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Circuits, Systems, and Signal Processing
https://doi.org/10.1007/s00034-018-0993-3
A Water Cycle Algorithm-Based Multilevel Thresholding
System for Color Image Segmentation Using Masi Entropy
Pankaj Kandhway1·Ashish Kumar Bhandari1
Received: 23 June 2018 / Revised: 10 November 2018 / Accepted: 13 November 2018
© Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract
In this paper, a recently developed metaheuristic water cycle algorithm (WCA)
is coupled with Masi entropy (Masi-WCA) to perform color image segmentation
over the optimal threshold value selection process. Masi entropy gives the non-
extensive/additive information that exists in an image by a tunable entropic parameter.
The water cycle algorithm is a newly established population-based method which has
been employed to exploit an optimal value of weighing factors for enforcement of con-
straints on individual components. The idea behind WCA is grounded on thought of
water cycle and how streams and rivers flow downward toward the sea in the real world.
The key feature of this paper is to exploit the modern optimization techniques such
as water cycle algorithm, monarch butterfly optimization, grasshopper optimization
algorithm, bat algorithm, particle swarm optimization, and wind-driven optimization
for the color image segmentation purpose. In this paper, two objective (fitness) func-
tions are exploited which are Tsallis and Masi entropy for a fair comparison of the
proposed method. The proposed scheme is examined intensively regarding quality, and
a statistical graph is included to compare the outcomes of the proposed Masi-WCA
method against similar algorithms. Different to other recently developed optimization
algorithms used for color image multilevel thresholding operations, WCA presents
a better performance in terms of superior quality and fast convergence rate. Experi-
mental evidence encourages the use of WCA for multilevel thresholding with Masi
entropy, while it concludes that Tsallis entropy does not outperform over the proposed
scheme.
Keywords Multilevel thresholding ·Masi entropy ·Tsallis entropy ·Water cycle
algorithm ·Grasshopper optimization ·Monarch butterfly optimization ·Color
image segmentation
BAshish Kumar Bhandari
bhandari.iiitj@gmail.com
Extended author information available on the last page of the article
Circuits, Systems, and Signal Processing
1 Introduction
Color image segmentation is the process of splitting a digital image into distinct non-
overlapping homogenous regions according to the similarity in some image features
such as color, pattern, intensity value, regional statistics. Segmentation methods are
involved as a preprocessing step in various computer vision systems and pattern recog-
nition applications such as edge detection, medical imaging, object detection, video
surveillance, classification, industrial production, traffic control system and agricul-
ture fields [1,5,12,15,22]. Based on the general principle of image segmentation,
the taxonomy of different segmentation algorithms can be built which distinguishes
the following categories: thresholding methods, edge-based techniques, and region-
based techniques. Image segmentation using multilevel thresholding (MT) is one of
the leading methods, and thresholding methods play a vital role to accomplish image
segmentation. However, most of the techniques are based on the image histogram
due to its simplicity, effectiveness, easy to implement, and rapidness. Generally, the
thresholding approaches are categorized as bi-level and multilevel thresholding. The
easiest one is bi-level thresholding, but for daily-life color images and remote sensing
images [9], it does not give optimum results. Among all the remarkable thresholding
methods, thresholding based on information entropy theory is a fascinating subject.
Entropy-based image segmentation methodology has enticed the devotion of numer-
ous researchers [2,6,13,32,35] and is considered as one of the prominent global
thresholding methods.
Sezgin et al. [42] presented an analysis of image thresholding algorithms and
reported that due to the advancements in the information theory, entropy-based thresh-
olding methods have an excessive influence on image segmentation. Pun et al. [34]
introduced the concept of entropy-based thresholding for the image segmentation.
Afterward, Otsu et al. [3] proposed a method that uses the optimum threshold val-
ues by using maximum values of the inter-class variance of gray levels. Kittler and
Illingworth have presented a method that optimizes Bayes risk factor to perform
thresholding-based segmentation [21]. Kapur et al. [10] proposed a technique that
uses the concept of entropy maximization to determine the homogeneousness between
classes. A moment preservation principal-based image segmentation approach was
developed by Tsai [44] in 1985. In 2015, Bhandari et al. [11] utilized Tsallis entropy,
which reports that the information in the image can be treated as either additive or non-
additive. A maximum entropy method has been proposed on the basis of non-extensive
Tsallis entropy [11] whose entropic parameter qenables the Tsallis entropy to han-
dle the nonadditive information. However, this technique is analogous to maximum
entropy method developed by Kapur et al. [10].
In 2006, Sahoo et al. [39] proposed a two-dimensional Tsallis–Havrda–Charvat
entropy-based thresholding selection technique. Furthermore, Sahoo [40] has pro-
posed a method for thresholding based on Renyi entropy and this entropy can handle
the additive property using the tunable entropic parameter α[20,37]. The value of α
and qcan be varied to maximize the Renyi’s and Tsallis entropies, respectively. The
thresholding approaches have used optimum threshold values by using maximum val-
ues of the inter-class variance of gray levels, and based on moment preserving principle
[11], cross entropy [23], fuzzy approach [26], and Renyi entropy [39] have also been
Circuits, Systems, and Signal Processing
successfully used in the segmentation of images. Masi et al. [27] have introduced a
new entropic measure, which is based on the analysis of thermodynamic entropies that
utilizes the complete probability distribution for image segmentation. Masi entropy
has an entropic parameter r, when the value of ris taken to be 1, the Masi entropy [27,
29] reduces to the Shannon entropy [36] which is also known as Boltzmann–Gibbs
entropy.
The entropy-based thresholding can be designed as a non-convex multifaceted opti-
mization problem. If the multiple thresholds are treated as the spatial dimensions of
the metaheuristics, their parallelism can efficiently address the issue of computational
cost of segmented color images by using multilevel color image thresholding-based
segmentation. The fitness function of the metaheuristics technique is a standard for
choosing the optimum solution. In recent years, entropy is used as a fitness function for
the optimization techniques which has drawn the attention of numerous researchers.
In this contexture, several optimization algorithm-based thresholding methods have
been developed which use varieties of evolutionary techniques such as artificial bee
colony (ABC) motivated from the searching behavior of swarm of honey bees [7,10],
differential evolution [43], genetic algorithm (GA) which is motivated from the notion
of survival of the fittest given by Darwin [38], PSO inspired by the public behavior of
fish schooling or bird flocking [13,14], FA which is based on flashing phenomenon
of fireflies of tropical areas during summer season [19], WDO other optimization
methodology which is grounded on the atmospheric behavior of earth [4,6], electro-
magnetism optimization (EMO) which has been inspired by the attraction–repulsion
mechanism of the electromagnetism theory [30]. In the recent years, a modified PSO
[25]-based multilevel thresholding is applied for the segmentation of medical image
segmentation. Recently, a new color image segmentation method has been proposed
using entropy thresholding and bat algorithm [28]. The bat algorithm is based on the
echolocation of the bats.
In 2017, a new color image multilevel thresholding has been proposed by exploit-
ing backtracking search algorithm (BSA) for satellite images [17]. In favor of color
image multilevel thresholding, another new approach has been given [33] through the
modification of fuzzy entropy, and further the proposed modified entropic parameters
are optimized by Levy flight firefly algorithm to get more accurate results for satellite
image segmentation. On the other hand, a new gray-level co-occurrence matrix has
been introduced, first time as an objective function to get accurate multilevel thresh-
olding for multiband remote sensing images as well as natural color images [31].
Cuckoo search (CS) [6,8] and differential evolution (DE) [16] have been reported
to solve many multilevel color image segmentation problems for the estimation of
optimal threshold values.
Motivated by the successful results in the aforementioned literature, this paper
applies different multilevel thresholding approaches for challenging real-life image
segmentation problem by using water cycle algorithm (WCA) [18], grasshopper opti-
mization technique (GOA) [41], monarch butterfly optimization (MBO) [46], bat
algorithm (BAT) [47], particle swarm optimization (PSO) [24], and wind-driven opti-
mization (WDO) and examines their feasibility for color image thresholding. The
WCA, GOA, and MBO are the latest and unexploited optimization techniques for the
color image segmentation; these optimization algorithms are inspired by nature. This
Circuits, Systems, and Signal Processing
paper has been exploited by two objective functions such as Tsallis and Masi entropy
criterion for color image segmentation.
Rest of the paper is organized as follows: In Sect. 2, related works are presented
through the formulation of Tsallis entropy and different optimization algorithms such
as BAT, PSO, WDO, MBO, and GOA. Section 3presents a detail description of the
proposed Masi-WCA-based segmentation approach. Section 4reports the visual and
quantitative results of the proposed technique which are supported by ME, MSE,
PSNR, SSIM, FSIM, and entropy. Finally, Sect. 5concludes the paper by highlighting
the main contribution and future scope of the proposed work.
2 Related Works
2.1 Thresholding Criterion for Multilevel Thresholding
Let Idenote a test image with an extreme of Lgray levels {0,1…L −1} and dimension
of the image be M×N.LetG={0,1…L −1} designates the set of intensity values of
the image. The count of pixels with gray level iis denoted by ni, and the dimension
of the image represents a total number of pixels. The pixels of a grayscale or colored
image are classified into regions or sets on account of their intensity level (L). This
system of arranging the pixels defines the thresholding. For the selection of fitting
neighborhoods in a test image, the optimum threshold value (th) should be obtained
in a routine, which obeys the simple law of following equations:
C0←iif 0 ≤i≤th
C1←iif th + 1 ≤i≤L−1(1)
where irepresents the intensity values of the grayscale image with L, which brings
the maximum intensity level. Crepresents the class of the image.
C0←iif 0 ≤i<th1
C1←iif th1+1≤i<th2
C2←iif th2+1≤i<th3
Cn←iif thn+1≤i<L−1
(2)
where {th1,th
2,th
3,…, thn} represents multiple thresholds.
Segmentation of pixels in their respective classes is done using Eqs. (1) and (2)for
bi-level and multilevel thresholding, respectively. In the image, the probability of gray
level iis estimated by the number of pixels representing the intensity level (frequency
of gray level i) occurred in the image, given by Eq. (3):
hini
M×Nwhere,hi≥0 and
L−1
0
hi1.(3)
Circuits, Systems, and Signal Processing
The complete probabilistic distribution Hof gray levels can be formulated as
H={ h0,h1, h2…hL−1}. The pixels in the image are separated into two classes (which
is bi-level segmentation) as C0and C1given by Eq. (1), and the pixels in the image are
divided into more than two classes (which is multilevel segmentation) as C0,C1,…,
Cnis given by Eq. (2), Each of the C0,C1,C2, and Cncorresponds to the different
object class and background class. Now, the probability of the classes defined for bi-
level can be extended for multilevel thresholding, which is formulated by following
the equations:
For bi-level thresholding,
w0
th
i0
hi,w
1
L−1
ith+1
hi.(4)
For multilevel thresholding,
w0
th1
i0
hi,w
1
th2
ith1+1
hi,w
2
th3
ith2+1
hi,..., w
n
L−1
ithn
hi.(5)
The above-defined probability distributions are further normalized. Consequently,
a vector of optimal thresholds {T∗
1,T∗
2,...,T∗
n}is determined using:
{T∗
1,T∗
2,...,T∗
n}arg max{fit(T1,T2,...,Tn)}
Subject to 0 ≤T0<T1<···<Tn≤L−1.(6)
where fit (T1,T2,…,Tn) represents the optimization criterion or objective function,
which determines the optimum thresholds for performing multilevel thresholding. Two
objective functions used in this paper to compute the optimum threshold values have
been discussed in this section.
2.2 Tsallis Thresholding Method
Albuquerque et al. proposed a concept based on Tsallis entropy [11,39]. The function
of thresholding is represented by Eqs. (7) and (8), and the concept of Tsallis entropy
has been proposed by Constantino Tsallis. Discrete probabilities make up the notion
of Tsallis entropy where the sum of all discrete probabilities is 1. Tsallis entropy
is capable of handling nonadditive or non-extensive information [2]. This degree of
nonadditivity is represented by the variable qwhich behaves as an entropic parameter.
When the value of qis taken to be 1, the Tsallis entropy reduces to the Shannon entropy
which implies that the Tsallis entropy is derived from Shannon entropy. In this paper,
the value of qis considered as 0.8 to maximize the Tsallis entropy.
Eq(I/th)Eq(C0/th)+Eq(C1/th)+(1−q)Eq(C0/th)Eq(C1/th)(7)
Circuits, Systems, and Signal Processing
where
Eq(C0/th)1
1−qth
i0hi
woq
−1
Eq(C1/th)1
1−qL−1
ith+1 hi
w1q
−1(8)
The maximized Tsallis entropy can be achieved by using Eq. (9) which can be
presented as
ETET0+ET1+ET2+···+ETn+(1−q)ETnET1ET3...ETn(9)
where
ET01
1−qth1
i0hi
w0q
−1,w
0
th1
i0
hi
ET11
1−q⎡
⎣
th2
ith1+1 hi
w1q
−1⎤
⎦,w
1
th2
ith1+1
hi
ET21
1−q⎡
⎣
th3
ith2+1 hi
w2q
−1⎤
⎦,w
2
th3
ith2+1
hi
ETn1
1−q⎡
⎣
L−1
ithn+1 hi
wnq
−1⎤
⎦,w
n
L−1
ithn+1
hi(10)
where ETirepresents the Tsallis entropy of ith class and the optimal multilevel seg-
mentation problem is solved by assuming n-dimensional problem of optimization. The
values of Eq. (10) give the Tsallis entropy of each region (or class), and these values
areusedinEq.(9) to get the maximum entropic value. Now, to solve multilevel thresh-
olding problem, n-dimensional optimal thresholds are obtained by Eq. (11), which is
used for the maximization of objective function:
(T∗)arg maxn
i0
ETi(11)
2.3 Bat Algorithm
Xin-She Yang proposed the bat algorithm which is used as a metaheuristic approach
for optimization at a global scale. This optimization method is stimulated from the
echolocation of micro-bats. The bats use the notion of SONAR echoes to detect their
prey and avoid obstacles. The bats transmit the sound waves in the presence of an
Circuits, Systems, and Signal Processing
object; these waves are reflected back. The time period between the reflection and
transmission of the wave impacts the movement of the bats. After the reception of
reflected wave, bats use their own pulse to determine the space between them and the
prey. The pulse rate ranges from 0 till 1, where 1 represents emission at maximum
level and 0 indicates no emission. The loudness of the sound wave and the distance of
bat from prey are proportional to each other. In other words, the loudness and pulse
rate are inversely proportional to one another [47].
2.4 Particle Swarm Optimization
Particle swarm optimization (PSO) improves the candidate solutions iteratively to
get the optimized solution for a problem. It has a population of dubbed particles (or
candidate solutions) where these particles are moved in and around in the search space
in accordance with few simple formulae which includes the position of the particle and
particle’s velocity. The movement of each particle is judged by its best-known localized
position which is updated when other particles find better positions. This gives high
expectation that the swarm is moving toward the optimum solutions. In PSO, search
space (possible set of solutions) and possible solution are called as particle position
and swarm, respectively. The position representing the best fitness is given as ‘Pbest,’
and ‘Gbest’ defines the best solution of all the particles [24].
2.5 Wind-Driven Optimization
WDO is motivated from the atmosphere of earth, where blowing wind attempts to
balance the horizontal air pressure. It is a nature-inspired global optimization method
that is created on the ideology of atmospheric motion [4]. It has been shown that wind-
driven optimization can be executed easily and is effective in solving the optimization
problems. In general terms, WDO has an ability to put in effect the constraints in the
search domain. This method is operational on the population-based recursive heuristic
global optimization algorithm for multimodal along with multidimensional challenges.
2.6 Monarch Butterfly Optimization
Monarch butterfly optimization (MBO) is a new type of metaheuristic algorithm and
inspired from nature; all these butterflies individually are placed in two distinct lands
(areas). In the paper [46], the locations of the monarch butterflies are modernized
in two techniques. Initially, the offsprings are produced or position-modernized by
migration operator and it is adjusted by the migration ratio. This migration behavior
of monarch butterflies addresses numerous optimization problems, and it is monitored
by some rules. Subsequently, the location of the butterflies is changed by the worth of
butterfly adjusting operation [46].
Circuits, Systems, and Signal Processing
2.7 Grasshopper Optimization Algorithm
Saremi proposed grasshopper optimization algorithm in 2016 which imitates the
swarming behavior of grasshoppers. Three components affect the flying route of
grasshopper in a swarm. They are a social relationship, gravity, and the horizontal
movement of wind. In the GOA algorithm, the most important searching mechanism
is a social relationship and the swarming behavior changes significantly when the
parameters are changed. The authors have proposed the mathematical model search
for grasshopper’s interaction and move the swarm closer to the target. In the GOA
algorithm, it is estimated that the target is the best solution. While the grasshoppers
interact and chase the target, the best solution gets updated if a better solution is found
[41].
3 The Proposed Method
In this section, the proposed Masi-WCA approach is demonstrated. Alfred Renyi
proposed a definition for the measure of information that preserves expansively for
independent events which later termed as Renyi entropy. This entropy quantifies the
randomness, uncertainty, or diversity of a system. Renyi entropy is used as a diversity
index in statistics and ecology and also essential in quantum information where it is
used to measure entanglement [20,40]. The entropic parameter αdefines the amount
of extensive information that is present in the image. In a system, the value of α
determines which events contribute in the calculation of Renyi entropy. For example,
if the value of αtends to 0, almost all the events are weighted equally by Renyi entropy
and if a value of αtends to infinity, Renyi entropy is evaluated using the events that
have the highest probability. Similarly, in the case of image processing, the events
represent the classes of the image whose probabilities are used to determine Renyi
entropy. When the value of αis closer to zero, regardless of the probability of each
class, Renyi entropy weighs all possible events more equally and when αis one, Renyi
entropy reduces to Shannon entropy. This entropy yields maximum result when the
value of αis taken to be 0.8.
The property of Tsallis entropy is examined when considering two systems with
different temperatures to be in contact with each other and to reach the thermal
equilibrium. It is verified that the total Tsallis entropy of the two systems cannot
decrease after the contact of the systems. It leads to a generalization of the principle of
entropy increase in the framework of non-extensive statistical mechanics. Therefore,
a maximum entropy method was proposed based on non-extensive Tsallis entropy
[11]. Tsallis entropy is the generalization of Shannon entropy. The pseudo-additivity
property of Tsallis entropy with entropic parameter qcan handle the non-extensive
information for statistically independent subsystems. Sahoo et al. [40] has proposed
a method for thresholding based on Renyi entropy. Renyi entropy can handle the
additive property using the tunable entropic parameter α[20,40]. However, Renyi’s
and Tsallis entropies cannot handle the additive and nonadditive information simul-
taneously. In the subsequent part, the illustration of the proposed scheme’s concept
Circuits, Systems, and Signal Processing
is done using flowchart in Fig. 1. A complete explanation of each step is presented
thereafter.
3.1 Masi’s Thresholding Method
Masi entropy combines the additivity of Renyi entropy and the non-extensivity of
Tsallis entropy. The main argument which deviates Renyi and Tsallis entropies from
the Masi entropy is the concordant parameter r. Unlike probability functions of Renyi
entropy and Tsallis entropy, where each state probability is raised to the power of
their entropic parameters αand q, respectively, in case of Masi, the entire probability
function is raised to the power r[27,29,36]. The parameter rrepresents the measure
of the degree of extensivity/non-extensivity that might be existent in the system. The
entropy-based thresholding methodology is developed on the entropic measure, which
is further presented by Masi [27] for gray-level images [29]. Successively, all the
entropies directly or indirectly are the generalization of well-established Shannon
entropy. The entropic parameter gives the flexibility to achieve different results as the
demand entertains. According to the concept of Masi entropy, to obtain an optimal
threshold value th, for the bi-level thresholding-based image segmentation is expressed
by Eqs. (12) and (13):
Er(I/th)Er(C0/th)+Er(C1/th)(12)
where
Er(C0/th)1
1−rlog1−(1 −r)
th
i0hi
w0loghi
w0,
Er(C1/th)1
1−rlog1−(1 −r)
L−1
ith+1 hi
w1loghi
w1(13)
The entropy between the two classes C0and C1is maximized, and the gray level at
which this holds true is treated to be the optimal threshold. The complete procedure of
the optimal multilevel color image thresholding task is addressed. Basically, entropy is
defined as a measurement of randomness, which follows the concept that homogeneous
regions will have minimal unpredictability and the non-homogeneous regions will have
maximum unpredictability. The higher the value of entropy, the better is the separation
between objects and background. The information which exists in pixels of an image
has either the additive property or the nonadditive property. In this paper, the value of
ris considered as 1.18 to maximize the Masi entropy.
The Masi entropy algorithm is selected for several thresholds that are multilevel
thresholding (MT) by maximizing the Masi entropy, and the maximized Masi entropy
can be achieved by using Eq. (14) which can be presented as
ErT ErT
0+ErT
1+ErT
2+···+ErT
n,(14)
Circuits, Systems, and Signal Processing
Fig. 1 Flowchart of the
Masi-WCA approach for the
multilevel (thresholding)
segmentation
Yes
Yes
Yes
No
Start
Select the Initial parameters
Determine the cost of each raindrop using Eq. (19)
Create initial population that is random in
nature by Eqs. (18), (20), and (21)
Obtain the intensity of the flow using Eq. (22)
Get the stream flow to rivers by Eq.(24)
Get the rivers flow to sea by Eq.(25)
If the stream has lower function
(objective function for stream)
value than corresponding river
Interchange the locations (positions) of
stream with the corresponding river
If the river has lower function
(objective function for river)
value than sea
Interchange the locations (positions) of
river with the sea
Generate cloud and start raining
process using Eqs. (27) and (28)
Evaporation condition fulfilled?
Reduce the value of the dmax using Eq. (26)
Testing the convergence norms
Exit
Yes
No
No
No
Compute the histogram of the input image
Get the best (optimal) solution and segmented
the input image using optimal solution
Circuits, Systems, and Signal Processing
where
ErT
01
1−rlog1−(1 −r)
th1
i0hi
w0loghi
w0,w
0
th1
i0
hi
ErT
11
1−rlog⎡
⎣1−(1 −r)
th2
ith1+1 hi
w1loghi
w1⎤
⎦,w
1
th2
ith1+1
hi
ErT
21
1−rlog⎡
⎣1−(1 −r)
th3
ith2+1 hi
w2loghi
w2⎤
⎦,w
2
th3
ith2+1
hi
ErT
n1
1−rlog⎡
⎣1−(1 −r)
L−1
ithn+1 hi
wnloghi
wn⎤
⎦,w
n
L−1
ithn+1
hi(15)
where ErT
irepresents the Tsallis entropy of ith class and the optimal multilevel seg-
mentation problem is solved by assuming n-dimensional problem of optimization. The
values of Eq. (15) give the Masi entropy of each region (or class), and these values are
used in Eq. (14) to get the maximum entropic value. Now, to solve multilevel thresh-
olding problem, n-dimensional optimal thresholds are obtained by Eq. (16), which is
used for the maximization of objective function:
(T∗)arg maxn
i0
ErT
i.(16)
3.2 Water Cycle Algorithm
Water cycle algorithm (WCA) [18] is a recently proposed metaheuristic technique
for optimizing constrained functions and is used for engineering problems. The key
objective of WCA is to introduce a new global optimization approach for finding the
constrained optimization complications. Hence, a new population-based algorithm is
proposed named as the water cycle algorithm (WCA) method and the idea behind
the WCA is grounded on a thought of water cycle and how streams and rivers flow
downward toward the sea in the real world. WCA is motivated by nature, and it includes
four important sub-parts, e.g., create the initial population, a stream flow to the rivers
or sea, evaporation condition, and raining process.
3.2.1 Create the Initial Population
For solving an optimization problem via population-based metaheuristic techniques,
the values of problem variables are designed as an array. This array is known as
‘Raindrop’ for a single solution. For a Mvar dimensional optimization problem, the
Circuits, Systems, and Signal Processing
values are represented as an array for a raindrop of 1×Mvar. This array is expressed
in Eq. (17).
Raindrop [y1,y2,y3,...,yM].(17)
For the optimization approach, a matrix of raindrops of size Mpop ×Mvar is created
that is represented as the population of raindrops. Therefore, the matrix Yis produced
randomly, where rows and column are given as the population size (Mpop) and the
design variable size (Mvar), respectively.
Population of raindrops ⎡
⎢
⎢
⎢
⎣
raindrop1
raindrop2
.
.
.
raindropMpop
⎤
⎥
⎥
⎥
⎦⎡
⎢
⎢
⎢
⎢
⎣
y1
1y1
2... y1
Mvar
y2
1y2
2... y2
Mvar
.
.
..
.
..
.
..
.
.
yMpop
1yMpop
2... yMpop
Mvar
⎤
⎥
⎥
⎥
⎥
⎦
.(18)
The values of each decision variable (y1,y2,...,yMvar ) are signified as floating
point number, that is, real values or a predefined set for discrete and continuous prob-
lems, respectively. A raindrop cost is achieved by the calculation of cost function (C)
expressed in Eq. (19).
CjCost jfxj
1,xj
2,...,xj
Nvar j1,2,3,...,Mpop.(19)
The best individuals are chosen from rivers and sea that gives a number of Msr,
and the raindrops contained minimum value among others that is considered as a sea,
where Msr is the summation of user parameter, that is, the number of rivers and a
single sea. The remaining population, that is, raindrops from the streams which flow
to the rivers or may directly flow to the sea, is obtained and represented in Eq. (21).
Msr Number of Rivers + 1
Sea
(20)
MRaindrops Mpop −Msr.(21)
The assigned raindrops to the sea and rivers depend on the flow intensity and are
expressed in Eq. (22).
NSnround
Costm
Msr
j1Cost j×MRaindrops,m1,2,...,Msr ,(22)
where NSnrepresents the number of streams which flow to the specific rivers or sea.
3.2.2 A Stream Flow to the Rivers or Sea
The streams are formed by the raindrops, and these streams are connected to each
other to form new rivers. Some of the streams may also flow directly to the sea, and
Circuits, Systems, and Signal Processing
all streams and rivers finish in a sea that gives best optimal solution (point). The
connecting line of a stream flows to river and uses randomly chosen distance, which
is defined as follows in Eq. (23).
Y∈(0,C×d),C>1,(23)
where dis represented as the current distance between stream and river. The value of
Yin Eq. (23) corresponds to a distributed random number between 0 and (C×d). A
value of Cis between 1 and 2; for the best solution, it should be 2 or near to 2. The
Cvalue greater than 1 enables streams to flow in altered directions toward the rivers.
Hence, the new position for rivers and streams is defined as:
Yj+1
Stream Yj
Stream + rand ×C×Yj
River −Yj
Stream(24)
Yj+1
River Yj
River + rand ×C×Yj
Sea −Yj
River,(25)
where the value of the function rand is a generated random number between 0 and 1
that is uniformly distributed. The solution of the stream is better than its connecting
river, the locations of stream and river are swapped, and this criterion is also applied
for rivers and sea.
3.2.3 Evaporation Condition
The most important factor of the WCA is evaporation that prevents the method from
rapid convergence or the immature convergence. The evaporated (vaporized) water
is carried into the atmosphere to produce clouds and then condenses in the colder
atmosphere, liberating the water back to earth in the form of rain. This rain generates
the new streams and follows the conditions which have been mentioned above. The
complete process or cycle is called water cycle. The pseudocode gives an idea about
how to define whether or not river flows to the sea.
if Yj
Sea −Yj
River<dmax
j1,2,3, ...,Msr −1
Evaporation and raining process
end
where the value of dmax is close to zero (small number). The distance between sea and
a river is less than dmax. This condition specifies that the river has linked the sea, and
the evaporation and raining process are applied. Otherwise, it reduces the search and
a small value inspires the search intensity near the sea. The value of dmax adaptively
decreases as:
dj+1
max dj
max −dj
max
max iteration.(26)
Circuits, Systems, and Signal Processing
3.2.4 Raining Process
In this part, the new raindrops formed streams in the different positions or location.
The new positions of the recently formed streams can be found exactly and clearly
with the help of the following equation.
Ynew
Stream LB + rand ×(UB −LB),(27)
where LB and UB are represented as lower and upper bounds, respectively. Equa-
tion (29) is described only for streams which directly flow to the sea, and the core idea
of this equation is to inspire the generation of the stream to improve the exploration
near the sea in the possible state for difficulties.
Ynew
stream Ysea +√μ×randn(1,Mvar ),(28)
where μrepresents a coefficient which gives the range of searching states (region)
near the sea, randndefines a random number which is normally distributed, and the
value for µis set to 0.1. From this equation, the created individuals with variance µ
are distributed around the best-obtained optimum point.
The detailed steps of the proposed Masi-based color image multilevel thresholding
operation are described, and the flowchart of the pseudocode of Masi-WCA method
is shown in Fig. 1.
Step 1 An input image Iis taken, if Iis a color image then it is separated into three
bands (Red–Green–Blue) and grayscale image is directly used.
Step 2 Compute the histogram of each band of the color image.
Step 3 Allocate the control parameters of WCA such as population size (Mpop), number
of design variable (Mvar), number of iterations (stopping criterion), objective function,
threshold levels.
Step 4 Generate the optimum thresholds by maximizing an objective function (or
minimizing the negative of the entropy value) following the below pseudocode of
WCA:
Circuits, Systems, and Signal Processing
Step 5 The search intensity near the sea (the optimum solution) is used. The current
best solution (optimum solution) for each of the color channels represents the set of
optimal threshold values (TR,TG, and TB,) with best maximum objective function
value.
Step 6 Each color channel is segmented individually using the corresponding threshold
values. The segmented color channels are then concatenated to form the segmented
color image.
4 Experimental Results and Discussion
In this section, experiment results have been discussed to evaluate the performance of
the proposed Masi-WCA method over other methods for multilevel thresholding of
the color images. The segmented results have been evaluated over 10 daily-life color
images, and each color image is a multidimensional image with multimodal nature
due to the presence of different bands (RGB). Input images and histogram plots of
each band are shown in Fig. 2. Moreover, the presence of dense and complex features
requires a sophisticated and accurate multilevel thresholding algorithm for the detec-
tion and identification of the region of interest. Each of the test images is segmented
into four different thresholding levels: 3-level, 5-level, 8-level, and 12-level thresh-
olding to achieve segmentation. All the algorithms are implemented using MATLAB
R2017a on a personal computer with 3.4 GHz Intel Core-i7 CPU, 8 GB RAM running
on Windows 10 system. Each of the test images is independently run 50 times using
Circuits, Systems, and Signal Processing
Fig. 2 Test color images (IMG1, IMG2, IMG3, IMG4, IMG5, IMG6, IMG7 IMG8, IMG9, and IMG10) and
corresponding histogram plot of each frame (R–G–B) of the color images (Color figure online)
all algorithms to avoid any stochastic discrepancy due to the random nature of the
optimization algorithm. For all the optimization algorithms, the population size and
the number of iterations are set as 15 and 200 to provide fairness and convenience for
performance comparison between WCA and other compared optimization algorithms.
4.1 Image Quality Measures
The objective function value depends on the mathematical modeling of the objective
function and the architecture and search strategy of the optimization algorithms. The
objective function value indicates toward the best or worst segmentation quality of
the algorithm. The computation time of an algorithm depends on the complexity of
the method. The complexity of the method depends upon the mathematical structure
of its objective function and the structure of the optimization algorithm. Therefore,
to test the efficiency of any algorithm, computation time is an important parameter to
be computed. The time taken by any technique to perform the segmented output is
directly proportional to the complexity of that algorithm. The computation time also
increases as the number of thresholding level increases.
Circuits, Systems, and Signal Processing
Table 1 Fidelity parameters considered to test the efficiency of the proposed method with other algorithms
S. No. Parameters Formula Remarks
1. Misclassification error
(ME) [29]
ME 1−|BO∩BT|+|FO∩FT|
|BO|+|FO|ME relates to the wrong
assignment of
foreground pixels to the
background or
background pixels to
the foreground
2. Mean square error
(MSE) [10]
MSE M
i1N
j1I(i,j)−I(i,j)2
MN Calculates the difference
between expected value
and the actual value
3. Peak signal-to-noise
ratio (PSNR) [11]
PSNR(in dB) 20 log10255/√MSEIt is the ratio of maximum
power of a signal to the
power of noise
4. Structural similarity
index (SSIM) [45]
SSIM(I,I)(2μIμI+l1)(2σII+l2)
(μ2
I+μ2
I+l1)(σ2
I+σ2
I+l2)Evaluates the similarity
between the segmented
image and the original
image
5. Feature similarity index
(FSIM) [48]
FSIM N
c1SD(c)PCmax(c)
N
c1PCmax(c)Calculates the feature
similarity of segmented
image and the original
image
6. Entropy [29]EntropyiPilog2PiIndicates the average
information of an image
To provide a comprehensive performance assessment of each algorithm, different
fidelity parameters such as best objective function values, computation time (in sec-
onds), misclassification error (ME), peak signal-to-noise ratio (PSNR), mean square
error (MSE), feature similarity index (FSIM), structural similarity index (SSIM), and
entropy are included in Table 1. A comprehensive discussion of the experimental
results has been presented in this section. Therefore, the computed segmented results
for each test image using Tsallis and Masi entropies as fitness function have been eval-
uated using BAT, PSO, WDO, MBO, GOA, and WCA. Keeping the view of Tables 2,
3,4,5,6,7,8, and 9, Masi-WCA has outperformed in comparison with other optimiza-
tion techniques (BAT, PSO, WDO, MBO, and GOA) as well as Tsallis entropy-based
fitness function.
4.2 Experiment 1: Tsallis Entropy
The results achieved for each test image by using Tsallis entropy as an objective (or
fitness) function are discussed and evaluated in this section. In the case of Tsallis
entropy, it is easy to confirm which optimization technique has produced superior per-
formance due to an optimal value of ME, MSE, PSNR, SSIM, FSIM, and entropy for
the maximum number of cases. The metaheuristic methods are stochastic in behaviors
Circuits, Systems, and Signal Processing
Table 2 Comparison of best objective values (entropy) and ME computed by different algorithms using Tsalli entropy
Test Images mBest objective values ME
BAT PSO WDO MBO GOA WCA BAT PSO WDO MBO GOA WCA
IMG1 3 23.6625 24.0533 23.9536 23.6781 24.0617 24.0600 0.3524 0.4162 0.5643 0.4184 0.4184 0.4184
5 30.8518 31.0922 31.1790 30.8247 31.2057 31.1844 0.2458 0.2313 0.3681 0.1495 0.2607 0.1595
8 38.5848 39.7181 39.0562 38.1332 39.7252 39.9561 0.1024 0.1436 0.1555 0.1630 0.0952 0.0932
12 47.4674 48.4684 49.2492 47.3112 49.4166 49.8127 0.0854 0.0683 0.1506 0.1355 0.0828 0.0827
IMG2 3 22.4017 22.6702 22.6698 22.5428 22.6706 22.6705 0.2954 0.2771 0.2731 0.2229 0.2749 0.2733
5 28.8023 29.0605 29.1217 28.6063 29.1560 29.1555 0.1547 0.1499 0.1913 0.1410 0.1508 0.1508
8 35.4117 36.6402 37.2726 36.0717 37.3011 37.1190 0.1245 0.0857 0.1045 0.0657 0.1025 0.0549
12 42.8853 45.0334 45.6237 43.0182 45.6693 45.8490 0.0824 0.0309 0.0190 0.0153 0.0110 0.0090
IMG3 3 23.5911 23.9655 23.9135 23.6810 23.9882 23.8882 0.2845 0.3254 0.3061 0.2515 0.2175 0.2175
5 30.3331 31.1887 31.3589 31.0935 31.3577 31.3164 0.1859 0.1750 0.1496 0.1439 0.1610 0.1368
8 38.0343 39.7592 40.3236 39.4158 40.3273 40.2197 0.0842 0.0984 0.1063 0.0904 0.0917 0.0710
12 47.4535 48.6091 49.4703 47.4568 49.5093 49.2301 0.0524 0.0332 0.0954 0.0597 0.0604 0.0332
IMG4 3 21.9057 22.0597 22.0894 21.8900 22.9145 22.0783 0.3148 0.3103 0.2986 0.2712 0.4393 0.3340
5 27.9494 28.5781 28.5787 28.4820 28.6246 28.5626 0.2497 0.1875 0.2657 0.1612 0.3413 0.0609
8 35.0129 36.4026 36.8213 35.7984 36.8392 37.0327 0.1145 0.0879 0.1336 0.0602 0.0455 0.0401
12 43.2593 44.1975 45.2234 42.3860 45.7143 45.3410 0.0345 0.0060 0.0193 0.0213 0.0361 0.0162
IMG5 3 23.1362 23.3392 23.3464 23.1079 23.3465 23.3459 0.6248 0.6221 0.6278 0.6603 0.6231 0.6201
5 28.7548 30.2588 30.2993 29.8523 30.2997 30.2850 0.4912 0.4819 0.4818 0.4549 0.4711 0.4438
8 36.3362 38.1366 38.7655 37.9012 38.8626 38.5558 0.3586 0.3438 0.1161 0.3101 0.0311 0.0303
12 43.7241 46.5987 47.3371 44.9737 48.0974 48.2165 0.0957 0.1399 0.2230 0.0500 0.0278 0.0278
Circuits, Systems, and Signal Processing
Table 2 continued
Test Images mBest objective values ME
BAT PSO WDO MBO GOA WCA BAT PSO WDO MBO GOA WCA
IMG6 3 24.1001 24.3211 24.3169 24.1788 24.3219 24.3217 0.5472 0.5830 0.5625 0.5484 0.5858 0.5843
5 31.1547 31.9538 31.9820 31.4907 32.0148 31.9846 0.3254 0.4799 0.5117 0.5147 0.4984 0.4974
8 40.0675 41.1692 41.2524 40.7387 41.4379 41.2775 0.3885 0.4184 0.4442 0.4383 0.4295 0.4209
12 49.4834 50.4927 51.1964 48.9085 51.2882 51.0154 0.2568 0.3315 0.4337 0.3829 0.4044 0.3976
IMG7 3 23.3164 23.4230 23.4276 23.0121 23.4276 23.4276 0.6154 0.5047 0.5009 0.6136 0.4978 0.4978
5 29.9767 30.2605 30.2963 29.9847 30.2986 30.2841 0.3824 0.3555 0.3672 0.3155 0.3537 0.3530
8 37.4742 38.0977 38.4448 36.9008 38.5395 38.3252 0.2659 0.2503 0.3129 0.3515 0.2536 0.2291
12 41.1657 45.8514 47.0507 41.1935 47.1958 46.8605 0.1607 0.1737 0.1995 0.2498 0.1882 0.1391
IMG8 3 23.5163 24.0351 23.8610 23.6665 24.0208 24.0520 0.0425 0.0291 0.0442 0.0299 0.0238 0.0308
5 30.1625 31.4701 31.4361 31.0079 31.4722 31.5756 0.0249 0.0101 0.0149 0.0120 0.0184 0.0119
8 38.6747 39.7974 40.2670 38.9761 40.4634 40.4327 0.0092 0.0053 0.0071 0.0094 0.0060 0.0048
12 48.0959 48.8381 49.6706 46.8829 49.6887 49.6197 0.0056 0.0019 0.0046 0.0059 0.0051 0.0023
IMG9 3 25.0622 25.1968 25.1974 25.0745 25.1998 25.1990 0.3627 0.3313 0.3382 0.1342 0.3172 0.3118
5 32.2101 32.8962 32.9332 32.4852 32.9338 32.9339 0.2004 0.1223 0.1164 0.0941 0.1111 0.1111
8 40.4392 41.8573 42.1557 41.5616 42.2111 42.1812 0.1027 0.0913 0.0835 0.0875 0.0784 0.0740
12 49.9727 50.9584 51.7581 50.1221 52.1058 51.9546 0.9624 0.0466 0.0666 0.0449 0.0402 0.0385
IMG10 3 24.1035 24.3549 24.2254 24.2075 24.3568 24.3566 0.3499 0.3455 0.2054 0.4106 0.3476 0.3476
5 30.8900 31.4720 31.5701 31.1903 31.5722 31.4965 0.2498 0.1517 0.1552 0.1582 0.1684 0.1486
8 39.2608 39.9224 40.2683 39.2633 40.4068 40.1706 0.0998 0.0950 0.0732 0.0282 0.0708 0.0259
12 48.1145 48.3231 49.6991 46.7666 49.7289 49.7500 0.0339 0.0339 0.0321 0.0289 0.0346 0.0214
Circuits, Systems, and Signal Processing
Table 3 Comparison of best objective values (entropy) and ME computed by different algorithms using Masi entropy
Test Images mBest objective values ME
BAT PSO WDO MBO GOA WCA BAT PSO WDO MBO GOA WCA
IMG1 3 26.6810 27.4789 27.2242 27.1595 27.2242 27.4859 0.3456 0.4148 0.4240 0.4245 0.4167 0.4167
5 33.6287 33.9236 33.9193 33.3791 33.9986 33.7726 0.2393 0.1384 0.2867 0.1929 0.1532 0.1384
8 34.4078 41.9836 42.3678 41.4078 42.3893 42.1377 0.0882 0.1111 0.1475 0.1086 0.1018 0.0873
12 47.9780 50.5366 51.2376 50.0426 51.3765 52.5470 0.0598 0.0904 0.1413 0.0853 0.0871 0.0537
IMG2 3 25.1811 25.3319 25.3338 25.2388 25.3337 25.3339 0.2709 0.2753 0.2664 0.2671 0.2741 0.2608
5 30.1201 31.3222 31.3330 31.1011 31.3620 31.3663 0.1444 0.1467 0.1499 0.1691 0.1446 0.1405
8 38.2523 38.5286 38.8355 37.9862 38.9740 38.8282 0.1044 0.0496 0.0686 0.0219 0.0499 0.0179
12 44.2169 45.9221 46.9592 45.3514 46.4144 47.2531 0.0146 0.0438 0.0356 0.0164 0.0417 0.0092
IMG3 3 27.0690 27.2621 27.3025 27.2844 27.1133 27.3035 0.2691 0.2629 0.2513 0.2628 0.2195 0.2190
5 33.1745 34.0943 33.8430 33.8468 34.2392 34.1157 0.1275 0.1245 0.1294 0.1185 0.1219 0.1132
8 41.8906 42.2943 42.1705 41.7210 42.7027 42.7085 0.0929 0.1075 0.1015 0.0883 0.0981 0.0867
12 48.7221 50.3378 51.1508 49.0052 51.1461 51.0335 0.0720 0.0503 0.0734 0.0282 0.0659 0.0272
IMG4 3 24.0474 24.3120 24.3200 24.3103 24.3594 24.3195 0.2865 0.2896 0.2928 0.2848 0.3348 0.2803
5 29.5679 30.5902 30.7264 30.5064 30.8314 30.6697 0.1986 0.1598 0.1848 0.1278 0.2634 0.1205
8 37.2637 38.0770 37.6447 37.9510 38.3390 38.4531 0.0551 0.0358 0.0627 0.0335 0.0346 0.0332
12 43.5186 45.8086 46.6270 44.4961 46.5681 45.7637 0.0168 0.0194 0.0176 0.0168 0.0201 0.0123
IMG5 3 26.1574 26.2372 26.2482 26.2123 26.2519 26.2589 0.6598 0.6205 0.6224 0.6071 0.6259 0.6201
5 31.6403 32.6291 32.7057 32.6483 32.6981 32.7331 0.2726 0.2500 0.4602 0.4260 0.2418 0.2402
8 39.2784 40.4329 40.6107 40.2742 40.8125 40.7968 0.0450 0.2545 0.1204 0.1693 0.1377 0.0363
12 47.0336 48.5701 48.9930 47.9102 49.5631 49.4614 0.0149 0.0036 0.2367 0.0383 0.0003 0.0002
Circuits, Systems, and Signal Processing
Table 3 continued
Test Images mBest objective values ME
BAT PSO WDO MBO GOA WCA BAT PSO WDO MBO GOA WCA
IMG6 3 27.4615 28.0444 28.0507 27.9091 28.0509 28.0516 0.5429 0.5450 0.5419 0.5490 0.5414 0.5385
5 34.6827 35.1996 35.1593 34.8721 35.1654 35.1626 0.4845 0.4523 0.4557 0.4684 0.4474 0.4474
8 40.3257 43.7379 44.0565 43.4377 44.1106 44.0160 0.4249 0.3986 0.4378 0.4362 0.4177 0.4162
12 52.2337 52.7060 53.1992 51.9110 53.2519 53.2351 0.3847 0.3570 0.4133 0.3435 0.3894 0.3425
IMG7 3 26.7832 26.5221 26.5233 26.4987 26.5256 26.5256 0.5960 0.4896 0.4972 0.5096 0.4887 0.4887
5 33.0155 32.8431 32.8819 32.7158 32.8823 32.8806 0.3800 0.3717 0.3670 0.3643 0.3561 0.3511
8 39.0899 40.2794 40.5465 39.8385 40.5075 40.5660 0.2601 0.2376 0.3037 0.2619 0.2761 0.2513
12 45.4787 47.9411 47.9221 45.7391 48.6288 48.5744 0.1552 0.1335 0.3088 0.1570 0.1568 0.1513
IMG8 3 27.7280 27.1908 27.4399 27.3665 27.4013 27.3773 0.0246 0.0422 0.0458 0.0242 0.0492 0.0406
5 33.2309 34.2908 34.2769 33.7945 34.4285 34.2825 0.0229 0.0117 0.0180 0.0153 0.0116 0.0162
8 41.8751 42.3056 42.6670 41.8956 42.6795 42.6635 0.0027 0.0032 0.0081 0.0023 0.0034 0.0031
12 50.2470 50.0783 51.5005 50.0429 51.4546 51.4045 0.0087 0.0023 0.0048 0.0034 0.0026 0.0020
IMG9 3 29.2846 29.3033 29.3062 29.2946 29.3067 29.3067 0.1228 0.1269 0.1467 0.1284 0.1269 0.1206
5 20.1897 36.4456 36.4737 36.2666 36.4759 36.4743 0.0871 0.1092 0.1189 0.1528 0.0811 0.0801
8 44.2452 44.7442 45.0135 44.3202 45.0497 45.0507 0.0618 0.0898 0.0807 0.0789 0.0709 0.0591
12 53.6070 53.3127 54.0538 52.4540 54.0444 54.0928 0.0576 0.0467 0.0821 0.0454 0.0310 0.0307
IMG10 3 27.7508 27.8115 27.7321 27.7441 27.5486 27.8170 0.3513 0.3460 0.1835 0.3599 0.1740 0.1476
5 34.3078 34.2824 34.4133 34.0951 34.3000 34.3048 0.0882 0.1051 0.1694 0.1258 0.1507 0.0817
8 42.3127 42.1838 42.3750 41.7415 42.5671 42.5470 0.0241 0.0658 0.0728 0.0299 0.0451 0.0207
12 50.3471 50.2649 51.1724 49.4832 51.3857 51.4777 0.0194 0.0201 0.0273 0.0087 0.0279 0.0073
Circuits, Systems, and Signal Processing
Table 4 Comparison of MSE and PSNR computed by different algorithms using Tsallis entropy
Test Images mMSE PSNR
BAT PSO WDO MBO GOA WCA BAT PSO WDO MBO GOA WCA
IMG1 3 2329.117 1657.781 1592.075 1657.840 1658.782 1651.283 14.4588 15.9355 16.1111 15.9353 15.9329 15.9525
5 966.3367 646.6796 750.4083 506.9183 632.8722 483.6508 18.2795 20.0239 19.3778 21.0814 20.1176 21.2854
8 597.2589 257.6861 231.4519 715.3292 231.6539 204.8888 20.3691 24.0198 24.4861 19.5857 24.4824 25.0156
12 246.0013 132.4347 145.5454 185.6143 178.5549 126.0360 24.2214 26.9107 26.5008 25.4446 25.6130 27.1258
IMG2 3 2477.346 1872.570 1827.429 1609.604 1844.907 1844.762 14.1909 15.4064 15.5123 16.0636 15.4710 15.4713
5 2812.490 773.8249 1047.408 930.7994 749.3698 747.4913 13.6398 19.2443 17.9296 18.4422 19.3838 19.3947
8 529.9086 247.4238 401.5091 446.4363 404.6433 224.3851 20.8887 24.1963 22.0938 21.6332 22.0600 24.6208
12 251.1864 134.1501 103.6165 228.6657 131.5480 94.9620 24.1308 26.8548 27.9765 24.5387 26.9399 28.3553
IMG3 3 4610.765 2163.111 1756.611 1265.359 1157.106 1156.930 11.4930 14.7800 15.6840 17.1086 17.4970 17.4977
5 3409.098 845.0531 613.2652 695.0360 759.1298 714.7138 12.8044 18.8619 20.2543 19.7107 19.3276 19.5894
8 558.9698 379.2637 274.1164 321.8267 311.0264 252.4922 20.6569 22.3413 23.7514 23.0545 23.2028 24.1083
12 494.7088 167.8417 207.1465 212.0112 157.3247 129.9917 21.1873 25.8818 24.9680 24.8672 26.1628 26.9916
IMG4 3 2999.194 2208.992 2179.095 2244.713 3095.866 2167.387 13.3607 14.6888 14.7480 14.6191 13.2229 14.7714
5 2043.011 1473.985 1552.021 1176.138 1864.624 578.3216 15.0280 16.4458 16.2218 17.4262 15.4248 23.9410
8 672.9349 445.6058 721.6192 337.6573 292.6481 262.4043 19.8510 21.6412 19.5477 22.8460 23.4673 20.5091
12 293.9376 185.0830 132.2968 237.5685 168.8779 127.8007 23.4482 25.4571 26.9153 24.3729 25.8550 27.0654
IMG5 3 2032.603 2178.213 2216.885 2488.122 2193.867 2173.083 15.0502 14.7497 14.6733 14.1720 14.7186 14.7600
5 1635.736 1348.851 1347.347 1283.824 1309.381 1280.253 15.9936 16.8311 16.8360 17.0457 16.9601 22.7945
8 450.2605 817.4483 427.3445 735.1722 388.4176 341.6838 21.5961 19.0062 21.8230 19.4669 22.2378 17.0578
12 308.8621 319.8442 473.5711 327.1159 217.2874 192.0187 23.2331 23.0814 21.3769 22.9837 24.7604 25.2973
Circuits, Systems, and Signal Processing
Table 4 continued
Test Images mMSE PSNR
BAT PSO WDO MBO GOA WCA BAT PSO WDO MBO GOA WCA
IMG6 3 987.0456 830.2028 829.7392 901.0295 830.1584 829.6667 18.1874 18.9389 18.9413 18.5834 18.9391 18.9417
5 538.0868 373.6915 397.4516 579.3526 436.8545 377.2228 20.8222 22.4056 22.1379 20.5013 21.7274 22.3648
8 308.5379 204.9673 207.8084 279.2073 190.3819 186.0385 23.2377 25.0139 24.9541 23.6715 25.3345 25.4347
12 198.7499 141.5556 129.1789 161.8157 126.0390 115.0182 25.1477 26.6215 27.0188 26.0405 27.1257 27.5231
IMG7 3 695.2308 711.5475 710.0882 1250.853 704.0629 704.0629 19.7095 19.6087 19.6176 17.1587 19.6546 19.6546
5 355.8365 355.0479 350.2766 363.6682 346.3414 343.9169 22.6182 22.6279 22.6866 22.5237 22.7357 22.7662
8 569.8280 183.5467 209.9983 390.1004 163.5987 161.9772 20.5733 25.4933 24.9086 22.2190 25.9930 26.0362
12 358.0817 103.5121 98.6003 302.0821 93.1860 81.5990 22.5909 27.9808 28.1920 23.3295 28.4372 29.0139
IMG8 3 2676.075 1415.517 1155.465 1410.582 1376.921 1361.884 13.8558 16.6216 17.5032 16.6368 16.7417 16.7894
5 1273.115 862.1839 657.3221 720.5909 802.3219 761.9887 17.0821 18.7748 19.9530 19.5539 19.0873 19.3113
8 1119.215 371.4364 292.1027 347.9131 291.0752 286.2890 17.6416 22.4319 23.4754 22.7160 23.4907 23.5627
12 246.4926 215.8021 161.8803 185.3232 171.9952 139.5100 24.2127 24.7902 26.0388 25.4515 25.7756 26.6847
IMG9 3 1538.622 1546.127 1512.029 1415.415 1508.786 1404.460 16.2594 16.2383 16.3352 16.6219 16.3445 16.6557
5 1295.339 595.3676 582.8319 710.3856 571.1262 566.8527 17.0069 20.3829 20.4753 19.6158 20.5634 20.5961
8 991.9354 310.3972 274.5883 323.1550 275.9448 272.0461 18.1659 23.2116 23.7439 23.0366 23.7225 23.7843
12 328.8292 211.8525 178.1365 198.9144 126.4611 123.5397 22.9610 24.8704 25.6232 25.1441 27.1112 27.2127
IMG10 3 3853.107 1920.841 1453.017 2217.726 1916.256 1913.173 12.2726 15.2958 16.5080 14.6717 15.3062 15.3132
5 1032.793 756.5383 836.2672 848.2029 829.2271 747.2794 17.9906 19.3424 18.9073 18.8458 18.9440 19.3959
8 429.2603 373.0270 322.7719 499.6533 325.1023 264.6941 21.8035 22.4134 23.0418 21.1441 23.0106 23.9033
12 237.4513 184.4520 149.7612 261.4616 154.8862 119.8224 24.3750 25.4719 26.3768 23.9567 26.2306 27.3454
Circuits, Systems, and Signal Processing
Table 5 Comparison of MSE and PSNR computed by different algorithms using Masi entropy
Test Images mMSE PSNR
BAT PSO WDO MBO GOA WCA BAT PSO WDO MBO GOA WCA
IMG1 3 1907.897 1659.598 1206.761 1194.943 1166.368 1658.454 15.3252 15.9307 17.3145 17.3573 17.4624 15.9337
5 582.3666 547.8266 723.4511 503.7277 488.2661 385.5977 20.4788 20.7443 19.5367 21.1088 21.2442 22.2694
8 775.0387 243.5058 229.1289 337.3675 191.8945 179.6851 19.2375 24.2657 24.5300 22.8497 25.3001 25.5856
12 156.888 105.4432 141.5649 169.5977 116.8384 109.9004 26.1749 27.9006 26.6212 25.8366 27.4549 27.7208
IMG2 3 1994.531 1860.240 1778.729 1790.225 1840.429 1667.177 15.1323 15.4351 15.6297 15.6017 15.4816 15.9109
5 1024.982 737.1750 754.2279 831.5745 737.0215 717.8710 18.0236 19.4550 19.3557 18.9317 19.4560 19.5703
8 402.7459 237.5100 266.9797 259.9666 220.4617 211.1958 22.0804 24.3739 23.8660 23.9816 24.6974 24.8839
12 153.5339 125.5113 110.1747 197.4294 114.5827 105.0374 26.2687 27.1439 27.7099 25.1766 27.5396 27.9173
IMG3 3 1781.675 1595.124 1570.839 1588.655 1138.436 1569.627 15.6225 16.1028 16.1694 16.1205 17.5677 16.1728
5 1166.486 751.4732 588.9525 687.1949 624.3157 616.7650 17.4620 19.3716 20.4300 19.7600 20.1767 20.2296
8 398.7838 336.1622 265.1657 278.3702 290.5861 263.2799 22.1234 22.8653 23.8956 23.6845 23.4980 23.9266
12 256.3238 143.4300 170.4635 181.1443 155.9730 131.3008 24.0429 26.5644 25.8144 25.5505 26.2003 26.9481
IMG4 3 2645.895 2188.003 2150.925 2158.003 2694.728 2131.081 13.9050 14.7303 14.8045 14.7902 13.8256 14.8448
5 1727.866 1374.457 1328.548 1125.023 1656.627 801.7596 15.7557 16.7494 16.8970 17.6191 15.9385 19.0903
8 375.1304 273.6131 324.2423 286.7381 271.1008 229.9580 22.3889 23.7594 23.0221 23.5559 23.7994 24.5143
12 276.1277 156.0622 119.4704 172.3331 136.8360 109.8640 23.7197 26.1978 27.3582 25.7671 26.7687 27.7222
IMG5 3 2440.744 2166.883 2202.042 2090.797 2218.364 2194.301 14.2555 14.7724 14.7025 14.9276 14.6704 14.7178
5 880.9027 1051.734 1284.765 1169.207 1026.750 1014.176 18.6815 17.9117 17.0425 17.4518 18.0161 18.0696
8 405.4003 574.8633 415.9868 611.7878 496.1246 355.3255 22.0519 20.5351 21.9400 20.2647 21.1748 22.6245
12 251.3370 173.3319 469.3152 274.5418 120.8373 105.5106 24.1282 25.7420 21.4161 23.7447 27.3087 27.8978
Circuits, Systems, and Signal Processing
Table 5 continued
Test Images mMSE PSNR
BAT PSO WDO MBO GOA WCA BAT PSO WDO MBO GOA WCA
IMG6 3 1952.355 947.8553 933.7278 858.8974 940.0461 938.9922 15.2252 18.3633 18.4286 18.7913 18.3993 18.4041
5 486.1624 452.7884 379.5472 447.6889 405.5670 383.4907 21.2629 21.5718 22.3381 21.6210 22.0501 22.2932
8 267.9626 207.4237 205.4541 208.7744 200.6867 193.1228 23.8500 24.9622 25.0036 24.9340 25.1056 25.2724
12 150.6573 131.2338 123.5594 155.2184 111.5679 107.8503 26.3509 26.9503 27.2120 26.2213 27.6554 27.8025
IMG7 3 943.3454 698.3214 709.2467 717.6921 703.9962 703.9962 18.3840 19.6902 19.6228 19.5714 19.6551 19.6551
5 477.7652 338.8332 348.8553 348.3117 355.3824 350.3098 21.3386 22.8309 22.7043 22.7111 22.6238 22.6862
8 260.2893 178.9940 205.6641 208.3290 192.9579 164.2313 23.9762 25.6024 24.9992 24.9433 25.2761 25.9762
12 177.9725 93.7994 201.5786 123.2034 82.6243 80.9288 25.6272 28.4088 25.0863 27.2245 28.9597 29.0497
IMG8 3 1658.768 1238.750 1153.625 1405.235 1140.479 1139.377 15.9329 17.2009 17.5101 16.6533 17.5599 17.5641
5 536.6176 833.4880 621.4000 608.6535 673.0879 605.6759 20.8341 18.9218 20.1970 20.2871 19.8500 20.3084
8 315.1776 351.2660 276.7228 302.4039 271.2235 264.2881 23.1452 22.6744 23.7103 23.3249 23.7975 23.9100
12 162.7694 168.7567 143.8844 137.0611 147.1909 140.3064 26.0150 25.8581 26.5506 26.7616 26.4519 26.6600
IMG9 3 1301.492 1254.169 1183.938 1240.940 1236.931 1205.920 16.9863 17.1472 17.3975 17.1932 17.2073 17.3176
5 637.4961 574.7344 564.1440 597.7608 567.3688 561.5978 20.0860 20.5361 20.6169 20.3655 20.5921 20.6365
8 322.5220 282.9169 269.7142 314.9966 272.4066 264.5908 23.0452 23.6142 23.8217 23.1477 23.7786 23.9050
12 199.7894 164.0251 159.8043 186.6064 132.8069 128.9092 25.1250 25.9817 26.0949 25.4215 26.8985 27.0279
IMG10 3 2163.980 1914.501 1472.855 2007.900 1281.432 1921.749 14.7782 15.3102 16.4492 15.1033 17.0538 15.2938
5 775.0387 694.3670 827.7491 745.8902 729.7852 626.8542 19.2375 19.7149 18.9518 19.4040 19.4988 20.1591
8 363.3477 301.1411 317.1194 336.9517 310.4724 290.9004 22.5275 23.3431 23.1185 22.8551 23.2105 23.4933
12 243.0949 150.2649 138.4998 203.3034 142.9496 133.3641 24.2730 26.3622 26.7163 25.0493 26.5789 26.8804
Circuits, Systems, and Signal Processing
Table 6 Comparison of SSIM and FSIM computed by different algorithms using Tsallis entropy
Test Images mSSIM FSIM
BAT PSO WDO MBO GOA WCA BAT PSO WDO MBO GOA WCA
IMG1 3 0.9245 0.9467 0.9452 0.9523 0.9466 0.9468 0.7885 0.8155 0.8122 0.8016 0.8156 0.8156
5 0.9690 0.9788 0.9738 0.9847 0.9787 0.9852 0.8248 0.8587 0.8430 0.8611 0.8656 0.8714
8 0.9797 0.9919 0.9925 0.9878 0.9928 0.9936 0.8397 0.8702 0.8791 0.8490 0.8838 0.8860
12 0.9921 0.9960 0.9951 0.9941 0.9940 0.9979 0.8645 0.9014 0.8847 0.8750 0.8981 0.8996
IMG2 3 0.9280 0.9452 0.9466 0.9557 0.9460 0.9467 0.7843 0.8032 0.8037 0.7945 0.8033 0.8039
5 0.9160 0.9770 0.9678 0.9749 0.9774 0.9774 0.7902 0.8611 0.8584 0.8358 0.8689 0.8699
8 0.9876 0.9932 0.9876 0.9884 0.9875 0.9939 0.8870 0.9290 0.9134 0.8803 0.9164 0.9418
12 0.9935 0.9965 0.9974 0.9944 0.9972 0.9976 0.9242 0.9607 0.9681 0.9212 0.9619 0.9767
IMG3 3 0.8681 0.9377 0.9477 0.9624 0.9667 0.9667 0.6128 0.7861 0.8050 0.8188 0.8294 0.8288
5 0.9016 0.9748 0.9722 0.9801 0.9780 0.9795 0.6780 0.8664 0.8612 0.8663 0.8682 0.8741
8 0.9848 0.9892 0.9918 0.9907 0.9908 0.9926 0.8779 0.9188 0.9339 0.9213 0.9302 0.9385
12 0.9839 0.9952 0.9935 0.9939 0.9952 0.9964 0.8884 0.9547 0.9482 0.9436 0.9595 0.9649
IMG4 3 0.9302 0.9352 0.9363 0.9353 0.9074 0.9419 0.6684 0.7140 0.7122 0.7126 0.6810 0.6915
5 0.9407 0.9561 0.9503 0.9635 0.9400 0.9856 0.6965 0.7754 0.7709 0.7737 0.7534 0.7675
8 0.9790 0.9860 0.9760 0.9912 0.9917 0.9929 0.7913 0.8290 0.8262 0.8384 0.8532 0.8644
12 0.9922 0.9953 0.9967 0.9938 0.9954 0.9975 0.8481 0.8912 0.9147 0.8768 0.9049 0.9152
IMG5 3 0.9315 0.9232 0.9218 0.9134 0.9226 0.9233 0.7884 0.8004 0.8005 0.7848 0.8027 0.8033
5 0.9404 0.9505 0.9504 0.9528 0.9518 0.9529 0.8396 0.8530 0.8575 0.8587 0.8595 0.8636
8 0.9860 0.9693 0.9844 0.9729 0.9901 0.9880 0.8692 0.8772 0.9180 0.8765 0.9165 0.9212
12 0.9897 0.9882 0.9820 0.9883 0.9923 0.9931 0.9185 0.9138 0.9142 0.9097 0.9506 0.9542
Circuits, Systems, and Signal Processing
Table 6 continued
Test Images mSSIM FSIM
BAT PSO WDO MBO GOA WCA BAT PSO WDO MBO GOA WCA
IMG6 3 0.9709 0.9762 0.9758 0.9753 0.9761 0.9762 0.7585 0.7840 0.7897 0.7794 0.7843 0.7843
5 0.9848 0.9880 0.9878 0.9842 0.9872 0.9883 0.8236 0.8537 0.8514 0.8225 0.8450 0.8542
8 0.9898 0.9936 0.9934 0.9914 0.9940 0.9944 0.8699 0.9133 0.9142 0.8896 0.9197 0.9200
12 0.9940 0.9955 0.9956 0.9950 0.9957 0.9961 0.9055 0.9355 0.9449 0.9189 0.9514 0.9524
IMG7 3 0.9752 0.9751 0.9752 0.9567 0.9755 0.9755 0.7859 0.7928 0.7954 0.7484 0.7957 0.7957
5 0.9882 0.9874 0.9875 0.9873 0.9877 0.9878 0.8728 0.8757 0.8755 0.8719 0.8798 0.8788
8 0.9905 0.9934 0.9922 0.9859 0.9942 0.9943 0.8075 0.9222 0.9153 0.8586 0.9336 0.9351
12 0.9985 0.9963 0.9964 0.9897 0.9966 0.9971 0.8649 0.9492 0.9528 0.8833 0.9570 0.9656
IMG8 3 0.9442 0.9760 0.9689 0.9760 0.9764 0.9767 0.7290 0.8322 0.8142 0.8120 0.8259 0.8322
5 0.9762 0.9856 0.9862 0.9870 0.9866 0.9873 0.7888 0.8552 0.8434 0.8429 0.8491 0.8605
8 0.9784 0.9937 0.9947 0.9935 0.9946 0.9948 0.7841 0.8709 0.8805 0.8678 0.8804 0.8819
12 0.9947 0.9960 0.9972 0.9963 0.9968 0.9973 0.8918 0.8967 0.9127 0.9096 0.9119 0.9172
IMG9 3 0.9568 0.9530 0.9537 0.9600 0.9544 0.9555 0.7810 0.8119 0.8059 0.7937 0.8115 0.8144
5 0.9635 0.9825 0.9829 0.9801 0.9835 0.9836 0.7975 0.9043 0.9080 0.8806 0.9098 0.9102
8 0.9706 0.9906 0.9921 0.9903 0.9917 0.9919 0.8311 0.9457 0.9574 0.9452 0.9587 0.9600
12 0.9908 0.9938 0.9944 0.9938 0.9964 0.9965 0.9475 0.9622 0.9707 0.9551 0.9836 0.9843
IMG10 3 0.9004 0.9442 0.9425 0.9331 0.9441 0.9441 0.6718 0.7514 0.7502 0.7450 0.7518 0.7518
5 0.9745 0.9782 0.9752 0.9756 0.9751 0.9787 0.7728 0.8208 0.8213 0.8109 0.8217 0.8217
8 0.9901 0.9886 0.9911 0.9852 0.9907 0.9935 0.8338 0.8639 0.8808 0.8564 0.8819 0.8862
12 0.9936 0.9951 0.9960 0.9942 0.9955 0.9970 0.8885 0.9085 0.9262 0.8869 0.9295 0.9393
Circuits, Systems, and Signal Processing
Table 7 Comparison of SSIM and FSIM computed by different algorithms using Masi method
Test Images mSSIM FSIM
BAT PSO WDO MBO GOA WCA BAT PSO WDO MBO GOA WCA
IMG1 3 0.9401 0.9468 0.9600 0.9606 0.9617 0.9469 0.8001 0.8156 0.8258 0.8238 0.8259 0.8269
5 0.9815 0.9829 0.9758 0.9849 0.9851 0.9883 0.8379 0.8742 0.8515 0.8255 0.8730 0.8733
8 0.9790 0.9922 0.9928 0.9897 0.9941 0.9947 0.8108 0.8753 0.8749 0.8739 0.8863 0.8858
12 0.9951 0.9968 0.9953 0.9945 0.9963 0.9989 0.8717 0.8946 0.8948 0.8915 0.9012 0.9055
IMG2 3 0.9425 0.9459 0.9485 0.9477 0.9464 0.9486 0.7935 0.8033 0.8035 0.7972 0.8038 0.8042
5 0.9694 0.9785 0.9778 0.9745 0.9783 0.9789 0.8382 0.8658 0.8664 0.8514 0.8679 0.8684
8 0.9883 0.9937 0.9924 0.9937 0.9945 0.9946 0.8961 0.9334 0.9254 0.9287 0.9400 0.9423
12 0.9961 0.9968 0.9972 0.9943 0.9969 0.9980 0.9501 0.9639 0.9681 0.9311 0.9725 0.9778
IMG3 3 0.9519 0.9546 0.9558 0.9543 0.9671 0.9674 0.7914 0.7935 0.8000 0.8004 0.8219 0.8295
5 0.9673 0.9791 0.9834 0.9812 0.9827 0.9829 0.8458 0.8789 0.8801 0.8750 0.8811 0.8832
8 0.9880 0.9898 0.9920 0.9916 0.9914 0.9929 0.9188 0.9221 0.9315 0.9323 0.9305 0.9357
12 0.9923 0.9959 0.9950 0.9945 0.9953 0.9972 0.9338 0.9618 0.9553 0.9512 0.9612 0.9664
IMG4 3 0.9256 0.9362 0.9364 0.9369 0.9213 0.9469 0.6790 0.7154 0.7095 0.7127 0.7082 0.7195
5 0.9468 0.9586 0.9592 0.9671 0.9478 0.9865 0.7402 0.7758 0.7765 0.7721 0.7706 0.7884
8 0.9909 0.9923 0.9918 0.9922 0.9927 0.9939 0.8348 0.8574 0.8454 0.8533 0.8606 0.8714
12 0.9929 0.9959 0.9969 0.9956 0.9965 0.9975 0.8671 0.9013 0.9190 0.8924 0.9123 0.9270
IMG5 3 0.9145 0.9237 0.9224 0.9261 0.9219 0.9287 0.7747 0.8036 0.8017 0.8033 0.8000 0.8039
5 0.9703 0.9644 0.9528 0.9573 0.9631 0.9656 0.8522 0.8559 0.8632 0.8597 0.8697 0.8703
8 0.9866 0.9788 0.9849 0.9776 0.9820 0.9895 0.9052 0.8914 0.9098 0.8892 0.9052 0.9323
12 0.9912 0.9942 0.9820 0.9901 0.9961 0.9966 0.9307 0.9599 0.9168 0.9329 0.9764 0.9806
Circuits, Systems, and Signal Processing
Table 7 continued
Test Images mSSIM FSIM
BAT PSO WDO MBO GOA WCA BAT PSO WDO MBO GOA WCA
IMG6 3 0.9495 0.9739 0.9740 0.9759 0.9741 0.9753 0.7359 0.7848 0.7843 0.7870 0.7853 0.7876
5 0.9847 0.9868 0.9888 0.9873 0.9880 0.9890 0.8467 0.8472 0.8620 0.8495 0.8540 0.8587
8 0.9919 0.9940 0.9934 0.9934 0.9937 0.9949 0.9013 0.9148 0.9127 0.9133 0.9143 0.9184
12 0.9953 0.9959 0.9959 0.9951 0.9963 0.9969 0.9312 0.9409 0.9480 0.9274 0.9574 0.9577
IMG7 3 0.9666 0.9759 0.9754 0.9749 0.9756 0.9759 0.7455 0.7991 0.7960 0.7920 0.7982 0.7982
5 0.9833 0.9879 0.9876 0.9877 0.9874 0.9886 0.8534 0.8763 0.8759 0.8753 0.8760 0.8792
8 0.9911 0.9937 0.9924 0.9926 0.9930 0.9946 0.8917 0.9284 0.9185 0.9169 0.9254 0.9362
12 0.9938 0.9966 0.9923 0.9958 0.9970 0.9978 0.9200 0.9549 0.9173 0.9440 0.9646 0.9659
IMG8 3 0.9711 0.9771 0.9770 0.9754 0.9778 0.9778 0.8159 0.8157 0.8013 0.8278 0.8388 0.8392
5 0.9892 0.9861 0.9895 0.9888 0.9890 0.9892 0.8432 0.8588 0.8427 0.8470 0.8451 0.8471
8 0.9934 0.9939 0.9950 0.9945 0.9951 0.9956 0.8815 0.8729 0.8781 0.8755 0.8812 0.8834
12 0.9969 0.9967 0.9976 0.9974 0.9972 0.9979 0.9126 0.9083 0.9232 0.9252 0.9174 0.9178
IMG9 3 0.9645 0.9655 0.9677 0.9659 0.9662 0.9682 0.8078 0.8143 0.8223 0.8142 0.8174 0.8236
5 0.9702 0.9834 0.9836 0.9821 0.9836 0.9838 0.8105 0.9078 0.9088 0.8964 0.9098 0.9111
8 0.9912 0.9915 0.9925 0.9909 0.9920 0.9929 0.9415 0.9536 0.9592 0.9464 0.9591 0.9602
12 0.9941 0.9951 0.9949 0.9945 0.9962 0.9969 0.9648 0.9723 0.9723 0.9696 0.9830 0.9852
IMG10 3 0.9381 0.9438 0.9623 0.9414 0.9680 0.9437 0.7425 0.7509 0.7502 0.7516 0.7604 0.7615
5 0.9790 0.9809 0.9753 0.9787 0.9792 0.9841 0.8108 0.8202 0.8227 0.8157 0.8224 0.8297
8 0.9906 0.9920 0.9911 0.9917 0.9914 0.9920 0.8575 0.8774 0.8829 0.8684 0.8826 0.8875
12 0.9933 0.9959 0.9965 0.9948 0.9961 0.9975 0.8978 0.9204 0.9290 0.9110 0.9398 0.9403
Circuits, Systems, and Signal Processing
Table 8 Comparison of entropy obtained by using different algorithms for each sample image
Test Images mTsallis Masi
BAT PSO WDO MBO GOA WCA BAT PSO WDO MBO GOA WCA
IMG1 3 1.9422 2.1818 2.0008 2.1426 2.1799 2.1841 2.3799 2.1838 2.4193 2.3956 2.3894 2.3795
5 2.8596 3.1280 2.8217 2.8930 3.0711 3.1943 3.0674 3.1898 2.9132 3.2783 3.1810 3.4653
8 3.4118 3.7515 3.7336 3.1421 3.9023 3.9498 3.5278 3.8485 3.7724 3.6225 3.9303 3.9390
12 4.0120 4.1843 3.9345 3.9301 4.2374 4.2899 4.4197 4.4042 4.2372 3.9842 4.3301 4.3901
IMG2 3 2.8538 2.9800 2.9852 2.9712 2.9830 2.9852 2.9919 2.9850 2.9902 3.0089 2.9881 2.9882
5 3.3284 3.6817 3.6641 3.6018 3.6338 3.7327 3.7026 3.5651 3.6718 3.6239 3.7209 3.7372
8 4.0388 4.3197 4.2429 4.1293 4.2852 4.4144 4.2188 4.3470 4.3318 4.3448 4.3296 4.3808
12 4.5660 4.8632 4.8044 4.4154 4.7727 4.8868 4.6194 4.5330 4.8801 4.5952 4.8679 4.8895
IMG3 3 2.3741 2.2732 2.5199 2.3830 2.5393 2.5423 2.5703 2.5105 2.5265 2.5162 2.7657 2.5456
5 2.9604 3.1777 3.1182 3.2159 3.1682 3.2239 3.2158 3.2691 3.3013 3.2882 3.3058 3.3108
8 3.7628 3.7898 4.0479 3.3065 3.9616 4.1293 3.9196 3.9009 4.0128 3.4903 4.0204 4.0394
12 4.1955 4.4984 4.3090 4.2109 4.4124 4.5608 4.4060 4.4868 4.4122 4.3840 4.5409 4.5733
IMG4 3 2.9187 2.9262 2.9478 2.9853 2.3984 2.9484 2.7696 2.9580 2.9573 2.9506 2.7474 2.9575
5 3.2990 3.5075 3.4257 3.5937 3.1778 3.4438 3.3467 3.5738 3.5465 3.5406 3.3981 3.5932
8 4.0849 4.1796 4.2000 4.0410 4.2036 4.2235 4.1295 4.2723 4.2120 4.0714 4.2387 4.2839
12 4.3088 4.5967 4.6241 4.4790 4.6651 4.6739 4.3951 4.6798 4.6811 4.6021 4.6732 4.7601
IMG5 3 1.9236 1.9329 1.8482 1.7365 1.9279 1.9425 1.7565 1.8988 1.8628 1.9834 1.8559 1.8720
5 2.1226 2.6362 2.6439 2.5835 2.6309 2.7151 3.1996 3.0799 2.7174 2.8069 3.0296 3.0864
8 3.2516 3.2573 3.7160 3.4405 3.8128 3.8506 3.4600 3.4473 3.6425 3.5227 3.4989 3.7792
12 3.8070 4.0997 4.0013 3.4760 4.0018 4.0698 3.9443 4.2096 4.0716 4.0789 4.3584 4.3595
Circuits, Systems, and Signal Processing
Table 8 continued
Test Images mTsallis Masi
BAT PSO WDO MBO GOA WCA BAT PSO WDO MBO GOA WCA
IMG6 3 1.8882 2.0809 2.0614 1.9854 2.0827 2.0882 2.2350 2.2061 2.2218 2.1973 2.2233 2.2364
5 2.6701 2.7176 2.6977 2.5088 2.6890 2.7500 2.7567 2.8339 2.8891 2.7367 2.9050 2.9582
8 2.8529 3.2919 3.2706 2.5831 3.3367 3.3940 3.3327 3.4061 3.2927 3.0933 3.3595 3.3816
12 3.7973 3.6368 3.5439 3.6197 3.6746 3.6863 3.7659 3.7995 3.6415 3.7318 3.8134 3.8341
IMG7 3 2.0502 2.2496 2.2341 1.7696 2.2744 2.2744 1.9175 2.3333 2.2841 2.0121 2.3075 2.3094
5 2.5503 2.6626 2.8984 2.9334 2.9215 2.9898 3.0094 2.8083 2.7922 2.9846 2.9879 3.0113
8 2.9939 3.6501 3.2606 2.9130 3.6112 3.8012 3.6771 3.7130 3.4079 3.3774 3.6227 3.7384
12 3.8763 4.3077 4.1894 3.5902 4.2274 4.4705 4.0807 4.1718 3.6762 4.2892 4.2876 4.3807
IMG8 3 2.3589 2.5167 2.4674 2.0374 2.4684 2.5341 2.5033 2.6574 2.7615 2.5249 2.7313 2.7646
5 2.8176 2.8599 3.0978 2.9846 2.9370 2.9425 3.1392 2.9088 3.1426 3.1766 3.1204 3.1443
8 2.9354 3.5490 3.7403 3.6087 3.6964 3.7058 3.7641 3.5663 3.8517 3.4546 3.8097 3.7535
12 3.9459 4.0187 4.0771 3.1308 3.8809 4.0814 4.0785 4.2344 4.2212 4.1823 4.2564 4.2819
IMG9 3 2.0989 2.8102 2.8111 2.2475 2.8469 2.8666 3.1278 3.0737 3.0934 3.0711 3.0932 3.1030
5 3.4640 3.6144 3.5284 3.6288 3.5002 3.6511 3.7654 3.6650 3.5679 2.8701 3.5774 3.5782
8 3.6468 4.1783 4.2835 4.0379 4.2420 4.2943 4.3116 4.2314 4.3069 4.2222 4.2620 4.3138
12 4.4145 4.4858 4.6682 4.3004 4.9027 4.9423 4.4674 4.6532 4.6917 4.3856 4.8401 4.8519
IMG10 3 2.6599 2.6017 2.6237 2.6725 2.7444 2.7427 2.7045 2.7491 2.8597 2.7087 3.1567 2.7426
5 3.5833 3.5738 3.5240 3.3134 3.4657 3.6130 3.5278 3.5485 3.5266 3.6117 3.6673 3.6860
8 3.9873 4.2767 4.2026 3.8701 4.2786 4.3662 4.2144 4.3587 4.3902 4.1550 4.2350 4.3344
12 4.6380 4.6900 4.6899 4.3264 4.7955 4.8357 4.6415 4.6819 4.7807 4.5147 4.7972 4.8629
Circuits, Systems, and Signal Processing
Table 9 Comparison of CPU timing (seconds) obtained by using different algorithms for each sample image
Test Images mTsallis Masi
BAT PSO WDO MBO GOA WCA BAT PSO WDO MBO GOA WCA
IMG1 3 3.064241 2.042758 1.846648 1.132360 3.043941 2.172922 2.729718 1.886554 1.647181 1.133710 2.858181 1.560761
5 3.127268 2.136245 1.917924 1.150149 3.596862 2.229018 2.864532 1.983129 1.698554 1.210767 3.347778 1.600512
8 3.264451 2.217329 1.973822 1.266951 4.31361 2.314626 3.144932 2.114758 1.796637 1.376532 3.917079 1.693484
12 3.629263 2.390708 2.157474 1.400572 5.208445 2.440397 3.011516 2.283915 1.943955 1.563293 4.998499 1.844940
IMG2 3 3.376751 2.323213 2.098434 1.415802 3.291224 2.543869 3.039097 2.290330 1.966063 1.425660 3.172987 1.651330
5 3.593653 2.420012 2.246969 1.492039 3.871242 2.496350 3.173747 2.298351 2.065051 1.628909 3.680925 1.699563
8 3.475856 2.551533 2.301698 1.590663 4.536667 2.514353 3.255377 2.431390 2.119484 1.711671 4.165527 1.810793
12 3.692507 2.703696 2.421820 1.597828 5.418761 2.718983 3.524278 2.498293 2.157093 1.879737 5.276797 1.855314
IMG3 3 3.417266 2.456090 2.105522 1.414200 3.345031 2.415680 3.035354 2.153402 1.991174 1.424668 3.154539 1.697284
5 3.496797 2.440508 2.212915 1.479021 3.937596 2.549610 3.241645 2.343459 2.024723 1.523116 3.703274 1.736053
8 3.688308 2.578551 2.434279 1.548385 4.527138 2.657685 3.252934 2.412454 2.139619 1.695077 4.288098 1.869568
12 3.940363 2.760373 2.511613 1.670923 5.681480 2.820902 3.422530 2.631573 2.271785 1.862901 5.403997 2.034918
IMG4 3 3.007738 1.961775 1.780454 1.038381 2.968650 2.078354 2.688919 1.831287 1.628306 1.071300 2.790454 1.333753
5 3.105255 2.083076 1.896287 1.107947 3.520348 2.131320 2.741506 1.920795 1.662501 1.259352 3.321177 1.391788
8 3.142912 2.162715 1.986756 1.197775 4.148654 2.280560 2.980492 2.091781 1.812234 1.323585 3.902595 1.486456
12 3.324912 2.292093 2.118102 1.342651 5.207378 2.420468 2.998342 2.244436 1.941663 1.510050 4.997951 1.619551
IMG5 3 3.313723 2.341023 2.092476 1.410977 3.328259 2.446729 3.042230 2.202260 1.993396 1.388393 3.180866 1.783246
5 3.415731 2.432804 2.172593 1.457714 3.850123 2.467152 3.111195 2.269911 2.086373 1.489396 3.700526 1.875028
8 3.612044 2.545550 2.340386 1.587909 4.402758 2.548041 3.223933 2.507097 2.134811 1.690285 4.203067 1.909466
12 3.902244 2.745547 2.404682 1.742962 5.550993 2.791429 3.267751 2.581133 2.163601 1.891578 5.252055 1.956325
Circuits, Systems, and Signal Processing
Table 9 continued
Test Images mTsallis Masi
BAT PSO WDO MBO GOA WCA BAT PSO WDO MBO GOA WCA
IMG6 3 3.560528 2.322217 2.121196 1.393874 3.296211 2.589151 3.051498 2.157469 1.930124 1.424629 3.139574 1.757205
5 3.639132 2.442749 2.218921 1.441985 3.916344 2.534493 3.148693 2.258748 2.016930 1.596126 3.655890 1.913091
8 3.587664 2.557660 2.343472 1.589342 4.472489 2.622218 3.368548 2.413964 2.120368 1.685067 4.208352 1.973476
12 3.664967 2.648303 2.461953 1.722925 5.559611 2.870621 3.431996 2.627925 2.274186 1.792280 5.359747 2.075318
IMG7 3 3.462354 2.311418 2.106100 1.389536 3.313738 2.397093 3.145764 2.187313 1.949887 1.414667 3.102823 1.838840
5 3.361374 2.420589 2.200434 1.454229 3.892024 2.476877 3.413669 2.287456 2.035837 1.517981 3.739177 1.949340
8 3.524405 2.619137 2.307788 1.556151 4.441887 2.641374 3.298022 2.475883 2.130472 1.692463 4.240713 2.073395
12 3.683715 2.705419 2.472781 1.734490 5.641763 2.742141 3.500602 2.658646 2.272981 1.902274 5.391924 2.261910
IMG8 3 3.445493 2.285896 2.171136 1.349684 3.369772 2.438821 3.107408 2.195035 1.957011 1.381983 3.168380 1.903105
5 3.437417 2.339092 2.259347 1.410219 3.932368 2.475432 3.172579 2.322453 1.997903 1.484205 3.661069 2.022841
8 3.460181 2.515218 2.281119 1.481569 4.431888 2.586342 3.100623 2.401727 2.160108 1.573004 4.172257 2.145962
12 3.681267 2.667378 2.480032 1.657273 5.587312 2.883095 3.454415 2.573330 2.318785 1.843208 5.399538 2.188881
IMG9 3 3.361825 2.339379 2.126145 1.396423 3.348080 2.513841 3.231031 2.173848 2.021824 1.423619 3.117242 1.828056
5 3.587134 2.414801 2.247124 1.453879 3.916358 2.487642 3.152700 2.366204 2.073128 1.524638 3.676654 2.071436
8 3.555435 2.553502 2.293884 1.557555 4.457825 2.592829 3.199074 2.456286 2.093650 1.669190 4.225314 2.055384
12 3.663386 2.841852 2.532738 1.851879 5.542501 2.769589 3.494165 2.605643 2.266900 1.848286 5.268247 2.169543
IMG10 3 3.485532 2.352906 2.131865 1.386404 3.329477 2.486543 3.046821 2.170185 1.939523 1.424374 3.149625 1.971243
5 3.570638 2.424722 2.255303 1.495900 3.908645 2.446009 3.144932 2.247742 2.009180 1.521099 3.747954 1.980714
8 3.739055 2.483903 2.317718 1.481708 4.464439 2.744119 3.214741 2.472509 2.073827 1.629913 4.273872 2.174626
12 3.734478 2.652656 2.514762 1.775900 5.563878 2.772494 3.465809 2.612613 2.216497 1.835042 5.285317 2.369175
Circuits, Systems, and Signal Processing
or nature; the best solution or result created at each run may not be same or identi-
cal. The complete analysis of the Tsallis-based optimization techniques has revealed
that the WCA technique produces optimal value for a maximum number of fidelity
parameters among all the techniques for almost every image. The objective function of
Tsallis method is maximized to determine the thresholding results. The performance
of different objective functions has been compared by using the quantitative results
such as best objective function and ME values obtained by Tsallis which is reported in
Table 2. Tables 4,6,8, and 9depict the statistical evaluation of the quality parameters
such as MSE, PSNR, SSIM, FSIM, entropy, and computational time, respectively.
Figure 3depicts the segmented results for all the images obtained at 3-level, 5-level,
8-level, and 12-level thresholding for each optimization algorithm (BAT, PSO, WDO,
MBO, GOA, and WCA) (Fig. 4). Figures 5,6,7,8,9, and 10 show the plots of ME,
MSE, PSNR, SSIM, FSIM, and entropy of 8-level MT using Tsallis entropy corre-
spondingly, and it can be clearly noticed from the plots that the Tsallis-WCA method
outperforms all other optimization (BAT, PSO, WDO, MBO, and GOA) techniques.
4.3 Experiment 2: Masi Entropy
In this section, several experimental results are reported using WCA, BAT, PSO,
WDO, MBO, and GOA based on Masi entropy for multilevel color image threshold-
ing segmentation. The ME and fitness function values using Masi entropy scheme are
presented in Table 3. It has been tested that the misclassification error evaluated by
the WCA coupled with Masi entropy harvests the lowest value among all the tech-
niques for the maximum number of cases. The architecture of the algorithm explores
the search space more efficiently and locates the thresholding values accurately, and
fitness function value is influenced by the architecture and complexity of the methods.
The proposed Masi-WCA method has generated optimum value and the second best
value with a very small margin for the fitness function as shown in Table 3. Compari-
son of PSNR and MSE values obtained by BAT, PSO, WDO, MBO, GOA, and WCA
with Masi entropy is depicted in Table 5. Masi entropy-based method has reported
best performance in comparison with Tsallis entropy, whereas WCA has found best
among BAT, PSO, WDO, MBO, and GOA. Table 7compares the similarity in features
of the original image and segmented image from the results obtained by using BAT,
PSO, WDO, MBO, GOA, and WCA with Masi entropy, respectively. Tables 8and
9compare the entropy and computational time (seconds) for the Tsallis entropy and
Masi entropy coupled with optimization techniques. Figure 4depicts the segmented
results of each image at threshold level L=3, 5, 8, and 12 acquired for Masi-BAT,
Masi-PSO, Masi-WDO, Masi-MBO, Masi-GOA, and Masi-WCA. The main limita-
tion of the proposed Masi-WCA method is that it takes more processing time for
segmentation as compared to Masi-MBO methods, but Masi entropy coupled with
different optimization techniques (Masi-BAT, Masi-PSO, Masi-WDO, Masi-MBO,
Masi-GOA, and Masi-WCA) is always faster than Tsallis entropy-based optimization
techniques (Tsallis-BAT, Tsallis-PSO, Tsallis-WDO, Tsallis-MBO, Tsallis-GOA, and
Tsallis-WCA) which is reported in Table 9. However, Masi-WCA computes the best
Circuits, Systems, and Signal Processing
L-3 L-5 L-8 L-12 L-3 L-5 L-8 L-12
BAT-Tsallis
PSO-Tsallis
WDO-Tsallis
MBO-Tsallis
GOA-Tsallis
WCA-Tsallis
BAT-Tsallis
PSO-Tsallis
WDO-Tsallis
MBO-Tsallis
GOA-Tsallis
WCA-Tsallis
Fig. 3 Results showing 3-level, 5-level, 8-level, and 12-levelsegmented images using Tsalli-based multilevel
thresholding approaches
quality of segmented images over Masi-MBO. In some cases, the proposed Masi-
WCA method is not produced best or optimal values, but it has maintained the second
best value of those cases. The entropy value of the segmented image of the proposed
Circuits, Systems, and Signal Processing
L-3 L-5 L-8 L-12 L-3 L-5 L-8 L-12
BAT-Tsallis
PSO-Tsallis
WDO-Tsallis
MBO-Tsallis
GOA-Tsallis
WCA-Tsallis
BAT-Tsallis
PSO-Tsallis
WDO-Tsallis
MBO-Tsallis
GOA-Tsallis
WCA-Tsallis
Fig. 3 continued
Masi-WCA method is a maximum value for 5-level, 8-level, and 12-level threshold-
ing, whereas it holds the best and the second best value for 2-level thresholding for
almost all cases.
Circuits, Systems, and Signal Processing
BAT-Tsallis
PSO-Tsallis
WDO-Tsallis
MBO-Tsallis
GOA-Tsallis
WCA-Tsallis
L-3 L-5 L-8 L-12 L-3 L-5 L-8 L-12
Fig. 3 continued
Figures 11,12,13,14,15, and 16 depict the plots of the ME, MSE, PSNR, SSIM,
FSIM and entropy of 8-level MT using Masi entropy, respectively, from which it can
confirm that the performance of the proposed Masi-WCA approach is superior among
BAT, PSO, WDO, MBO, GOA when coupled with Masi entropy. Figures 3and 4
show the segmented results of each test image at 3-level, 5-level, 8-level, and 12-level
of thresholding from which it can be visually investigated that proposed Masi-WCA
method surpasses the Tsallis-WCA-based results. From Tables 2,3,4,5,6,7,8, and
9, it can be clearly identified that Masi entropy overcomes the Tsallis entropy-based
methods and the statistical analysis of the proposed algorithm outclasses with respect
to all methods.
4.4 Comparison of WCA-Masi with Other Algorithms
The accuracy of the proposed Masi-WCA multilevel thresholding algorithm is the
highest among all other compared algorithms in terms of image segmentation process.
Moreover, compared methods are not very effective to perform the optimal thresholds.
The Masi-MBO method is an average method where the accuracy is a major concern,
but Masi-GOA method is the second best approach for the multilevel thresholding
of the color image segmentation. WCA, MBO, and GOA are the recently introduced
optimization techniques which are not exploited for the image segmentation purpose.
However, Tsallis entropy is a very efficient algorithm but has slightly lower per-
formance than Masi entropy (objective function). Since the objective function holds
Circuits, Systems, and Signal Processing
L-3 L-5 L-8 L-12 L-3 L-5 L-8 L-12
BAT-Masi
PSO-Masi
WDO-Masi
MBO-Masi
GOA-Masi
WCA-Masi
BAT-Masi
PSO-Masi
WDO-Masi
MBO-Masi
GOA-Masi
WCA-Masi
Fig. 4 Results showing 3-level, 5-level, 8-level, and 12-level segmented images using Masi-based multilevel
thresholding approaches
the major importance in determining the threshold values, the mathematical model of
Masi entropy is able to provide better thresholded results because of the non-extensive
or additive information properties. WDO has fairly better results than PSO and BAT.
Circuits, Systems, and Signal Processing
L-3 L-5 L-8 L-12 L-3 L-5 L-8 L-12
BAT-Masi
PSO-Masi
WDO-Masi
MBO-Masi
GOA-Masi
WCA-Masi
BAT-Masi
PSO-Masi
WDO-Masi
MBO-Masi
GOA-Masi
WCA-Masi
Fig. 4 continued
This is due to the poor capability of these algorithms in searching the accurate thresh-
olding levels to separate the image pixels into homogenous regions. BAT has generated
poorly segmented outputs at the lower and higher thresholding levels; GOA produces
better outputs but not as better than WCA. In few cases, the GOA gives similar results
Circuits, Systems, and Signal Processing
L-3 L-5 L-8 L-12 L-3 L-5 L-8 L-12
BAT-Masi
PSO-Masi
WDO-Masi
MBO-Masi
GOA-Masi
WCA-Masi
Fig. 4 continued
Fig. 5 Plots of 8-level ME using
Tsallis
as the WCA. In some cases, WDO, PSO, and MBO optimization techniques have
achieved better segmentation. But, the BAT and MBO are not very efficient in deter-
mining the threshold values accurately. The results of Masi-GOA have followed the
performance of Masi-WCA, and Masi-GOA approach is fair at classifying pixels at
high thresholding levels.
Circuits, Systems, and Signal Processing
Fig. 6 Plots of 8-level MSE
using Tsallis
Fig. 7 Plots of 8-level PSNR
using Tsallis
The results presented in this paper by proposed algorithm are superior in almost all
the cases followed by some of the other recently developed metaheuristics algorithm.
This action occurs because each image contains diverse features that characterize a
particular optimization problem. Besides, the randomness of metaheuristic algorithms
(ECA) produces some fluctuations in the results. For instance, if a threshold value is
chosen through the metaheuristic that is not suitable, the segmented image probably
will be not the best. Such condition merely can be identified by PSNR, MSE, SSIM,
FSIM, and entropy because the objective function (Tsallis and Masi entropies) only
offer information about how the intensities values are distributed in different regions.
Circuits, Systems, and Signal Processing
Fig. 8 Plots of 8-level SSIM
using Tsallis
Fig. 9 Plots of 8-level FSIM
using Tsallis
The analysis of convergence rate of IMG1 and IMG7 images using BAT, PSO, WDO,
MBO, GOA, and WCA based on Tsallis and Masi entropies is shown in Figs. 17,18,
19, and 20 for each R-G-B color channel individually. It is concluded that WCA con-
vergence is faster as compared to other algorithms with respect to maximum objective
values.
One of the advantages of the proposed Masi-WCA method is that the function
values are reduced to near-optimum point in the early iterations. This may be due to the
searching criteria and constraint handling approach of WCA where it initially searches
a wide region of problem domain and rapidly focuses on the optimum solution. In
general, the WCA offers competitive solutions compared with other metaheuristic
Circuits, Systems, and Signal Processing
Fig. 10 Plots of 8-level entropy
using Tsallis
Fig. 11 Plots of 8-level ME using
Masi
optimizers. However, the computational efficiency and quality of solutions given by
the WCA depend on the nature and complexity of the underlined problem. This applies
to the efficiency and performance of numerous metaheuristic methods. The WCA
is used for solving the real-world optimization problems, in terms of the optimum
solution; it provides better or close to the best value compared with BAT, PSO, WDO,
MBO, and GOA. In terms of the convergence function evaluations, the WCA reached
the best solution faster than other algorithms. Therefore, the use of WCA in multilevel
thresholding adds more accuracy and flexibility in determining the optimal threshold
values that can vibrantly distinguish different objects present in the image and produces
Circuits, Systems, and Signal Processing
Fig. 12 Plots of 8-level MSE
using Masi
Fig. 13 Plots of 8-level PSNR
using Masi
high-quality segmented color images. The proposed algorithm effectively deals with
the uncertainties in color images with high randomness and multiple small targets with
full of inherent uncertainty.
5 Conclusion
In this paper, a new color image multilevel thresholding method based on water cycle
algorithm is proposed for segmentation. In this work, recently proposed optimization
Circuits, Systems, and Signal Processing
Fig. 14 Plots of 8-level SSIM
using Masi
Fig. 15 Plots of 8-level FSIM
using Masi
algorithms such as BAT, PSO, WDO, MBO, GOA, and WCA have been employed to
maximize Tsallis and Masi entropies to solve the problem of image segmentation by
determining the optimum multilevel threshold values. From the results, it can be con-
cluded that the proposed Masi-WCA method can be efficiently and effectively be used
in color image thresholding operation. The qualitative and quantitative illustrations for
almost all test images exhibit that the WCA outperforms the BAT, PSO, WDO, MBO,
and GOA. The study also reports about the performance of the two entropy-based
objective functions with each optimization technique, which confirms the superiority
of the Masi over Tsallis entropy. In order to measure the performance of the pro-
posed approach, entropy, MSE, ME, SSIM, FSIM, and PSNR have been utilized to
Circuits, Systems, and Signal Processing
Fig. 16 Plots of 8-level entropy
using Masi
Fig. 17 Convergence plots of 12-level MT using Tsallis entropy for each R-G-B channel of the color IMG1
Fig. 18 Convergence plots of 12-level MT using Masi entropy for each R-G-B channel of the color IMG1
assess the quality of segmentation by considering the coincidences among the original
images and respective segmented images. Even if individual entropies are considered,
WCA outperforms all other optimization techniques considered in this paper. Hence,
WCA has been proven to be superior to the rest of algorithms and also the Masi-WCA
beats Tsallis-WCA in terms of efficiency and robustness. The experimental outcomes
are encouraging and motivate futuristic research areas to apply WCA to other image
Circuits, Systems, and Signal Processing
Fig. 19 Convergence plots of 5-level MT using Tsallis entropy for each R-G-B channel of the color IMG7
Fig. 20 Convergence plots of 5-level MT using Masi entropy for each R-G-B channel of the color IMG7
processing applications such as image enhancement, image denoising, image classifi-
cation, and various computer-related problems. Furthermore, the performance of other
traditional entropies can be estimated using WCA concept for multilevel color image
segmentation.
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Affiliations
Pankaj Kandhway1·Ashish Kumar Bhandari1
Pankaj Kandhway
pankaj.kandhway@gmail.com
1Department of Electronics and Communication Engineering, National Institute of Technology
Patna, Patna 800005, India
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