000E-
04 0.8975 2.7000E-
02 0.8987 1.8000E-
03
5 0.9523 4.4155E-
04 0.9526 4.3000E-
03 0.9372 1.3000E-
03 0.9372 5.4000E-
03 0.8839 2.6500E-
02 0.9581 1.0000E-
04 0.9286 1.1000E-
03 0.9265 1.5400E-
03
Continued
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Table 11. Mean and STD values of FSIM results obtained by Otsu’s method. Signicant values are in bold.
Test
Image n
IHBO HBO SSA MFO GWO SCA HS EMO
Mean STD Mean STD Mean STD Mean STD Mean STD Mean STD Mean STD Mean STD
Test 9
20.8038 4.3129E-
04 0.8029 4.2000E-
03 0.7291 4.2200E-
05 0.7291 4.3200E-
05 0.7664 8.0000E-
04 0.7662 4.5500E-
05 0.7762 4.5000E-
05 0.7662 1.0000E-
04
30.8740 2.5672E-
04 0.8476 2.5000E-
03 0.8029 1.4000E-
03 0.8029 1.5700E-
04 0.8256 2.6000E-
02 0.8335 2.2500E-
16 0.8335 2.2500E-
16 0.8351 2.3000E-
03
40.9129 9.3445E-
04 0.8853 9.1000E-
03 0.8498 1.9000E-
03 0.8498 2.0000E-
04 0.8754 2.7000E-
02 0.8858 2.0100E-
02 0.8858 5.1100E-
06 0.8872 2.2000E-
03
5 0.8886 1.7457E-
04 0.8998 1.7000E-
03 0.8782 3.4000E-
03 0.8789 2.4000E-
03 0.8914 1.5800E-
02 0.9154 8.0000E-
04 0.9148 5.0000E-
04 0.9116 2.4000E-
03
Test 10
2 0.8247 5.7809E-
16 0.8351 5.6300E-
16 0.7573 1.4000E-
03 0.7573 5.6300E-
16 0.7261 1.0000E-
03 0.7291 5.6000E-
16 0.7291 5.3300E-
16 0.7275 6.0000E-
04
30.9214 5.7809E-
16 0.8935 5.6300E-
16 0.8357 1.3000E-
03 0.8357 7.2500E-
04 0.7896 2.4300E-
02 0.8589 5.6300E-
16 0.8015 1.2200E-
02 0.8052 3.3000E-
03
4 0.9271 1.2322E-
04 0.9281 1.2000E-
03 0.9281 2.1200E-
03 0.8935 1.1000E-
02 0.8211 2.2700E-
02 0.8491 3.0500E-
16 0.8494 1.1000E-
03 0.8511 3.1000E-
03
50.9679 4.5182E-
04 0.9483 4.4000E-
03 0.9581 3.7000E-
03 0.9387 4.3200E-
03 0.8734 2.2400E-
02 0.8782 7.5400E-
05 0.8702 3.4000E-
03 0.8776 6.1000E-
03
Table 12. Mean and STD values of SSIM results obtained by Otsu’s method. Signicant values are in bold.
Test Image n
IHBO HBO SSA MFO GWO SCA HS EMO
Mean Std Mean Std Mean Std Mean Std Me an Std Mean Std Mean Std Mean Std
Test 1
2 0.6499 3.2010E-04 0.6279 3.1000E-03 0.6459 4.0100E-16 0.7679 4.5100E-16 0.2855 5.0000E-04 0.7679 3.0400E-16 0.7679 4.3100E-16 0.7879 4.4000E-03
3 0.7011 1.2804E-04 0.6774 1.2400E-02 0.7218 8.0000E-04 0.6459 8.6200E-05 0.2858 9.5000E-03 0.8166 7.8900E-16 0.8006 7.8900E-16 0.8066 2.3000E-03
4 0.6318 4.1304E-05 0.7028 4.0000E-04 0.7647 5.0000E-04 0.7262 3.9200E-04 0.6585 7.4000E-03 0.8312 4.5100E-16 0.8212 2.0000E-04 0.8103 1.7000E-03
5 0.7759 1.9619E-04 0.7497 1.9000E-03 0.8089 5.2000E-03 0.7647 1.5000E-03 0.6912 1.2600E-02 0.8575 2.3000E-03 0.8590 1.3000E-03 0.8576 3.1000E-03
Test 2
20.7683 1.3424E-04 0.6554 1.3000E-03 0.7587 1.1300E-16 0.7587 1.1500E-16 0.6474 2.9000E-03 0.6559 1.1300E-16 0.6469 1.1300E-16 0.6473 1.4000E-03
30.8216 5.2662E-04 0.7938 5.1000E-03 0.7928 1.2000E-03 0.7693 4.5100E-16 0.7095 2.5100E-02 0.7218 4.5100E-16 0.7218 4.5100E-16 0.7224 2.7000E-03
40.9006 1.0326E-05 0.8701 1.0000E-04 0.8612 1.0000E-04 0.8614 6.3500E-03 0.7512 2.0300E-02 0.7647 4.5100E-16 0.7648 1.0000E-04 0.7647 1.6000E-03
50.9131 2.0032E-03 0.8822 1.9400E-02 0.8840 1.0800E-02 0.8837 9.0000E-03 0.7698 1.7000E-02 0.7897 8.8000E-03 0.7993 1.2100E-02 0.7813 1.0500E-02
Test 3
2 0.7791 8.3640E-05 0.7899 8.1000E-03 0.7886 1.0300E-16 0.7886 1.1300E-16 0.7585 3.5000E-03 0.7886 1.1300E-16 0.7876 1.0100E-16 0.7887 5.0000E-04
30.8713 2.0652E-04 0.8617 2.0000E-03 0.8574 1.5000E-03 0.8562 7.8900E-16 0.7312 2.0700E-02 0.8544 7.8900E-16 0.8522 3.0100E-05 0.8533 2.0000E-03
40.9147 8.2607E-05 0.9138 8.0000E-04 0.9123 1.1000E-03 0.9109 9.0000E-04 0.8922 3.3500E-02 0.9101 3.0000E-04 0.9026 7.0000E-04 0.8288 2.1000E-03
50.9701 2.1684E-04 0.9691 2.1000E-03 0.9327 2.5000E-03 0.9346 3.4000E-03 0.9038 2.6400E-02 0.9328 6.8400E-05 0.9343 0.0000E+00 0.9357 2.7000E-03
Test 4
20.7953 2.0652E-05 0.7684 2.0000E-04 0.7679 2.2500E-16 0.7679 2.2500E-16 0.6065 1.1000E-03 0.7953 2.2500E-16 0.6615 2.2500E-16 0.7582 6.0000E-04
30.8772 8.2607E-05 0.8475 8.0000E-04 0.8066 1.0000E-03 0.8066 5.6300E-16 0.7169 3.6000E-03 0.8466 5.6300E-16 0.8465 5.6300E-16 0.8275 1.1000E-03
40.9189 1.9619E-04 0.8878 1.9000E-03 0.8312 3.2000E-03 0.8312 6.0000E-04 0.7214 1.7000E-02 0.8886 3.0000E-04 0.8326 2.8000E-03 0.8875 2.5000E-03
50.9548 3.0978E-05 0.9225 3.0000E-04 0.8584 4.0000E-04 0.8582 2.2900E-04 0.7761 1.4300E-02 0.9225 6.7300E-05 0.9212 2.1000E-03 0.9206 1.7000E-03
Test 5
2 0.8452 2.0652E-05 0.8166 2.0000E-04 0.8492 4.0000E-04 0.8292 2.2500E-16 0.7014 2.5000E-03 0.8382 2.2500E-16 0.8382 2.2500E-16 0.8396 1.0000E-03
30.9070 1.6521E-04 0.9053 1.6000E-03 0.8894 3.7000E-03 0.8894 2.2500E-16 0.7869 2.0800E-02 0.8794 2.2500E-16 0.8513 1.7000E-03 0.8426 3.2000E-03
4 0.9072 8.2607E-05 0.9075 8.0000E-04 0.9110 1.2000E-03 0.9112 5.4500E-04 0.8062 1.4900E-02 0.9131 4.5100E-16 0.9105 2.0000E-04 0.9099 2.1000E-03
5 0.9162 5.1630E-04 0.9459 5.0000E-03 0.9457 7.0000E-04 0.9457 8.0000E-03 0.8176 1.3600E-02 0.9435 4.0000E-04 0.9405 5.0000E-04 0.9441 2.6000E-03
Test 6
2 0.5776 2.6847E-04 0.5836 2.6000E-03 0.6137 3.3800E-16 0.6137 2.8100E-04 0.5596 1.8000E-03 0.7589 3.3800E-16 0.7577 3.3800E-16 0.7546 5.0000E-04
3 0.6836 3.6141E-03 0.6907 3.5000E-03 0.7992 2.5600E-05 0.7572 5.0000E-04 0.6077 7.7000E-03 0.7985 0.0000E+00 0.7979 1.0200E-02 0.7981 1.3000E-03
4 0.7549 1.0326E-04 0.7628 1.0000E-04 0.8458 7.3100E-05 0.8478 4.0100E-04 0.6439 2.5500E-02 0.8312 1.8100E-05 0.8412 0.0000E+00 0.8611 1.9000E-03
5 0.8201 3.0978E-04 0.8286 3.0000E-04 0.8961 5.0000E-04 0.8720 1.5800E-04 0.7691 2.5400E-02 0.8856 8.4300E-05 0.8841 1.2000E-03 0.8841 1.0700E-02
Test 7
2 0.8203 9.2933E-04 0.8288 9.0000E-04 0.7315 0.0000E+00 0.7316 0.0000E+00 0.7149 1.2000E-03 0.7310 2.0000E-04 0.7313 2.2000E-03 0.7309 3.0000E-04
30.8359 1.0326E-04 0.8076 1.0000E-04 0.8006 1.0000E-04 0.8016 4.2800E-04 0.6385 2.3100E-02 0.8076 2.2500E-16 0.8056 2.2500E-16 0.8066 7.0000E-04
40.8879 5.1630E-03 0.8879 5.0000E-03 0.8853 9.0000E-04 0.8876 1.6000E-03 0.7796 3.6900E-02 0.8849 5.6300E-16 0.8835 5.6300E-16 0.8759 1.8000E-03
50.9366 1.4456E-03 0.9146 1.4000E-03 0.9281 1.4000E-03 0.9212 1.6400E-03 0.8942 2.6400E-02 0.9108 3.6100E-05 0.9174 9.0000E-04 0.9195 3.1000E-03
Test 8
2 0.7661 5.2662E-04 0.7015 5.1000E-03 0.7684 4.5100E-16 0.7584 4.3100E-16 0.6049 1.2000E-02 0.7012 4.5100E-16 0.7035 4.5100E-16 0.7014 2.0000E-04
30.8839 1.2391E-04 0.8830 1.2000E-03 0.8465 2.3500E-16 0.8465 2.7700E-03 0.6443 3.1700E-02 0.8830 2.2500E-16 0.8726 2.6400E-16 0.8727 3.5000E-03
40.9250 1.3424E-04 0.9227 1.3000E-03 0.8861 6.0000E-04 0.8861 2.0000E-03 0.7845 6.1400E-02 0.9026 4.0000E-04 0.9197 1.1300E-16 0.9018 3.3000E-03
5 0.9315 2.6847E-04 0.9419 2.6000E-03 0.9222 3.0000E-04 0.9222 4.7000E-03 0.8944 4.2100E-02 0.9419 2.0000E-04 0.9317 3.0000E-04 0.9290 3.9000E-03
Test 9
2 0.5878 1.0326E-05 0.5679 1.0000E-04 0.5991 1.0000E-04 0.5991 3.0000E-04 0.5146 9.0000E-04 0.5137 1.0000E-04 0.52285 1.1000E-03 0.5128 1.2000E-03
30.7641 2.5815E-04 0.6523 2.5000E-03 0.7641 8.0000E-03 0.7635 1.7300E-04 0.5483 5.7300E-02 0.6571 2.2500E-16 0.6532 2.2500E-16 0.6645 9.4000E-03
40.8882 3.7173E-04 0.8582 3.6000E-03 0.8413 5.1000E-03 0.8419 3.6000E-04 0.6294 4.4200E-02 0.8434 4.5100E-16 0.8438 1.7700E-05 0.8481 4.6000E-03
50.8970 3.0978E-05 0.8667 3.0000E-04 0.8567 5.3000E-03 0.8683 2.8500E-03 0.7585 2.9200E-02 0.8911 5.9600E-05 0.8952 4.4000E-03 0.8907 3.3000E-03
Test 10
20.7165 2.2097E-02 0.6923 2.1400E-02 0.7015 1.6000E-03 0.7005 1.1300E-16 0.5516 4.8000E-03 0.5991 1.1300E-16 0.53991 1.1300E-16 0.6505 2.8000E-03
3 0.8799 1.5592E-03 0.8801 1.5100E-02 0.8641 3.0700E-05 0.8339 2.1700E-03 0.6494 5.9100E-02 0.7341 4.5100E-16 0.7504 1.0300E-02 0.7307 6.0000E-03
4 0.8780 2.0652E-04 0.8896 2.0000E-03 0.9229 9.0000E-04 0.9228 1.8400E-02 0.7995 4.1300E-02 0.8431 5.6300E-16 0.8412 6.0000E-04 0.8323 4.0000E-03
50.9476 2.2717E-04 0.9156 2.2000E-03 0.9412 2.0000E-04 0.9418 1.0000E-03 0.8187 3.1900E-02 0.8567 7.0000E-04 0.8571 1.8000E-03 0.8618 1.1000E-03
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Table 13. Comparison of the p-values obtained through the Wilcoxon signed-rank test between the pairs of
IHBO vs HBO, IHBO vs SSA, IHBO vs MFO, IHBO vs GWO, IHBO vs SCA, IHBO vs HS, and IHBO vs EMO
for tness results using Otsu’s method.
Test Image n HBO SSA MFO GWO SCA HS EMO
Test 1
2 1.532E-07 3.280E-01 3.280E-01 3.280E-01 1.390E-08 3.280E-01 3.280E-01
3 2.303E-06 6.570E-02 4.840E-01 4.230E-01 2.090E-13 5.890E-03 1.430E-15
4 1.510E-03 6.310E-01 1.300E-01 4.380E-01 1.370E-12 1.090E-07 1.510E-06
5 7.098E-10 2.700E-05 2.850E-02 1.860E-03 6.440E-13 2.910E-05 2.130E-05
Test 2
2 6.635E-01 4.250E-01 4.250E-01 4.250E-01 6.020E-07 4.250E-01 4.200E-01
3 2.561E-10 5.910E-03 3.650E-01 5.890E-03 2.260E-13 5.890E-03 1.380E-15
4 6.119E-15 1.390E-03 4.290E-02 1.230E-04 5.400E-13 4.120E-08 3.280E-06
5 1.036E-11 3.200E-03 4.100E-02 1.340E-07 9.140E-13 1.130E-06 4.300E-01
Test 3
2NaN NaN NaN NaN 2.610E-07 NaN 1.110E-16
3 4.408E-04 5.230E-03 3.610E-01 3.670E-04 3.890E-13 3.670E-04 2.010E-03
4 6.954E-14 1.700E-01 7.180E-02 4.580E-05 6.310E-12 1.310E-07 1.280E-01
5 7.836E-15 6.52E-05 2.960E-01 7.280E-09 7.110E-13 1.400E-11 7.610E-05
Test 4
2 2.557E-05 1.580E-01 1.580E-01 1.580E-01 2.320E-04 1.590E-01 2.590E-16
3 6.954E-02 1.220E-02 9.140E-01 3.010E-03 6.310E-11 3.010E-03 1.890E-15
4 6.789E-12 4.250E-01 3.650E-01 1.510E-04 6.160E-10 1.790E-06 3.300E-01
5 4.684E-01 4.610E-01 1.110E-03 4.290E-05 4.250E-09 1.420E-08 3.760E-10
Test 5
2NaN NaN 3.290E-01 NaN 2.460E-07 NaN 1.120E-16
3 9.820E-15 2.180E-01 8.190E-02 7.340E-04 8.910E-13 7.420E-04 7.420E-05
4 3.868E-12 9.880E-06 1.900E-03 3.870E-09 3.510E-10 8.290E-11 1.590E-05
5 7.439E-15 8.890E-01 5.310E-02 7.130E-04 6.750E-13 1.490E-06 7.210E-02
Test 6
2 6.205E-06 1.580E-01 1.580E-01 9.630E-01 5.630E-04 1.580E-01 4.130E-13
3 2.348E-05 3.640E-04 1.580E-03 2.590E-02 2.130E-13 3.590E-04 3.390E-15
4 5.775E-04 1.120E-02 3.820E-06 6.860E-01 5.240E-13 1.280E-07 4.740E-02
5 5.720E-04 4.300E-02 7.510E-01 2.690E-02 5.190E-13 3.120E-07 2.290E-06
Test 7
2 1.034E-06 8.140E-02 8.140E-02 8.140E-02 9.100E-06 8.140E-02 8.140E-02
3 9.331E-13 3.510E-04 1.620E-03 3.330E-01 8.210E-13 3.550E-04 3.260E-15
4 2.318E-04 4.530E-01 1.850E-01 2.360E-01 2.040E-13 7.560E-04 7.280E-04
5 7.035E-14 1.820E-01 6.520E-02 8.480E-03 6.190E-13 1.610E-09 2.120E-07
Test 8
2 6.876E-08 NaN NaN NaN 6.050E-07 NaN NaN
3 2.182E-14 1.100E-02 1.100E-02 1.656E-03 1.920E-13 1.130E-02 1.430E-07
4 2.784E-14 7.600E-02 3.450E-01 1.070E-01 2.450E-13 4.120E-03 1.380E-02
5 7.421E-13 1.460E-01 1.410E-05 4.680E-01 6.530E-13 1.510E-13 4.510E-09
Test 9
2NaN NaN NaN NaN 1.640E-05 NaN NaN
3 5.239E-13 1.770E-04 2.720E-01 9.630E-04 4.610E-13 1.800E-04 4.230E-15
4 6.756E-04 4.510E-07 4.710E-02 5.650E-07 6.130E-13 2.120E-09 5.080E-14
5 5.279E-04 3.890E-01 3.560E-01 6.300E-01 4.790E-13 3.900E-01 4.040E-04
Test 10
2 2.215E-05 8.120E-02 6.580E-01 8.200E-02 2.010E-05 8.060E-02 3.900E-16
3 2.425E-16 3.290E-01 1.050E+00 5.800E-02 2.200E-14 3.290E-01 1.120E-14
4 1.929E-03 4.120E-02 8.570E-05 5.990E-03 1.750E-12 6.360E-07 6.790E-14
5 7.572E-05 4.750E-03 1.890E-01 6.670E-08 6.870E-13 8.130E-11 1.390E-05
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Table 14. Implementation results of IHBO-Kapur over the set of test images.
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Table 15. Implementation results of IHBO-Kapur over the set of test images.
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Table 16. e best thresholds values obtained by Kapur’s method.
Test Image n IHBO HBO SSA MFO GWO SCA HS EMO
Test 1
2 95 164 96 163 96 163 96 163 96 164 96 163 96 163 97 162
3 24 96 164 23 96 163 23 96 163 23 96 163 23 109 161 23 96 163 23 96 163 23 95 164
4 23 80 125 173 23 80 125 173 23 80 125 173 23 80 125 173 23 74 130 184 23 80 125 173 23 80 125 173 47 48 139 140
5 23 62 94 135 177 23 77 119 159 190 23 62 94 135 177 23 62 94 135 177 21 76 113 135 166 23 71 109 144 180 23 71 109 144 180 23 111 115 119
138
Test 2
2 66 143 66 143 66 143 66 143 67 140 66 143 66 143 66 143
3 61 111 161 62 112 162 62 112 162 62 112 162 58 108 159 62 112 162 62 112 162 55 108 159
4 61 111 161 227 62 112 162 227 62 112 162 227 61 111 161 227 65 124 172 223 48 88 131 175 62 112 162 227 65 122 172 220
5 45 85 125 169 227 61 108 144 179
227 61 109 152 196
227 45 85 128 173 227 33 62 102 169 227 41 79 110 167 227 51 90 146 196 227 30 62 100 169 228
Test 3
2 79 143 79 143 79 143 79 143 80 142 79 143 79 143 79 143
3 78 143 233 79 143 233 49 101 153 49 101 153 47 103 157 49 101 153 49 101 153 49 100 153
4 49 102 151 233 49 101 153 233 33 73 113 159 49 101 153 233 36 91 154 234 33 73 113 159 33 73 114 160 36 91 155 234
5 33 73 113 160 233 33 73 115 161 233 33 73 113 159 233 33 72 115 159 233 29 59 107 166 237 33 69 104 138 172 33 69 104 138 172 33 69 104 135 170
Test 4
2 128 196 128 196 128 196 128 196 128 196 124 196 128 196 124 196
3 44 103 196 44 103 196 44 103 196 44 103 196 42 99 198 44 103 196 41 103 197 42 100 198
4 44 96 146 196 44 96 146 196 44 96 146 196 44 96 146 196 44 89 135 196 44 95 143 196 42 94 143 196 43 95 143 195
5 24 60 98 146 196 44 96 145 192 217 24 60 98 146 196 23 60 98 145 196 34 74 100 141 195 31 81 118 154 195 27 81 118 154 195 34 72 100 142 195
Test 5
2 70 171 70 171 70 171 70 171 70 169 70 171 70 170 70 171
3 68 126 182 68 126 182 68 126 182 68 126 182 70 131 182 68 126 182 69 126 182 66 126 182
4 69 115 163 232 68 125 182 232 68 126 182 232 68 125 182 232 68 131 183 232 63 97 136 183 63 97 136 185 69 131 183 230
558 100 138 182
233 64 104 143 184
232 64 104 144 185
232 63 102 142 184
232 14 52 102 153 186 16 64 104 143 185 16 60 104 145 185 17 60 105 145 184
Test 6
2 94 171 94 171 94 171 94 171 96 171 94 171 94 171 94 171
3 65 126 187 65 126 187 65 126 187 65 131 195 68 129 191 57 122 187 56 122 187 68 129 190
4 53 98 147 199 53 102 151 203 53 102 151 199 53 98 147 199 51 88 142 190 53 105 145 196 50 105 145 195 53 105 145 197
5 40 81 122 167 207 45 89 131 171 207 45 85 131 171 211 40 85 131 171 211 40 84 124 181 218 40 77 114 155 203 42 78 114 155 201 42 78 112 155 204
Test 7
2 94 175 94 175 89 170 94 175 94 171 89 170 89 170 89 170
3 47 103 175 50 103 175 47 103 175 47 103 175 47 102 172 47 103 175 45 104 176 40 104 176
4 47 98 149 197 47 99 153 197 46 98 145 197 46 98 149 197 48 100 156 195 46 98 145 195 40 98 142 195 39 97 143 195
5 42 85 124 162 197 42 85 124 162 197 42 85 124 162 197 42 85 124 162 197 45 80 126 164 194 46 79 113 149 189 43 79 112 149 185 23 80 124 164 196
Test 8
2 91 179 91 179 91 179 91 179 92 179 91 179 91 179 91 179
3 60 118 179 60 118 179 60 118 179 60 118 179 61 122 178 60 118 179 62 118 180 62 118 181
4 45 90 134 181 45 90 134 181 45 90 134 181 44 90 134 181 40 92 140 183 44 89 133 180 40 89 130 182 39 92 140 184
5 45 88 132 179 220 44 89 133 179 220 44 89 133 179 220 41 86 130 179 220 47 87 119 145 184 33 69 105 141 181 30 69 104 140 182 30 69 102 141 183
Test 9
2 125 226 124 222 94 151 124 222 120 222 124 222 124 222 124 222
3 92 154 226 94 151 222 94 151 222 94 151 222 95 154 222 114 155 223 113 154 223 114 152 220
4 19 93 150 222 19 82 142 222 19 94 151 222 73 114 157 222 69 106 151 226 89 129 158 226 87 126 158 225 68 106 150 226
5 19 74 115 157 222 19 74 116 158 222 19 74 115 157 222 61 95 135 174 222 63 92 127 184 224 19 73 113 155 222 20 70 113 155 225 18 70 113 155 223
Test 10
2 124 204 125 203 125 203 125 203 125 204 109 206 125 204 124 204
3 66 133 204 65 134 203 65 134 203 65 134 203 60 146 205 65 134 203 64 133 203 66 132 203
4 65 113 155 203 65 113 155 203 65 113 155 203 65 113 155 203 64 116 164 207 65 106 147 203 62 104 147 201 61 104 145 203
5 65 98 131 166 202 65 100 134 168
203 65 113 155 203
229 65 97 131 166 203 65 98 146 177 203 65 120 155 203
224 63 123 155 200
225 67 98 145 177 220
Content courtesy of Springer Nature, terms of use apply. Rights reserved
28
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Table 17. e computational time values of comparison algorithms obtained by Kapur’s method. Signicant
values are in bold.
Test Image n
IHBO HBO SSA MFO GWO SCA HS EMO
Mean STD Mean STD Mean STD Mea n STD Mean STD Mean STD Mean STD Mea n STD
Test 1
20.6211 3.2201E-01 0.6836 7.0752E-01 0.7857 5.3643E-01 0.6930 3.1476E-01 0.6549 3.1269E-01 0.6528 6.4282E-01 0.6422 2.6673E-01 0.6527 3.1368E-01
30.5093 4.3560E-02 0.5254 1.0785E-01 0.6369 2.1898E-01 0.6749 1.2499E-01 0.7524 1.0979E-01 0.6887 3.6489E-01 0.6254 1.1956E-01 0.6886 1.4060E-01
40.5634 5.0861E-02 0.9697 2.7778E-01 0.6134 1.7778E-01 0.6296 5.1239E-02 0.6121 3.1738E-02 0.7898 9.4604E-02 0.5834 3.9039E-02 0.7898 4.5911E-02
50.5653 5.2965E-02 0.9085 1.0697E-01 0.6058 2.0219E-01 0.6571 1.3897E-01 0.6398 4.3965E-02 0.7935 8.0408E-02 0.6090 5.1128E-02 0.7934 6.0127E-02
Test 2
2 0.7779 3.9933E-02 0.8627 3.0127E-01 0.7142 2.5375E-01 0.8898 9.0833E-02 0.6877 1.3219E-02 0.9225 2.6054E-01 0.8899 1.2951E-01 0.9965 1.5230E-01
3 0.9947 1.2210E-01 0.8889 1.2891E-01 0.9523 3.9247E-01 1.0086 1.9930E-01 0.6890 1.1056E-02 0.9347 2.5006E-01 1.0087 4.4735E-01 0.9281 5.2608E-01
4 0.8699 1.6830E-02 0.8588 1.8140E-01 0.9645 8.1876E-01 0.9650 1.2422E-01 0.8503 1.3099E-02 0.9623 2.0064E-01 0.9651 1.6877E-01 0.9554 1.9003E-01
50.7865 1.2499E-02 0.8645 9.5719E-02 0.9652 8.5805E-01 1.0000 9.5602E-02 0.7865 1.5981E-02 0.9771 1.6492E-01 1.0002 4.6753E-01 0.9702 5.2643E-01
Test 3
2 0.6717 1.1017E-02 0.6617 1.4953E-01 0.8472 5.8609E-01 0.8830 6.5657E-02 0.8550 2.6329E-02 0.9388 2.8028E-01 0.8831 2.9681E-01 0.8628 3.0682E-01
3 0.6306 9.4267E-03 0.6216 9.8255E-02 0.8025 5.2226E-01 0.9609 1.0703E-01 0.8747 1.8732E-02 0.9235 2.3095E-01 0.9610 1.5853E-01 0.8488 1.6387E-01
40.5589 9.4267E-03 0.5589 3.0645E-01 0.8296 4.7428E-01 1.1662 2.8681E-01 0.9157 5.3339E-01 0.5589 1.5967E-01 1.1663 1.0969E-01 0.9181 1.1339E-01
50.5909 2.6733E-02 0.5909 2.2728E-01 0.9605 9.9428E-01 0.9246 1.0181E-01 0.9706 3.9691E-02 0.9635 1.5405E-01 0.9247 1.6923E-01 0.8855 1.7494E-01
Test 4
2 0.4941 7.1552E-02 0.4641 1.9497E-01 0.8276 5.2441E-01 0.8285 4.4177E-02 0.5565 6.3007E-02 0.8104 2.8270E-01 0.7678 2.7930E-02 0.7435 2.9292E-02
30.4993 3.6084E-02 0.4993 1.2151E-01 0.8512 5.1131E-01 0.8736 7.2636E-02 0.5781 3.4843E-02 0.7371 1.4383E-01 0.8096 3.8484E-02 0.6763 4.0360E-02
4 0.5497 3.6538E-02 0.5390 1.1557E-01 0.9829 6.7516E-01 0.8855 4.0486E-02 0.5390 1.2968E-02 0.8286 2.6200E-01 0.8206 3.1534E-02 0.7602 3.3073E-02
5 0.5363 9.1571E-02 0.5562 2.3257E-01 1.2723 1.4614E+00 0.8818 5.5846E-02 0.6361 2.6198E-02 0.7207 1.9359E-01 0.8172 1.1146E-01 0.6612 1.1690E-01
Test 5
2 0.7228 3.6311E-02 0.6647 7.3112E-02 0.8580 5.3215E-01 0.8007 4.4415E-02 0.9133 1.4802E-02 0.7437 3.2203E-01 0.7420 7.6839E-02 0.6755 8.6981E-02
30.6075 2.2467E-02 0.7664 2.1040E-01 0.8947 8.0661E-01 0.8520 5.4253E-02 0.9414 1.1582E-02 0.9284 4.7313E-01 0.7895 6.5899E-02 0.9200 7.4597E-02
40.6222 5.9007E-03 0.6222 6.0431E-02 1.0681 8.9282E-01 0.8611 7.2294E-02 0.9758 1.6728E-02 1.0533 4.2204E-01 0.7980 2.0362E-01 1.0437 2.3049E-01
50.6001 5.3445E-02 0.6913 7.6421E-02 1.1622 8.4519E-01 0.8857 7.1385E-02 1.0463 6.6057E-02 1.0656 1.6732E-01 0.8208 2.8715E-01 1.0559 3.2505E-01
Test 6
2 0.4836 2.5644E-02 0.7428 3.6359E-02 0.9585 1.8394E-01 1.2662 7.5695E-01 0.4436 1.5728E-02 0.9087 3.3413E-01 1.1734 5.4196E-02 0.9005 5.9941E-02
3 0.5435 5.6509E-02 0.6535 7.3002E-02 0.8827 6.4412E-01 1.0701 3.3902E-01 0.5279 2.3592E-02 0.8868 2.6062E-01 1.0807 1.7295E-01 0.8787 1.9128E-01
4 0.5504 2.1446E-02 0.5395 8.7401E-02 0.6977 3.0568E-01 0.8458 2.1584E-01 0.5395 1.6157E-02 0.9269 3.2188E-01 0.8542 6.3354E-02 0.9186 7.0070E-02
50.5677 2.4963E-02 0.6194 1.0559E-01 0.7069 1.2637E-01 0.8570 7.8913E-02 0.6364 5.8908E-02 0.8848 1.0022E-01 0.8654 1.2911E-01 0.8768 1.4280E-01
Test 7
2 0.5455 3.7339E-02 0.5939 1.2669E-01 0.4780 3.6799E-02 0.5795 1.4316E-01 0.5105 1.6504E-02 0.8684 1.9612E-01 0.6316 4.5877E-02 0.9387 5.0740E-02
30.4883 2.8560E-02 0.5818 1.1185E-01 0.5335 3.4044E-02 0.6467 1.8119E-01 0.6021 1.2470E-01 0.9573 3.0246E-01 0.6984 1.1541E-01 1.0349 1.2764E-01
40.5849 1.5257E-01 0.6108 1.9344E-01 0.7632 1.6912E-01 0.9253 1.0241E-01 0.8054 1.8470E-02 0.9063 1.0705E-01 0.9993 4.0029E-02 0.9798 4.4271E-02
50.5458 1.8743E-01 0.5655 4.1189E-02 1.0039 1.0460E-01 1.2502 6.7362E-02 0.8526 4.7710E-01 0.9619 2.3782E-01 1.3143 1.3659E-01 1.0399 1.5107E-01
Test 8
2 0.5915 1.3711E-01 0.5551 6.2732E-02 0.5924 7.9398E-02 0.8981 2.2399E-01 0.7822 3.0141E-02 0.9030 8.2909E-01 0.8908 4.2758E-02 0.8217 4.7290E-02
30.5216 7.4430E-02 0.5548 6.4964E-02 0.6459 5.7744E-02 0.9793 1.2907E-01 0.8038 2.3314E-02 1.0378 4.8812E-01 0.9713 8.0057E-02 0.9443 9.0945E-02
4 0.6999 2.9661E-01 0.5630 3.8956E-02 0.6904 8.1345E-02 1.0467 1.1226E-01 0.8315 1.1721E-02 1.0638 8.5800E-01 1.0382 7.6029E-02 0.9679 8.6369E-02
50.5329 1.1733E-01 0.7009 2.2693E-01 0.8285 9.7747E-02 1.2561 2.6571E-01 1.0010 1.1077E-02 1.0808 6.5415E-01 1.2458 5.9913E-02 0.9834 6.8062E-02
Test 9
2 0.4615 4.5870E-02 0.7689 3.5663E-01 0.4458 1.0675E-01 0.5404 1.0088E-01 0.5355 1.8306E-02 0.8187 6.2703E-02 0.5408 3.4623E-02 0.8039 3.5065E-02
30.4833 3.2517E-02 0.7936 9.7089E-02 0.5035 2.0462E-01 0.6104 1.3673E-01 0.5638 1.8436E-02 0.8210 9.1884E-02 0.6645 4.0029E-02 0.8062 4.0540E-02
40.5012 6.3550E-02 0.7529 4.2596E-01 0.5015 2.7837E-01 0.7292 1.8009E-01 0.5955 3.4912E-02 0.8691 1.1840E-01 0.7939 3.1534E-02 0.8534 3.6487E-02
50.5241 2.2329E-02 0.7772 5.2625E-01 0.5241 2.3312E-01 0.8345 1.3417E-01 0.6217 3.0728E-02 0.9612 5.0023E-01 0.9086 3.4027E-02 0.9438 3.9372E-02
Test 10
20.4651 2.0153E-02 0.8459 3.1818E-01 0.7235 3.6587E-02 0.8950 1.4572E-01 0.4651 1.8426E-02 0.9310 1.3329E+00 0.8877 4.1338E-02 0.8471 5.6208E-02
30.4015 3.3950E-02 0.8162 4.8704E-01 0.7768 2.2043E-02 0.9610 2.2695E-01 0.4015 6.5846E-02 1.0508 1.2380E+00 0.9532 2.7673E-02 0.9561 3.7627E-02
40.5070 9.2734E-03 0.8488 8.6520E-02 0.8080 2.2270E-02 0.9996 4.1537E-01 0.5698 7.3988E-02 1.0611 6.9543E-01 0.9915 4.4210E-02 0.9654 5.8677E-02
50.5128 6.6147E-02 0.8712 1.8250E-01 0.9260 8.6580E-01 1.1455 1.5104E-01 0.5150 8.8414E-02 1.0858 4.0510E-01 1.1363 6.9128E-02 0.9879 9.1750E-02
Content courtesy of Springer Nature, terms of use apply. Rights reserved
29
Vol.:(0123456789)
Scientic Reports | (2023) 13:9094 | https://doi.org/10.1038/s41598-023-36066-8
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Table 18. Mean and STD values of the optimal tness obtained by Kapur’s method. Signicant values are in
bold.
Test Image n
IHBO HBO SSA MFO GWO SCA HS EMO
Mean STD Mean STD Mean STD Mea n STD Mean STD Mean STD Mean STD Mea n STD
Test 1
218.1417 3.4934E-18 17.8095 3.4500E-15 17.8096 3.4500E-15 17.7866 3.4000E-15 17.8096 3.9000E-15 17.8033 3.4000E-15 17.8019 2.2500E-15 17.8033 2.0300E-15
322.7497 3.6453E-14 22.3331 3.6000E-15 22.3330 3.6000E-15 22.2716 5.1900E-16 22.3330 3.6000E-15 22.1292 1.8100E-16 22.0993 2.6500E-15 22.0568 3.1300E-15
4 26.3699 2.3492E-03 26.6193 2.3200E-03 26.6193 2.3200E-03 26.5518 2.1200E-03 26.6123 6.1300E-04 26.7491 1.4100E-03 26.7421 2.1200E-03 26.7431 5.3100E-03
5 30.3417 2.3694E-03 30.5028 2.3400E-02 30.5017 1.9800E-02 30.4099 2.3100E-02 30.3931 1.0400E-02 30.5023 2.3100E-02 29.6273 4.1100E-02 30.1110 1.3500E-02
Test 2
218.5056 3.4731E-17 18.1667 3.4300E-16 18.1665 3.4200E-16 18.1665 3.4300E-16 18.1665 3.2200E-16 18.1279 3.5500E-14 18.0279 3.4200E-15 17.2279 1.6300E-15
323.0357 3.3516E-14 22.6139 3.3100E-13 22.6097 1.1900E-01 22.5809 4.7100E-02 22.6098 3.1300E-13 22.9852 8.1100E-12 22.9028 6.3300E-13 21.6341 3.7700E-14
427.5598 1.6303E-03 27.0551 1.6100E-03 27.0543 1.5300E-04 27.0461 1.1100E-01 27.0576 6.7700E-03 27.0879 6.1100E-03 26.6845n 3.1100E-04 25.3889 1.9600E-04
532.2244 4.5870E-01 31.3268 4.5300E-02 31.0742 1.1000E-02 31.0122 1.0600E-03 31.0743 1.3200E-02 31.3268 1.4900E-02 31.0123 7.4100E-02 31.0016 5.2300E-02
Test 3
2 18.1469 3.6453E-14 17.6414 3.6000E-15 18.6413 3.4200E-15 17.6141 3.6000E-15 17.6411 2.1300E-15 17.6255 1.2200E-15 17.6255 3.5600E-15 17.6251 6.3200E-12
3 22.6997 4.1921E-01 22.0674 4.1400E-02 23.0662 2.3000E-02 22.0418 1.1400E-03 22.0668 5.8000E-03 22.0384 3.1600E-02 22.0391 1.7000E-03 22.0398 5.1200E-02
4 27.3529 3.4731E-01 26.5910 3.4300E-02 26.5910 1.9800E-02 26.5910 4.7000E-03 26.5689 3.2200E-02 27.5168 3.5200E-03 26.1369 1.3600E-02 26.1437 5.2300E-03
531.9289 1.2353E-01 30.6514 1.2200E-02 30.7193 1.9600E-01 30.6091 8.3200E-03 30.4081 1.0300E-02 31.9289 2.5500E-03 29.9288 5.3600E-03 29.9106 1.6900E-03
Test 4
218.0880 2.1568E-14 17.5841 2.1300E-15 17.5834 1.1200E-15 17.5518 3.2200E-07 17.5832 1.8900E-15 17.5558 1.1200E-14 17.5558 1.1600E-14 12.5772 2.1200E-16
3 22.6379 2.5414E-05 22.0073 2.5100E-05 22.0054 1.0800E-05 21.9724 1.2200E-05 22.8691 2.3100E-05 21.9481 1.0800E-05 21.9685 1.1000E-05 17.5558 3.6800E-12
427.3481 3.4526E-04 26.5863 3.4100E-04 26.5626 4.9100E-05 26.5324 4.1600E-04 26.5812 3.1600E-04 26.5364 1.5800E-04 26.5264 1.9300E-04 21.9685 1.9800E-05
5 31.1370 3.2605E-02 31.5668 3.2200E-03 30.5469 2.2000E-03 30.4892 1.4700E-03 30.5541 2.9600E-03 30.5839 8.2300E-03 30.5023 4.5700E-03 30.5051 3.2900E-03
Test 5
2 17.9628 6.1970E-14 17.6339 6.1200E-15 17.9811 3.3200E-15 17.6312 3.6000E-15 17.6312 3.4100E-15 17.6174 2.6400E-15 17.6174 2.6400E-15 17.4348 1.3200E-15
3 22.8002 6.6425E-14 22.3827 6.5600E-14 22.3813 1.0800E-14 22.3470 1.8100E-14 22.3824 5.2300E-14 22.8676 1.3100E-14 22.3674 1.1800E-14 22.3674 1.1800E-14
427.3774 2.2581E-03 26.9820 2.2300E-03 25.9676 2.5200E-04 26.8941 1.6600E-04 26.9672 2.2000E-03 26.4152 2.0500E-03 26.9524 2.9800E-02 26.9525 3.0200E-02
531.3049 2.3897E-02 30.8528 2.3600E-02 30.6169 2.1500E-02 30.7064 1.3900E-02 30.7652 2.2600E-02 30.9643 2.8500E-02 30.9089 2.1200E-02 30.9936 3.0100E-02
Test 6
2 15.9637 3.5643E-14 15.9831 3.5200E-15 15.7306 3.0100E-15 15.7312 3.2000E-15 15.7132 3.4100E-15 17.8130 6.2100E-15 17.8119 5.0100E-15 17.8051 6.2100E-15
322.8561 3.6858E-03 22.5260 3.6400E-03 19.5334 1.3200E-03 19.5331 1.1800E-03 19.5332 1.2000E-03 22.4568 2.6000E-03 22.0993 2.1400E-03 22.4298 5.4100E-02
4 23.2270 3.6554E-03 22.8915 3.6100E-04 22.8903 2.9800E-04 22.8927 3.9400E-04 22.8903 2.6900E-04 26.7491 5.2200E-02 26.7421 3.2600E-02 26.7431 4.2000E-02
5 26.3719 1.7113E-02 25.9399 1.6900E-03 25.9323 1.1200E-03 25.9225 1.1000E-03 25.9179 1.0500E-03 30.5388 3.5900E-02 30.6273 3.8900E-02 30.6113 3.2300E-02
Test 7
2 18.1966 3.5438E-15 17.8985 3.5000E-15 17.8981 3.3800E-15 17.8981 3.3800E-15 17.8981 3.3800E-15 18.2279 6.5200E-15 18.2099 6.5200E-15 18.2171 6.5200E-15
3 22.1188 1.8834E-13 22.4376 1.8600E-14 22.4377 2.7800E-14 22.4377 2.7800E-14 22.4377 2.7800E-14 22.9852 3.6800E-14 22.9828 2.9800E-14 22.6343 2.6400E-14
427.9917 3.6048E-02 27.6491 3.5600E-03 26.6383 2.3600E-03 26.6481 2.1500E-03 26.6481 2.1500E-03 27.6491 6.1500E-03 26.6845 3.2300E-03 27.3889 2.1300E-03
532.1399 4.9515E-01 31.6134 4.8900E-02 30.5136 3.2300E-03 30.5124 3.1000E-03 30.5076 2.8900E-03 31.5273 2.9900E-02 31.3923 2.5200E-03 31.5016 3.0900E-02
Test 8
218.2946 2.0049E-13 17.9949 1.9800E-14 17.8949 1.3600E-14 17.8949 1.3600E-14 17.8949 1.3600E-14 17.9949 1.3600E-14 17.6255 1.3600E-14 17.6251 1.0900E-14
322.9549 3.1998E-04 22.5789 3.1600E-04 22.5790 3.2600E-04 22.5793 3.5200E-04 22.5791 3.1300E-04 22.0384 3.4600E-05 22.0391 3.9800E-05 22.0398 4.2300E-05
427.3468 2.3289E-02 26.8988 2.3000E-03 26.7988 2.1300E-03 26.7988 2.1300E-03 26.7984 2.3600E-04 26.5168 1.1600E-03 26.1369 2.1300E-04 26.1437 2.6800E-04
531.3574 2.6530E-01 30.8437 2.6200E-02 30.8458 2.9600E-02 30.8456 2.6400E-02 30.8132 6.6400E-03 29.9289 2.7400E-03 29.9288 2.6400E-03 29.9106 2.1400E-03
Test 9
217.9633 6.0856E-13 17.7739 6.0100E-14 17.7614 5.5400E-14 17.7614 4.4300E-14 17.7614 3.6400E-14 17.5558 4.0100E-14 17.5558 4.0100E-14 12.5772 4.7400E-14
322.4930 2.1467E-02 22.2559 2.1200E-03 22.4500 2.5100E-03 22.4503 2.5100E-03 22.4503 2.5100E-03 22.4503 2.1200E-03 21.9685 2.1900E-03 17.5558 2.9800E-05
4 26.3234 3.6959E-03 26.6969 3.6500E-03 26.6878 2.0100E-03 26.6031 1.8900E-03 26.6019 1.2400E-03 26.5364 1.5600E-04 26.5264 1.0900E-04 21.9685 1.7600E-04
5 30.4194 4.1536E-04 30.8510 4.6300E-03 30.4339 4.2200E-03 30.8509 4.5400E-03 30.8505 4.4100E-03 30.5839 4.0300E-03 30.5023 3.6400E-03 30.5051 3.3300E-03
Test 10
217.9739 1.2201E-12 17.7844 1.3600E-12 17.7841 6.1900E-12 17.7841 6.1900E-12 17.7841 6.1900E-12 17.6174 0.0000E+00 17.6174 1.2100E-03 17.4348 2.4200E-03
322.6818 3.1795E-03 22.4427 3.1400E-04 22.4421 1.4200E-04 22.4421 1.4200E-04 22.4421 1.4200E-04 22.4427 3.1200E-05 22.3674 3.0200E-04 22.3674 3.0200E-04
426.9854 3.3516E-01 26.4635 3.3100E-02 26.4609 1.8900E-02 26.4636 3.4300E-02 26.4605 1.7400E-02 26.4152 2.1000E-03 26.9524 2.8900E-01 26.9525 1.3500E-01
531.3269 4.1009E-01 30.9967 4.0500E-01 30.1383 2.3600E-01 30.1514 2.6900E-01 30.1919 3.1300E-01 30.9967 3.8900E-01 31.3269 2.4600E-01 30.9936 2.8800E+00
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Table 19. Mean and STD values of PSNR results obtained by kapur’s method. Signicant values are in bold.
Test Image n
IHBO HBO SSA MFO GWO SCA HS EMO
Mean Std Mean Std Mean Std Mean Std Me an Std Mean Std Mean Std Mean Std
Test 1
215.0365 8.7289E-14 14.6383 8.2100E-13 14.6373 6.1100E-13 14.6382 9.0100E-15 14.6383 4.5100E-15 13.9227 8.5200E-02 14.6383 1.1900E-01 14.6363 6.2000E-12
317.8278 3.4341E-14 17.3557 3.2300E-13 15.2182 3.4500E-14 16.2169 5.9700E-10 16.2189 5.2600E-11 17.2013 1.2400E-03 17.3557 5.4100E-15 16.2189 3.9300E-08
4 19.2329 3.3172E-05 19.2869 3.1200E-04 19.2866 3.0200E-04 19.2864 4.5600E-02 19.2846 4.4300E-13 17.4863 3.7800E+00 19.2860 1.8200E-10 19.2859 2.9300E-01
5 20.8324 2.1583E-04 20.8909 2.0300E-03 19.8789 2.0900E-04 20.7909 9.4300E-14 21.1109 3.2000E-02 17.7789 3.3300E-02 20.9807 1.8900E-10 20.4476 3.6500E-08
Test 2
217.2214 3.6255E-13 16.7654 3.4100E-12 16.2656 3.6900E-10 16.2654 5.4100E-15 16.4459 5.4100E-15 14.7513 1.1000E+00 16.2655 5.4100E-15 16.2655 6.3220E-15
318.8670 5.3266E-09 18.36738 5.0100E-08 18.36735 4.2100E-06 18.36718 5.7300E-01 16.4601 3.0300E-04 18.3863 3.4100E+00 18.3674 4.6100E-01 18.3674 9.3000E-01
4 18.9566 3.6149E-12 18.4546 3.4000E-11 16.3546 3.9700E-10 18.4096 5.3100E-14 17.3097 3.0200E-09 20.5965 5.0700E-11 20.5965 1.4700E-01 19.8412 6.0700E-14
521.3556 3.6149E-05 20.7901 3.4000E-04 19.4901 3.0900E-06 20.7819 2.1900E-06 19.1754 1.6900E-14 17.6409 3.0900E-02 19.8529 2.4900E-06 19.8412 3.5900E-01
Test 3
2 16.4529 3.2534E-15 16.0172 3.0600E-15 16.0171 3.1700E-15 16.0172 3.6000E-15 15.1254 2.1000E-08 16.6230 2.2800E+00 16.0493 3.6000E-15 16.0293 1.6300E-03
3 16.3862 4.6356E-04 15.9523 4.3600E-04 15.9423 3.1200E-05 15.9468 1.7500E-03 17.5652 3.0900E-15 16.89951 3.3600E+00 18.7642 2.2100E-01 18.6642 3.5600E-01
419.2745 3.2428E-02 18.7641 3.0500E-02 17.7641 2.1500E-02 18.7255 5.9200E-10 19.0701 2.3200E-12 17.1838 2.3100E+00 18.8583 5.6200E-09 18.5823 1.2200E-02
520.9717 4.1040E-02 20.4164 3.8600E-01 20.3369 3.4500E+00 20.4144 2.0200E-05 20.4742 6.2000E+01 17.4752 6.32E6-02 17.6935 6.2000E-06 17.2931 2.0000E-02
Test 4
213.9962 6.3898E-17 13.6256 6.0100E-16 13.6251 5.4100E-10 13.5356 5.8900E-16 13.6419 7.2100E-15 13.6419 2.1200E-01 13.6257 9.0100E-15 13.6312 8.3200E-09
3 14.8534 2.6580E-07 14.4601 2.5000E-06 12.4561 2.5000E-08 14.3691 1.1000E-02 14.3004 1.0200E+03 16.9059 2.4500E+00 14.6542 3.6000E-15 14.6012 2.0300E+02
420.7013 1.6480E-04 20.1531 1.5500E-03 20.1528 1.0500E-02 20.0639 1.3200E+02 20.1756 6.5600E-08 17.1663 6.4000E+01 20.1531 5.0200E-13 20.1234 3.9200E-01
521.2249 3.8488E-02 20.6629 3.6200E-01 20.6417 6.0100E-02 20.5698 3.2000E+01 20.6621 9.0600E-15 20.6629 3.2500E+03 20.6608 2.3200E+01 20.5308 9.0000E-01
Test 5
216.1870 8.5162E-16 15.7584 8.0100E-14 13.7574 4.6100E-15 15.6588 9.0100E-15 15.7533 3.6000E-15 13.6633 4.5000E+00 15.7584 9.0100E-15 15.7584 1.8400E-06
319.3218 2.8281E-14 18.8102 2.6600E-12 18.7322 2.0600E-10 18.8102 5.1600E-01 18.8384 7.0400E-10 18.8584 5.2300E+00 18.8102 3.6000E-15 18.8102 2.5400E-04
421.0439 4.8163E-06 20.4867 4.5300E-04 19.2899 3.9700E-01 20.3878 4.6900E-02 18.3942 1.4400E+00 14.1232 5.2200E+00 20.3208 6.0900E-14 18.8102 1.7400E-02
520.7689 3.8700E-02 20.2189 3.6400E+00 19.2089 3.6400E+01 20.0799 1.9400E-16 18.2268 3.1400E+02 14.2814 5.4100E-15 20.5404 5.1200E-02 20.4868 3.0100E-16
Test 6
2 13.2961 3.8275E-16 13.7299 3.6000E-15 11.3533 3.6000E-15 13.5298 3.6000E-15 13.5298 5.4100E-15 12.1402 2.8700E+00 13.5298 3.6000E-15 13.6298 3.9900E-02
3 16.7287 7.9102E-02 17.0229 7.4400E-01 10.6508 3.0900E-04 16.8065 9.5800E-01 16.8065 1.0900E-03 17.3447 3.0900E+00 17.1017 6.4000E-03 17.0017 4.1200E-12
419.6562 2.1902E-03 19.1357 2.0600E-02 13.6031 3.2200E-02 18.9026 2.8300E-12 18.9026 3.2100E-02 17.6229 3.0600E+01 19.0357 6.6700E-01 19.1125 3.0400E-06
5 21.1709 2.4985E-01 20.6103 2.3500E+00 12.6928 1.3400E+02 20.7048 2.7700E+02 20.6103 2.8900E-01 21.7048 3.9100E-07 20.2447 6.8100E-06 20.3441 5.6300E-13
Test 7
214.9932 5.5605E-15 14.5962 5.2300E-14 14.4886 1.7300E-02 14.5962 3.5800E-02 14.5962 5.4100E-15 10.1415 7.2800E-01 14.6310 8.2000E-03 14.6213 2.0800E-04
3 17.6125 3.3384E-15 17.1461 3.1400E-14 9.4044 2.4300E-10 17.1979 1.0800E-01 17.1461 3.4200E-08 17.8925 3.1800E+00 17.4714 3.6000E-15 17.4716 2.6900E-12
4 19.6647 5.3373E-03 19.1440 5.0200E-02 14.8254 1.9200E-04 17.1461 2.8300E-01 19.1447 3.1200E-08 17.9647 3.6100E+00 19.4588 1.0600E-01 19.6587 3.0900E-15
5 21.7843 5.3373E-01 21.8975 5.0200E+00 9.0705 3.1200E-01 21.1875 2.6900E-10 21.1341 5.0900E-02 21.2075 1.0900E+01 20.9991 3.4200E-08 20.5951 2.6900E-12
Test 8
215.6964 2.0094E-14 15.2029 1.8900E-13 15.1958 1.8000E-15 15.2029 1.2900E-04 15.2028 1.8000E-15 12.7136 5.6900E-10 15.2023 2.4500E-06 15.5967 6.0900E-10
319.0008 1.5097E-02 18.4977 1.4200E-02 13.0695 1.0000E-01 18.4977 7.3200E-01 18.4987 2.6900E-12 15.1063 3.2800E+00 18.4977 0.0000E+00 18.0481 3.4200E-08
421.7998 4.6887E-04 21.0278 4.4100E-02 11.8816 2.4200E-06 21.0279 3.0200E-18 21.0279 2.6200E+00 16.4222 3.4200E+00 21.0484 3.8900E-02 21.6258 1.0400E-01
5 23.4708 3.7850E-04 22.8493 3.5600E+01 12.3141 1.2000E+00 21.0654 3.4200E-08 22.8493 3.0200E-06 23.7483 6.2200E-02 22.8258 2.1200E-01 15.4654 3.9400E-05
Test 9
2 15.3868 1.6267E-13 14.9794 1.5300E-13 10.7742 2.6900E-12 14.2865 6.0200E-06 10.8308 1.8000E-15 17.5018 6.8900E-02 14.4657 1.2600E-14 13.7461 5.3100E+00
3 16.0658 2.0732E-12 17.8962 1.9500E-12 12.2133 1.1500E-04 14.2865 9.3000E-02 14.2865 6.0200E-02 17.8029 2.4400E+00 17.8462 3.8900E-02 15.8914 6.2200E-02
4 19.1575 2.2752E-04 19.6237 2.1400E-04 15.068 3.9800E-05 15.9906 9.1500E-02 15.9906 6.8900E-02 20.1125 2.7900E+00 15.9906 5.1100E-14 19.5751 2.3200E-03
519.9007 3.6787E-03 19.3737 3.4600E-01 12.2638 3.1000E-02 19.3733 2.6200E+00 19.3731 2.9600E+01 19.3737 3.0200E-13 19.1750 5.6100E-04 9.8521 1.9500E-12
Test 10
216.8790 5.4330E-15 12.2449 5.1100E-14 12.1690 3.0100E-12 12.2447 5.4100E-15 10.8308 3.8900E-15 12.2449 1.9500E-12 9.9525 6.8900E-02 16.8782 6.2900E-02
317.3375 2.2752E-11 16.8784 2.1400E-10 16.8809 5.3200E-14 16.8784 3.4200E-04 16.8784 2.5400E-01 16.4935 3.4200E-08 16.8784 1.9500E-12 16.2132 2.4600E-10
420.6547 6.5387E-05 20.1078 6.1500E-04 19.9473 2.5400E-01 20.1075 8.7300E-01 20.1078 2.9200E-06 16.7673 2.6200E+00 19.2037 3.4200E-04 19.2046 1.8100E-07
5 20.7243 7.3254E-03 21.9716 6.8900E-02 18.4898 2.3400E+00 20.1075 3.4200E-08 22.0537 2.0200E-14 16.9775 6.2200E-02 21.8813 1.0900E+00 20.8812 6.8700E-09
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Table 20. Mean and STD values of FSIM results obtained by kapur’s method. Signicant values are in bold.
Test Image n
IHBO HBO SSA MFO GWO SCA HS EMO
Mean STD Mean STD Mean STD Mea n STD Mean STD Mean STD Mean STD Mea n STD
Test 1
20.7647 3.6969E-04 0.7327 3.4000E-03 0.6624 3.3800E-16 0.7327 3.3800E-16 0.7327 0.0000E+00 0.6322 3.5800E-03 0.6739 2.2300E-02 0.6327 2.1600E-02
30.7794 2.7414E-04 0.7468 6.2000E-03 0.6615 2.2500E-16 0.7327 9.8000E-03 0.7479 6.5000E-03 0.2468 6.4900E-02 0.6911 2.2500E-16 0.6327 5.3000E-03
40.8524 1.3048E-03 0.8167 1.2000E-03 0.6987 4.5100E-16 0.7485 1.2100E-03 0.8165 2.9000E-03 0.5167 6.9600E-02 0.7621 4.0000E-04 0.6486 5.7000E-03
50.8643 2.0659E-03 0.8591 1.9000E-03 0.8224 2.3000E-03 0.8157 4.1000E-03 0.8319 1.9800E-02 0.8552 7.4200E-02 0.8174 4.8000E-03 0.7157 4.9000E-03
Test 2
2 0.7220 1.0873E-04 0.7364 1.0000E-04 0.6296 4.5100E-16 0.6684 4.5100E-16 0.7353 4.5100E-16 0.7351 2.9100E-02 0.7243 4.5100E-16 0.6689 2.0000E-04
3 0.7281 2.4465E-03 0.7364 2.2500E-02 0.7341 2.7000E-02 0.7353 2.7200E-02 0.7364 4.3000E-03 0.7362 8.1300E-02 0.7791 2.1800E-02 0.7353 2.7000E-02
40.8647 2.2073E-04 0.8477 2.0300E-02 0.8477 5.8100E-05 0.8108 7.1900E-03 0.8101 9.6000E-03 0.8577 8.7000E-02 0.7686 1.6000E-02 0.7108 2.5000E-03
5 0.8698 2.2834E-05 0.8469 2.1000E-03 0.6174 4.0000E-03 0.8108 6.8000E-03 0.8265 1.0200E-02 0.8709 8.5200E-02 0.8173 6.0000E-03 0.8108 9.2000E-03
Test 3
2 0.8530 2.7183E-05 0.8556 2.5000E-03 0.6437 4.5100E-16 0.7142 4.5100E-16 0.8557 5.6300E-16 0.8456 1.0500E-01 0.8651 4.5100E-16 0.7142 3.0000E-04
3 0.9030 3.1315E-04 0.8939 2.8800E-02 0.8945 9.6000E-03 0.855 9.6900E-06 0.8938 1.3000E-03 0.8639 1.4100E-01 0.9067 4.5100E-16 0.8254 4.0000E-04
40.9378 1.6310E-05 0.9273 1.5000E-03 0.9213 1.0400E-02 0.8942 1.6000E-03 0.9177 2.7200E-02 0.9362 1.8050E-01 0.9253 8.9000E-03 0.8942 4.3000E-03
50.9559 1.0873E-04 0.9446 1.0000E-03 0.9213 8.6000E-03 0.9178 1.3000E-03 0.9406 2.2800E-02 0.9432 1.7850E-01 0.8857 1.6200E-02 0.9078 6.7000E-03
Test 4
20.7067 1.7397E-04 0.6771 1.6000E-03 0.6693 0.0000E+00 0.5968 2.1500E-04 0.6771 0.0000E+00 0.6723 1.7500E-02 0.6136 0.0000E+00 0.5468 2.9000E-03
3 0.7771 5.3279E-04 0.8010 4.9000E-03 0.6494 5.6300E-16 0.6771 1.1400E-03 0.8002 4.0000E-03 0.8004 6.9100E-02 0.7994 5.6300E-16 0.5771 1.7000E-03
4 0.8146 3.6112E-05 0.8397 7.0000E-04 0.6916 8.0000E-04 0.8012 5.8800E-04 0.8383 4.8000E-03 0.8187 8.3800E-02 0.8414 5.0000E-04 0.8012 1.7000E-03
50.8863 6.3065E-04 0.8684 5.8000E-03 0.6801 1.4200E-02 0.8658 3.8600E-03 0.8622 9.3000E-03 0.8662 1.0800E-01 0.8617 1.6600E-02 0.8558 1.1800E-02
Test 5
2 0.7849 4.9038E-17 0.7904 4.5100E-16 0.5794 4.5100E-16 0.7308 4.5100E-16 0.7904 3.3800E-16 0.6423 1.0600E-01 0.7913 4.5100E-16 0.7413 7.0000E-04
3 0.8608 6.8501E-04 0.8602 6.3000E-03 0.7542 5.6300E-16 0.8602 1.1700E-02 0.8602 2.8000E-03 0.8612 1.1900E-01 0.8607 5.6300E-16 0.8407 3.0000E-04
40.9070 6.5239E-05 0.8978 6.0000E-04 0.7352 1.8700E-02 0.8603 5.3700E-04 0.8973 4.2000E-03 0.8875 1.1900E-01 0.8991 1.8900E-02 0.8607 1.0000E-02
5 0.9091 2.0007E-03 0.8902 1.8400E-02 0.7522 8.4000E-03 0.9098 1.5600E-02 0.8682 1.8200E-02 0.8856 1.2820E-01 0.8374 7.5000E-03 0.8484 9.2000E-03
Test 6
20.7958 4.1318E-04 0.7625 3.8000E-03 0.7591 3.3800E-16 0.7625 3.3800E-16 0.7625 5.6300E-16 0.4562 5.3500E-02 0.7625 3.3800E-16 0.7127 5.2300E-04
3 0.8468 3.9144E-04 0.8497 3.6000E-03 0.5891 3.9600E-04 0.8476 2.6600E-02 0.8473 1.5000E-03 0.8497 6.3700E-02 0.8476 1.6900E-04 0.7327 6.0000E-04
4 0.8986 5.1104E-04 0.8998 4.7000E-03 0.7352 7.5000E-03 0.8999 2.5700E-04 0.8997 1.4800E-02 0.8112 6.6500E-02 0.8975 1.6000E-02 0.7489 1.8700E-02
5 0.9245 1.1961E-04 0.9257 1.1000E-03 0.6903 3.6000E-03 0.9246 9.4700E-03 0.9265 1.5600E-02 0.8425 7.1400E-02 0.9258 4.5000E-03 0.8161 1.2500E-02
Test 7
2 0.7482 4.1216E-17 0.7169 5.6300E-16 0.6273 5.7700E-03 0.7168 1.9700E-03 0.7166 3.3800E-16 0.7342 4.6800E-02 0.7276 2.7400E-03 0.6681 4.4000E-03
3 0.7941 5.5453E-04 0.7992 5.1000E-03 0.7089 1.0700E-03 0.7993 2.2200E-03 0.7993 5.3000E-03 0.7345 1.0900E-01 0.7993 4.5100E-16 0.7453 4.0000E-03
40.8918 3.0890E-05 0.8545 5.6000E-03 0.7441 4.0000E-03 0.8545 3.2900E-03 0.8544 8.5000E-03 0.8137 1.2100E-01 0.8537 2.0400E-03 0.8008 5.4000E-03
50.9145 2.6096E-05 0.9050 2.4000E-03 0.6529 3.3000E-03 0.9028 1.2200E-02 0.9005 1.2600E-02 0.8426 1.3010E-01 0.8904 5.5000E-03 0.8108 2.8000E-03
Test 8
2 0.6907 1.4135E-03 0.7152 1.3000E-03 0.6758 1.1300E-16 0.7153 2.2600E-03 0.7153 0.0000E+00 0.8425 1.0700E-03 0.7153 4.5400E-02 0.5042 5.2100E-02
3 0.7956 6.6327E-03 0.8238 6.1000E-03 0.8210 4.5100E-16 0.8239 5.7500E-03 0.8239 9.9000E-03 0.8923 7.7300E-02 0.8239 4.5100E-16 0.6553 2.7000E-03
4 0.8273 1.7397E-03 0.8885 1.6000E-03 0.8868 6.0000E-04 0.8886 1.5800E-02 0.8905 6.3000E-03 0.9125 9.0200E-02 0.8886 1.0000E-03 0.7942 8.2000E-03
5 0.9297 2.9901E-04 0.8908 2.7500E-02 0.6242 2.2600E-02 0.8906 2.4600E-02 0.8918 3.2000E-02 0.9498 9.0700E-02 0.8981 2.4100E-02 0.8178 1.6500E-02
Test 9
2 0.6760 2.4465E-18 0.6477 2.2500E-16 0.6443 5.1900E-03 0.7332 4.1500E-03 0.6471 1.1300E-16 0.6123 5.4700E-02 0.6459 2.2500E-16 0.5468 6.9000E-03
30.8360 3.6771E-05 0.8010 8.9000E-03 0.7318 5.0000E-04 0.7332 1.7000E-03 0.7332 8.1000E-03 0.7312 1.0500E-01 0.6484 3.6500E-04 0.6771 3.9000E-03
4 0.7899 3.4366E-06 0.7568 5.0000E-04 0.8123 1.0000E-04 0.7554 1.0000E-03 0.7554 5.1000E-03 0.8353 1.1970E-01 0.7461 6.4800E-04 0.7012 1.7000E-03
5 0.8083 1.4135E-04 0.8344 1.3000E-03 0.6201 1.1000E-03 0.8341 1.7000E-03 0.8545 5.6000E-03 0.8468 1.2170E-01 0.8336 1.4000E-03 0.8358 2.2000E-03
Test 10
20.7673 1.1961E-03 0.6476 1.1000E-03 0.6467 2.0200E-02 0.6477 1.1300E-16 0.6477 1.1300E-16 0.7123 3.3200E-04 0.6477 2.7800E-02 0.7108 1.2800E-02
3 0.8721 5.4366E-04 0.8452 5.0000E-04 0.7455 3.7000E-03 0.7459 8.4000E-04 0.7459 1.8200E-02 0.8821 4.1000E-02 0.7459 3.2700E-02 0.7602 4.2600E-02
40.9468 2.6639E-02 0.9072 2.4500E-02 0.7264 9.3000E-03 0.8372 2.4900E-02 0.8372 1.0880E-02 0.8978 9.3200E-02 0.8279 1.8660E-02 0.8603 2.5600E-02
50.9535 4.7859E-03 0.9136 9.0000E-03 0.9535 6.5000E-03 0.8898 7.0400E-03 0.8384 2.8600E-02 0.8902 1.4100E-01 0.8888 1.1700E-02 0.8778 1.9700E-02
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Table 21. Mean and STD values of SSIM results obtained by Kapur’s method. Signicant values are in bold.
Test Image n
IHBO HBO SSA MFO GWO SCA HS E MO
Mean Std Mean Std Mean Std Mean Std Me an Std Mean Std Mean Std Mean Std
Test 1
2 0.5218 3.9233E-04 0.5044 3.8000E-03 0.5034 3.3800E-16 0.5017 3.3800E-16 0.5838 5.6300E-17 0.4799 3.5400E-03 0.5191 2.1700E-02 0.5017 2.0400E-02
3 0.6501 6.5044E-04 0.6284 6.3000E-03 0.5052 5.6300E-16 0.6687 6.1600E-03 0.7047 9.4000E-03 0.6158 1.5400E-01 0.6266 5.6300E-16 0.5687 6.3000E-03
40.8299 3.2272E-05 0.8022 7.0000E-04 0.5991 6.7600E-16 0.8005 8.1200E-04 0.7637 9.6000E-03 0.7279 1.5200E-01 0.7195 4.0000E-04 0.6005 3.5000E-03
5 0.8633 1.1357E-04 0.8786 1.1000E-03 0.8741 1.4000E-03 0.8741 2.5000E-03 0.7438 1.5400E-02 0.8139 1.5850E-01 0.7613 4.2000E-03 0.6741 2.7000E-03
Test 2
20.6319 1.3422E-04 0.5448 1.3000E-03 0.4549 1.1300E-16 0.4923 1.1300E-16 0.5279 5.6300E-17 0.1283 2.1900E-02 0.6319 1.1300E-16 0.4923 9.7100E-05
3 0.5424 3.9233E-03 0.5520 3.8000E-03 0.6239 1.0500E-02 0.7448 1.0600E-02 0.6305 8.3000E-03 0.6647 1.9600E-01 0.6601 8.5000E-03 0.6448 1.4500E-02
40.8166 3.9233E-03 0.6052 3.8000E-03 0.7796 4.0000E-04 0.8166 8.5900E-03 0.6187 1.6700E-02 0.7702 1.6000E-01 0.7013 1.7200E-02 0.7166 5.1000E-03
5 0.6884 1.3422E-03 0.6654 1.3000E-03 0.3887 6.8000E-03 0.8166 8.5000E-03 0.6381 1.3800E-02 0.8264 1.2340E-01 0.7202 2.8300E-02 0.8166 1.0500E-02
Test 3
2 0.6555 5.2655E-04 0.6336 5.1000E-03 0.4136 4.5100E-16 0.6262 4.5100E-16 0.7534 3.3800E-16 0.6436 1.2400E-01 0.6448 4.5100E-16 0.6262 2.2100E-05
3 0.8120 1.6519E-04 0.7269 1.6000E-03 0.8347 1.8100E-02 0.8139 1.4300E-04 0.6335 7.0000E-04 0.7669 1.6400E-01 0.7433 7.8900E-16 0.8139 3.0000E-04
40.8924 1.6519E-04 0.7659 1.6000E-03 0.8737 1.3400E-02 0.8894 1.6000E-03 0.8700 4.4100E-02 0.8759 2.0700E-01 0.7838 1.1200E-02 0.8794 5.9000E-03
50.9290 1.0325E-04 0.8692 1.0000E-03 0.8692 7.8000E-03 0.9185 9.0700E-04 0.9212 2.5000E-02 0.6592 2.0800E-01 0.7147 1.4500E-02 0.9185 9.7000E-03
Test 4
20.5555 3.0974E-05 0.5369 3.0000E-04 0.5197 1.1300E-16 0.1315 4.0000E-04 0.5316 0.0000E+00 0.5549 8.1100E-03 0.3546 1.1300E-16 0.1115 4.7000E-03
30.6699 3.0974E-05 0.6282 3.0000E-04 0.2821 3.3800E-16 0.6639 3.0800E-03 0.6148 5.9000E-03 0.6287 7.2700E-02 0.6247 3.3800E-16 0.6639 6.7000E-03
40.7790 1.3422E-04 0.6853 1.3000E-03 0.3141 1.1000E-03 0.7760 1.3000E-03 0.6803 1.2900E-02 0.6854 9.6500E-02 0.6879 9.0000E-04 0.6768 3.5000E-03
50.8456 9.2921E-04 0.8367 9.0000E-03 0.3995 1.9500E-02 0.8347 9.1800E-03 0.7054 2.2200E-02 0.8359 1.2600E-01 0.7081 2.1800E-02 0.8247 1.7100E-02
Test 5
2 0.7754 4.6563E-17 0.7495 4.5100E-16 0.3785 4.5100E-16 0.8224 4.5100E-16 0.8104 3.3800E-16 0.7595 1.9600E-01 0.7478 4.5100E-16 0.7468 9.0000E-04
3 0.8304 6.1888E-04 0.8027 8.9000E-03 0.7685 7.8900E-16 0.9226 7.7500E-03 0.8675 3.1000E-03 0.9236 1.9600E-01 0.8005 7.8900E-16 0.8105 4.0000E-04
4 0.9062 5.1947E-05 0.9406 6.0000E-04 0.7058 1.1200E-02 0.9226 1.9300E-03 0.8357 3.6000E-03 0.9406 1.9900E-01 0.8361 1.1400E-02 0.8215 5.5000E-03
5 0.9161 4.0531E-04 0.9508 7.8000E-03 0.7664 4.2000E-03 0.9452 7.0000E-03 0.7976 1.5800E-02 0.9508 2.1990E-01 0.7794 4.7000E-03 0.8348 3.3000E-03
Test 6
2 0.4244 5.1623E-05 0.4102 5.0000E-04 0.5247 4.5100E-16 0.5875 4.5100E-16 0.5879 1.1300E-16 0.5154 8.4500E-02 0.4103 4.5100E-16 0.5017 1.3000E-03
30.7983 1.5900E-03 0.6044 1.5400E-02 0.3652 4.7000E-04 0.7874 2.2000E-02 0.7883 3.8000E-03 0.6454 8.8400E-02 0.5901 1.0000E-04 0.6487 7.0000E-04
40.8693 7.2272E-04 0.7087 7.0000E-04 0.6089 9.5700E-03 0.8660 4.7800E-04 0.8601 1.3300E-02 0.8022 9.9500E-02 0.7134 1.5500E-02 0.8005 2.4000E-02
5 0.8921 5.1623E-04 0.7656 5.0000E-04 0.5617 7.4000E-03 0.9062 2.0500E-02 0.8794 1.6900E-02 0.9146 1.1500E-01 0.7656 7.3000E-03 0.8441 1.1000E-02
Test 7
2 0.4345 5.2655E-04 0.4831 5.1000E-03 0.4845 4.9000E-03 0.6839 1.7000E-03 0.5779 3.3800E-16 0.5442 3.3600E-02 0.4942 2.3000E-03 0.4923 3.3000E-03
3 0.6360 7.8466E-04 0.6148 7.6000E-03 0.6306 3.0000E-04 0.8244 6.3600E-04 0.7235 4.1000E-03 0.5532 1.5400E-01 0.6148 6.7600E-16 0.7148 2.5000E-03
4 0.6925 1.5487E-04 0.6694 1.5000E-03 0.6591 4.4000E-03 0.8634 5.0900E-03 0.7874 6.5000E-03 0.6052 1.6000E-01 0.6702 2.4900E-03 0.6066 5.4000E-03
5 0.7525 1.5487E-04 0.7274 1.5000E-03 0.5411 2.1000E-03 0.9054 1.0600E-02 0.9067 1.6500E-02 0.6124 1.6300E-01 0.7097 3.6000E-03 0.4136 1.7000E-03
Test 8
20.6538 1.0325E-05 0.3613 1.0000E-04 0.4099 1.1300E-16 0.6104 6.7400E-03 0.4361 1.7600E-18 0.6336 2.8200E-03 0.3613 9.8700E-02 0.6262 9.9400E-02
30.7617 4.0266E-04 0.4849 3.9000E-03 0.5846 5.6300E-16 0.7536 4.2300E-02 0.5909 2.8700E-02 0.5532 1.1900E-01 0.7648 5.6300E-16 0.7148 6.9000E-03
40.8486 5.8850E-04 0.5786 5.7000E-03 0.6865 1.0000E-04 0.8375 1.1200E-02 0.6964 1.6200E-02 0.6052 1.1630E-01 0.6702 7.5000E-04 0.6066 4.4000E-03
50.9023 1.7448E-03 0.5822 1.6900E-02 0.3601 1.4100E-02 0.8388 1.5800E-02 0.6972 2.8600E-02 0.6124 1.1040E-01 0.7097 1.3600E-02 0.4136 9.1000E-03
Test 9
2 0.6980 4.1239E-03 0.5561 6.9000E-03 0.3421 1.6400E-02 0.6373 1.3100E-02 0.3466 1.1300E-16 0.6124 5.9800E-02 0.7097 0.0000E+00 0.4136 3.8400E-02
30.7178 5.2979E-03 0.6282 6.1000E-03 0.5571 1.3000E-03 0.6373 3.6200E-03 0.5637 3.6600E-02 0.6336 1.2200E-01 0.7613 1.3900E-03 0.6262 9.1000E-03
4 0.7246 1.1357E-03 0.64562 1.1000E-03 0.7357 6.0000E-04 0.7556 2.7800E-03 0.6727 1.8900E-02 0.6785 1.4500E-01 0.7850 2.1800E-03 0.6339 3.8000E-03
5 0.7940 2.8909E-04 0.6708 2.8000E-03 0.4719 5.2000E-03 0.8719 4.7500E-03 0.8799 1.2300E-02 0.7642 1.4800E-01 0.8786 5.6000E-03 0.6694 5.6000E-03
Test 10
20.8690 2.5811E-04 0.5113 2.5000E-03 0.4771 5.4800E-02 0.5898 2.2500E-16 0.4826 2.2500E-16 0.8681 2.1500E-04 0.6029 7.5100E-02 0.8185 2.4300E-02
30.8785 8.2596E-04 0.6365 8.0000E-03 0.7206 2.0000E-02 0.8337 6.2600E-03 0.7226 4.9100E-02 0.8785 4.5700E-02 0.3678 7.4300E-02 0.1315 6.7800E-02
4 0.8837 2.3127E-02 0.7092 2.2400E-02 0.6881 8.8000E-03 0.8996 2.6300E-02 0.8199 6.5700E-02 0.4235 1.3000E-01 0.7005 3.1000E-02 0.6639 3.3900E-02
50.9360 1.2286E-02 0.7514 1.1900E-02 0.8106 1.9000E-02 0.9321 2.0600E-02 0.8197 5.7000E-02 0.6853 2.1820E-01 0.5919 1.3700E-02 0.6764 3.1200E-02
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Data availability
All data generated or analysed during this study are included in this published article71,72.
Received: 24 December 2022; Accepted: 29 May 2023
References
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Table 22. Comparison of the p-values obtained through the Wilcoxon signed-rank test between the pairs of
IHBO vs HBO, IHBO vs SSA, IHBO vs MFO, IHBO vs GWO, IHBO vs SCA, IHBO vs HS, and IHBO vs EMO
for tness results using Kapur’s method.
Test Image n HBO SSA MFO GWO SCA HS EMO
Test 1
2 1.413E-01 1.570E-01 1.600E-01 1.570E-01 9.640E-08 1.570E-01 1.570E-01
3 2.960E-03 3.290E-02 2.870E-01 2.390E-01 1.870E-13 2.970E-03 1.790E-15
4 7.495E-01 8.330E-01 4.570E-01 9.120E-01 4.340E-13 6.210E-07 1.170E-05
5 1.152E-02 1.280E-03 3.620E-02 6.830E-03 1.570E-12 1.250E-04 1.710E-05
Test 2
2 3.005E-01 3.340E-01 3.410E-01 3.430E-01 6.720E-06 3.380E-01 3.210E-01
3 1.988E-03 2.210E-02 7.540E-01 2.210E-02 8.330E-14 2.210E-02 7.920E-16
4 1.188E-01 1.320E-02 1.440E-01 2.930E-03 4.400E-13 1.480E-06 1.680E-04
5 2.411E-04 2.680E-04 8.390E-03 8.110E-09 6.480E-13 1.170E-07 2.630E-01
Test 3
2 3.770E-03 4.190E-02 4.180E-02 4.190E-02 1.870E-04 4.190E-02 4.220E-10
3 2.132E-02 2.370E-03 6.310E-01 1.780E-04 6.670E-13 1.810E-04 8.840E-04
4 1.134E-01 1.260E-02 5.300E-01 4.670E-07 8.680E-14 3.230E-14 1.010E+01
5 4.040E-04 4.490E-05 1.620E-01 4.570E-09 6.550E-13 7.260E-12 4.570E-05
Test 4
2NaN NaN NaN 1.630E-05 1.680E-05 NaN 1.120E-16
3 2.186E-02 2.430E-02 6.490E-01 5.870E-03 7.870E-12 5.920E-03 1.380E-15
4 5.191E-01 5.770E-01 1.640E-01 3.460E-04 4.700E-12 6.480E-06 3.080E-01
5 6.901E-03 7.670E-03 6.550E-06 1.170E-07 8.140E-11 5.510E-11 4.590E-11
Test 5
2 2.907E-01 3.231E-01 3.290E-01 4.520E-06 2.590E-07 4.120E-01 1.100E-16
3 2.402E-02 2.670E-02 3.620E-01 3.670E-05 5.720E-13 3.670E-05 1.400E-02
4 1.125E-07 1.250E-07 8.290E-05 1.910E-11 4.400E-10 7.400E-13 2.700E-09
5 7.171E-01 7.970E-01 1.960E-01 1.180E-02 5.140E-13 4.890E-05 4.310E-01
Test 6
2 1.825E-05 NaN 3.310E-01 NaN 3.290E-01 NaN 7.660E-15
3 5.299E-03 5.890E-03 2.340E-02 1.690E-01 1.510E-13 5.910E-03 1.470E-15
4 4.175E-01 4.640E-01 3.340E-02 9.190E-03 6.230E-13 1.480E-03 3.560E-03
5 5.578E-01 6.200E-01 4.500E-01 1.700E-03 5.720E-13 4.140E-08 2.300E-07
Test 7
2 1.988E-01 2.210E-02 2.210E-02 2.210E-02 2.120E-04 2.210E-02 2.210E-02
3 6.739E-03 7.490E-04 3.160E-03 3.470E-01 2.870E-13 7.490E-04 2.890E-15
4 5.290E-01 5.880E-01 2.990E-01 1.290E-01 1.850E-13 7.530E-04 7.270E-04
5 2.627E-01 2.920E-01 1.900E-02 4.790E-03 8.110E-13 1.340E-09 1.250E-07
Test 8
2 1.467E-01 1.630E-01 1.620E-01 1.620E-01 4.480E-05 1.620E-01 1.620E-01
3 1.359E-03 1.510E-03 1.510E-03 5.910E-03 2.650E-13 1.510E-03 1.190E-04
4 6.757E-01 7.510E-01 1.410E-02 9.850E-01 4.420E-13 4.580E-05 8.300E-01
5 2.564E-01 2.850E-01 2.760E-04 7.810E-01 9.130E-13 2.650E-12 1.120E-06
Test 9
2 5.560E-04 6.180E-04 4.200E-02 6.180E-04 1.640E-05 6.170E-04 6.170E-04
3 6.739E-04 7.490E-04 4.370E-01 3.100E-03 6.510E-13 7.500E-04 2.910E-15
4 5.236E-07 5.820E-07 6.130E-02 6.850E-07 5.950E-13 2.060E-09 5.100E-14
5 8.601E-01 9.560E-01 4.240E-04 3.700E-07 4.150E-13 4.240E-02 4.230E-02
Test 10
2NaN 4.240E-02 NaN NaN 2.380E-04 NaN NaN
3 3.842E-02 4.270E-02 1.650E-01 6.590E-01 1.900E-13 4.200E-02 3.910E-10
4 7.908E-01 8.790E-02 2.400E-04 1.370E-02 4.790E-13 1.480E-06 6.880E-14
5 3.239E-05 3.600E-06 4.600E-03 4.510E-09 1.890E-12 9.110E-12 1.600E-08
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Author contributions
E.H.H. performed the supervision, methodology, investigation, visualization; E.H.H. and G.M.M. develop the
soware; E.H.H.and I.A.I. and Y.M.W. participated in conceptualization, formal analysis, performed the experi-
ments and analyzed the results, and wrote the paper; I.A.I. and Y.M.W. performed the validation; G.M.M. wrote
the original dra, resources. All authors discussed the results and approved the nal paper.
Funding
Open access funding provided by e Science, Technology & Innovation Funding Authority (STDF) in coopera-
tion with e Egyptian Knowledge Bank (EKB).
Competing interests
e authors declare no competing interests.
Additional information
Correspondence and requests for materials should be addressed to E.H.H.
Reprints and permissions information is available at www.nature.com/reprints.
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