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Philippine Journal of Science
149 (3-a): 747-772, October 2020
ISSN 0031 - 7683
Date Received: 20 Apr 2020
Comparative Study of Heuristic Algorithms
for Electrical Impedance Tomography
Arrianne Crystal Velasco1,2*, Marion Darbas1, Renier Mendoza2,3,
Monica Bacon2, and John Cedrick de Leon2
1CNRS UMR 7352, LAMFA
Université de Picardie Jules Verne, 80000 Amiens, France
2Institute of Mathematics; 3Natural Sciences Research Institute
University of the Philippines Diliman, Quezon City 1101 Philippines
Based on electrical measurements from electrodes placed around the boundary of a body,
electrical impedance tomography (EIT) is an imaging procedure that recovers the spatial
distribution of the conductivities in the interior of a body. Recent studies have shown
promising results in reconstructing EIT images using heuristic algorithms. This work presents
a study of the applicability of six heuristic algorithms – firefly algorithm (FA), novel bat
algorithm (NBA), genetic algorithm with new multi-parent crossover (GA-MPC), success
history-based adaptive differential evolution with linear population size reduction with semi-
parameter adaptation hybrid with covariance matrix adaptation evolutionary strategy
(LSHADE-SPACMA), ensemble sinusoidal differential covariance matrix adaptation
(LSHADE-cnEpSin), and effective butterfly optimizer with covariance matrix adapted retreat
phase (EBOwithCMAR) – for the EIT image reconstruction problem. These algorithms have
never been employed to solve the EIT inverse problem. Series of numerical tests were carried
out to compare the performance of the selected algorithms.
Keywords: electrical impedance tomography, heuristic algorithms, inverse problem
INTRODUCTION
EIT is a non-invasive imaging technique in which the conductivity distribution within a body is reconstructed given
measurements of electrical current and voltage around its boundary. Most imaging modalities, such as computerized
tomography scan and magnetic resonance imaging scan, make use of ionizing radiation which can adversely affect
human health if not properly used or contained (Mettler 2012). This concern led to efforts to develop radiation-free
tomographic procedures, one of which is the EIT.
Image quality is a constant subject in EIT research, as EIT images are known to have poor spatial resolution due to
the technique’s soft-field property and a limited number of independent measurements (Miao et al. 2014). Image
reconstruction via EIT is also an ill-posed problem (Holder 2000). Nevertheless, due to its low cost, portability, and
non -invasive property, EIT is safe for long-term continuous monitoring – a trait that is lacking in most of the currently
747
available imaging technologies. In addition, different regularization techniques have been developed over the past
years to weaken its ill-posedness, resulting to more accurate and stable reconstructions.
EIT has wide-ranging applications in biomedical monitoring, geophysics, and industrial processes. For instance, it is
a potential tool for the detection of breast cancer (Zuo and Guo 2003); imaging of gastric emptying (Mangnall et al.
2003); and monitoring of pulmonary, brain, and cardiovascular functions (Isaacson et al. 2003). It can also be used to
obtain information about rock porosity and fracture formation (Parker 1984), detect leaks from buried pipes (Jordana
et al. 2001), and nondestructively test for material defects (Eggleston et al. 1990).
A typical EIT system consists of a set of electrodes attached around the boundary of a body, through which low-
frequency alternating currents are injected. The entire EIT operation can be divided into two parts: the forward
problem and the inverse problem. The forward model involves the calculation of the resulting voltages at the boundary
of a body after electric currents are injected. In this paper, the forward problem is solved using the finite element (FE)
method, considering the complete electrode model, which is currently the most accurate for impedance tomography
(Somersalo et al. 1992). On the other hand, the inverse problem, also known as the reconstruction problem, involves
the recovery of the conductivity distribution within a body using boundary voltage data. Yorkey et al. (1987) compared
several deterministic algorithms in solving the inverse EIT problem. They have shown that the modified Newton-
Raphson method can be an effective reconstruction algorithm. Recent studies have proposed the use of heuristic
approaches to solve the inverse problem. In the study of Rashid et al. (2010), differential evolution (DE) algorithm
has been shown to recover the geometry of the inclusions of a circular domain with improved results. In the study of
Ribeiro et al. (2014), a non-blind search is used to generate the first set of the population for GA and it obtained
reasonable reconstruction of one object in a circular domain after a hundred iterations. Heuristic algorithms are capable
of converging towards the global minimum, given sufficient computation time and an appropriate choice of
parameters. They are also very flexible and, therefore, are less restricted to certain forms of constraints (Maringer
2005). Furthermore, these methods are not dependent on the initial guess and the gradient of the cost functional. There
are several works on the use of heuristic algorithms in solving the inverse EIT problem. Popular heuristic optimization
methods like genetic algorithms (Mendoza and Lope 2012; Feitosa et al. 2014b; Barbosa et al. 2017), DE (Li et al.
2003; Barbosa et al. 2018), particle swarm optimization (Feitosa et al. 2014a), and simulated annealing (Martins et
al. 2012; Tavares et al. 2012) were used to solve the image reconstruction problem in EIT. A book chapter discussing
the implementation of several evolutionary and bioinspired algorithms on the EIT image reconstruction problem can
be found in the study of dos Santos et al. (2018).
The goal of this study is to present a comparative analysis of six heuristic algorithms – FA, NBA, GA-MPC, LSHADE-
SPACMA, LSHADE-cnEpSin, and EBOwithCMAR – for the EIT image reconstruction problem. Comparative
analysis of the above-mentioned algorithms is carried out in terms of the accuracy and precision of the produced
estimates, and the average cost of the said estimates for the case where the conductivity distribution of the body is
piecewise constant. Such an assumption arises, for instance, in medical imaging since various tissues in the body have
contrasting conductivities with discontinuities at the boundaries of organs or masses (Mendoza and Keeling 2016).
The algorithms are chosen in a way that a comparison between a standard heuristic algorithm (FA), improvements of
some popular heuristic algorithms (NBA, GA-MPC), and combinations of two or more algorithms and the more recent
ones (LSHADE-SPACMA, LSHADE-cnEpSin, EBOwithCMAR) is done.
This paper is organized as follows. Section 2 includes a brief overview of the forward and inverse problems. In Section
3, the proposed heuristic algorithms for the solution of the inverse problem are explained. The methodology used to
conduct numerical simulations and their results are then discussed in Section 4. Finally, Section 5 draws conclusions
about the results obtained and includes possible future works.
Philippine Journal of Science
Vol. 149 No. 3-a, October 2020
Velasco et al.: Comparative Study of Heuristic
Algorithms for EIT
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MATHEMATICAL FORMULATION OF THE EIT PROBLEM
Forward Problem
Consider a bounded simply connected domain with a smooth boundary . In the low-frequency regime
under consideration in EIT experiments, the electromagnetic field satisfies the quasi-static Maxwell equations where
the time derivative is neglected (Cheney et al. 1999; Borcea 2011). The governing (elliptic) equation of EIT is given
in by:
(1)
where is the conductivity distribution and is the potential function in the body (Calderón
1980). Moreover, assume that is piecewise constant, that is,σ
, where is the
background conductivity, is the characteristic function of the background domain
corresponds to the number of (possible) inclusions in Ω,
and is the
conductivity of the inclusion with for some constants
. This is the case, for example, in geophysics and medical imaging where the body under
investigation could be divided into different regions.
There are three known mathematical models for the EIT forward problem – namely. the continuum model, the point
electrode model, and the complete electrode model (CEM) (Borcea 2011). This work considers the CEM because it
is accordingly the most accurate mathematical forward model for real-life EIT (Somersalo et al. 1992). In CEM, a
finite number of electrodes denoted by
is attached to the boundary on which the current patterns
are injected. Resulting boundary potentials are measured. The electrode
contact impedance, denoted by, is the effect of a thin and highly resistive layer formed at the
electrode-object interface during electrode measurements (Somersalo et al. 1992) and it is assumed to satisfy
where is a positive constant. Using the CEM to model the electrodes, they are assumed
to be of finite length and are perfect conductors, which results in voltage measurements on these electrodes to be
constant. This contact impedance tends to be high for the frequencies used in EIT; thus, the voltage drop across the
impedance layer is large. Ignoring this voltage drop introduces a large modeling error, which results in reconstruction
errors. Accounting the voltage drop caused by the contact impedance, we have a Robin-type boundary condition.
Furthermore, assuming that the current flowing on each electrode is equal to the current injected and that there is no
current flow on the parts of the boundary where there is no electrode, we have:
(2)
(3)
(4)
Equations 1−4 constitute the CEM for EIT. Let us introduce the space
The CEM
forward problem is formulated as follows: find potentials
upon injecting current pattern
on the boundary of a body with known conductivity distribution in and electrode contact impedance
. The variational formulation of the CEM is given by (Somersalo et al. 1992)
(5)
Philippine Journal of Science
Vol. 149 No. 3-a, October 2020
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Algorithms for EIT
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for any test functions . The existence and uniqueness of a solution have been proved using the
Lax-Milgram Milgram theorem (Somersalo et al . 1992). The conservation of charge
assures the existence
of solutions while the arbitrary choice of ground
ensures the uniqueness of the solution.
We propose a discretization of the forward problem by means of two-dimensional
Lagrange FE (Rao 2005). We
use a triangulation of the domain , which consists of nodes. We consider the approach of Kaipio et al. (2000) and
we solve the following linear system:
(6)
where the matrix
is sparse, symmetric, and positive-definite (Kaipio et al. 2000; Crabb 2017).
The matrices and are defined, respectively, by:
and for :
if and
otherwise
Moreover,
is the vector of nodal values of the unknown potential and is the vector of currents through the
electrodes. The columns of the matrix
are the set of vectors
defined as
,
, . . . ,
. The above system uses a dummy variable
to force
the uniqueness condition. To transform back to an approximation
of the boundary potential , we have the relation:
(7)
Inverse Problem
The inverse problem (or the reconstruction problem) is the main part of the EIT problem where the conductivity
distribution of a body is recovered using voltage measurements at the boundary However, whereas the forward
problem is well-posed, the inverse problem of EIT is nonlinear and highly ill-posed. In the mathematical literature,
this is also known as Calderón’s problem (Calderón 1980). A strictly positive conductivity in the elliptic Equation 1
is uniquely determined in a bounded domain by the entire corresponding Dirichlet-to-Neumann (DtN) map on the
whole boundary of the domain. The main uniqueness, stability, and reconstruction results have been formulated using
the so-called continuum model (Astala and Päivärinta 2006). However, in several applications in EIT, one can only
measure currents and voltages on part of the boundary. Real-life data consist, essentially, of a finite-dimensional linear
electrode current-to-electrode voltage operator. Results have also been obtained on the problem of whether one can
determine the conductivity in the interior from only partial information on the DtN map (Uhlmann 2009; Hyvönen et
al. 2012; Borcea 2011). Furthermore, since voltage measurements are known to be noisy in nature, the solution can
be dominated by noise unless additional conditions are imposed. As such, EIT is a particularly difficult example of
attempting to recover a signal from noise (Holder 2000).
The challenging issues are thus to provide numerical methods for reconstructing the conductivity of a medium from a
finite number of boundary measurements. There are two primary types of algorithms in EIT: static imaging and
Philippine Journal of Science
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Algorithms for EIT
750
difference imaging. Static imaging attempts to recover the absolute conductivity distribution of a body, whereas
difference imaging aims to recover an image of the change in conductivity distribution between the acquisition times
of two data. In this work, we will focus on static imaging, which is suitable for the case when the electrical properties
of the body under study do not vary significantly during the time necessary for data collection (Herrera et al. 2007).
The inverse problem for EIT reconstructs the conductivity distribution inside the body from voltage measurements
on the electrodes
. The aim is to retrieve a finite number of inclusions of different conductivities in
. More precisely, the goal is to estimate by means of an iterative procedure a vector
of unknown parameters
and the vector
of conductivities, for which the error between the measured voltages and that predicted by
the CEM forward problem is minimum. The vector contains geometric attributes (e.g. center, radius, side length) of
the inclusions
, (of respective conductivities
). The objective function reads:
where the voltages
are computed by solving the forward problem (Equations 1−4) at a fixed conductivity
(described by the vectors P and S ) and
is the measured voltage at the electrodes.
is the Euclidean
norm.
PROPOSED HEURISTIC ALGORITHMS FOR THE INVERSE PROBLEM
The development of heuristic algorithms has experienced significant growth over the past two decades (Hussain et al.
2019). New algorithms, including improved variants of known methods, are continuously being proposed and applied
to various real-world problems. This is in part due to efforts directed at encouraging the creation of more advanced
methods, including those of the IEEE Congress on Evolutionary Computation, and Black-Box Optimization
Competition (Molina et al. 2018). Inspirations behind the methods are wide-ranging – from evolution and the behavior
of animals to physical processes. As such, the selection process of algorithms included in this study is an attempt to
balance the diverse inspirations involved in developing the methods and the recency of such methods.
As pointed out, estimation of the conductivity distribution based on boundary voltages and electric currents is an ill-
conditioned inverse problem. Minimizing the voltage error may then produce unsatisfactory results. Hence,
reconstruction requires some methods of improving the conditioning so that the wild variations causing the instability
are ruled out. The most common method is regularization, which involves applying further assumptions and
constraints based on a priori information. Typically, this means that the inverse problem is augmented with a side
constraint such as the minimum length solution, the minimum error with respect to a priori solution, or the smoothness
of the solution (Holder 2000).
The following heuristic approaches allow restrictions to the solution space and introduction of prior information
without using the classical regularization techniques described above. Moreover, no evaluation of objective function
derivatives is needed and no assumption on function continuity needs to be made. However, heuristic algorithms are
relatively expensive in terms of computing time and this limits their applicability to the field of difference imaging at
present. Nevertheless, the continuous and rapid advancement of computing technology makes the development of
real-time dynamic imaging applications based on heuristic methods conceivable in the near future. These heuristic
algorithms belong to a class of algorithms called metaheuristics (Siarry 2016).
We will use the following heuristic algorithms: FA, NBA, GA-MPC, LSHADE-SPACMA, LSHADE-cnEpSin, and
EBOwithCMAR. The detailed discussion of each algorithm and their pseudo-codes are presented in the appendices.
RESULTS AND DISCUSSION
Series of numerical tests were carried out to investigate and compare the performance of the proposed algorithms in
solving the inverse EIT problem.
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Presentation of the Test Configurations
Three different cases were considered (see Table 1). In Case 1, we study a CT (computer tomograph y) scan of a thorax
domain obtained from the study of Venkatratnam and Nagi (2017), as shown in Figure 1a. Solving the forward
problem using the FE method requires the parametrization of the boundaries of the lungs, the heart, and the whole
body. These parametric curves are approximated using the Fourier series. The coefficients of the Fourier series are
estimated by finding the parametric curve that fits the data points on the boundary curve. With this, any practical
domain or object may be studied for real-life applications of EIT. Also, Nissinen et al. (2011) discussed that the errors
due to the approximations of domain boundary affect the reconstruction of the conductivity.
Table 1. Different cases studied for the numerical test.
Case
Unknown variables
1 – The body Ω is a region representing the thorax with the lungs and the heart.
(geometry is fixed)
2 – The body Ω is a unit cir cle re gion with one ellipt ical i nclusion.
3 – The body Ω is a unit circle region with two rotated elliptical inclusions.
Now, for the numerical simulations of Case 1, only the conductivity values inside the inclusions, and , are
unknown. Indeed, the CT results already give the location and the geometry of the organs. We are only interested in
determining the conductivity values of the inclusions (i.e. lungs and heart). This method is particularly applicable for
lung or heart function monitoring to check if there is a deviation of the estimated conductivity values from the normal
values. For Case 2, six parameters are unknown, i.e. is the conductivity of the inclusion; is the center of the
ellipse; and are the lengths of the major and minor axes, respectively; and is the angle of rotation. In Case 3, we
aim in reconstructing two disjoint elliptical inclusions and the respective conductivity inside, i.e. 12 unknown
parameters. The conductivity of the background medium is assumed known in all configurations. In the first case,
it is equal to while in both the second and third case, the conductivity is. Both cases may
represent the domains used in the application of EIT for brain or breast tumor detection.
We work with synthetic data. We take electrodes. The contact impedance is constant across the electrodes
and it is equal to . Sixteen (16) current patterns are applied on the electrodes and the first current has the form
The remaining 15 current patterns are obtained by “rotating” the values of the first current
pattern, i.e. to get the second current pattern , we have , and
. This is repeated until we obtain the fifteen additional current patterns. The
synthetic voltage data are obtained by solving the forward problem (Equations 1−4) with the exact conductivity
distribution (see Section 2.1). A FEM mesh structure with triangular elements, nodes, and a mesh size
was used for the resolution of the forward problem in Case 1; and with triangular elements,
nodes, and for both Cases 2 and 3. In order to avoid an inverse crime [in the sense of Colton and Kress
(1998)], the inverse computations are done on a mesh with triangular elements, nodes, and
for Case 1; and on a mesh with triangular elements, nodes, andmesh size for Cases
2 and 3, which are different from the meshes used to solve the forward problems. To model possible experimental
errors, a random (additive) noise is added to the voltage data as
where) gives a vector of length where each element is uniformly distributed random number in the
interval[see Hintermüller and Laurain (2008)]. We note that to consider higher levels of noise, the use of
regularization methods or more assumptions are needed in the formulation of the inverse problem, especially for Cases
2 and 3. In our simulations, one noise seed is composed of 16 different noise vectors added to the corresponding 15
current-voltage measurements.
The study is based on independent runs of each proposed algorithm, with the same noise seed for all the runs and
a stopping criterion based on a pre-defined number of function evaluations. The only stopping criterion used for all
heuristic methods is when the maximum number of function evaluations is reached. In particular, for Case 1, we set
Philippine Journal of Science
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Algorithms for EIT
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the maximum number of function evaluations to
; in Cases 2 and 3, we have
function evaluations
with
as the number of unknown parameters. The search space was restricted differently for each case. In Case 1,
the bounds are given by and]. Case 2 has bounds] and
For Case 3, we fix and
. We implemented the numerical solver for the forward problem with FreeFEM++ (Hecht
2012) and the numerical optimization algorithms with Matlab. All experiments were executed in Matlab R2018a. The
parameter settings used for each algorithm can be found in Appendix VII.
Numerical Results
The comparative analysis of the heuristic methods is done by measuring the accuracy and precision of the solutions
generated, together with their respective average costs. This is because the inverse EIT problem is ill-conditioned,
which means that solutions tend to be extremely sensitive to perturbations, potentially making them inaccurate or
unstable. To quantitatively analyze the accuracy of recovered images, the average of the reconstruction errors for
runs of each algorithm was calculated. For the ith run, the relative reconstruction error is given by
where is a vector containing the values estimated by the algorithm at the run, whereas contains true values.
Based on Table 1, and are vectors in for Case 1, in for Case 2, and in for Case 3.
In order to measure the repeatability (reconstruction accuracy) of each algorithm, i.e. the degree to which the algorithm
produced similar results for runs, the standard deviation (std. dev.) of the reconstruction errors was also determined.
Table 2 compares the accuracy, repeatability, and average costs of each algorithm’s reconstructions for the three cases
considered. From now on, the “average reconstruction error” will be referred to as the “mean error” for simplicity.
The mean and standard deviation of the reconstruction errors of the generated solutions in the independent runs is
computed using the mean and std functions in Matlab.
In all the cases, the final solution or final reconstructed parameter values considered for each algorithm is the average
solution of the runs. Figures 1, 2, and 3 show the final solution of each algorithm for Cases 1, 2, and 3, respectively.
The difference in color and the shape of the inclusion between the original image the images generated by the proposed
heuristic algorithms is the difference between the true solution and the approximate from the true solution is. Table 3
shows the relative reconstruction error in of the final solution of each algorithm for the three cases, while Tables 4
and 5 display the solution. Note that we fixed the range of the conductivities in the plots and so the difference in the
color and the shape of the inclusion between the original image generated by the proposed heuristic algorithms is the
difference between the true solution and the approximate solution.
Table 2. Comparison of accuracy, repeatability, and average cost of the solutions generated by the proposed heuristic algorithms for all
cases.
Algorithm
Case 1 – thorax
Case 2 – one elliptical inclusion
Case 3 – two elliptic al inclusions
Mean error
Std. dev.
Ave.
cost
Mean
error
Std. dev.
Ave. cost
Mean error
Std. dev.
Ave. cost
FA
5.54E-02
7.77E-02
1.43E-03
0.2193
0.1046
0.6115
0.3071
0.0717
2.8463
NBA
7.74E-03
4.64E-03
1.43E-03
0.1742
0.0983
0.2562
0.3435
0.0926
1.9620
GA-MPC
2.23E-03
1.94E-03
1.43E-03
0.1527
0.0554
0.2422
0.1790
0.0530
0.2558
LSHADE-
SPACMA
6.08E-04
7.26E-08
1.43E-03
0.1143
0.0385
0.2407
0.1546
0.0674
0.2165
LSHADE-
cnEpSin
6.15E-04
4.32E-05
1.43E-03
0.1251
0.0762
0.2423
0.2033
0.0675
0.2219
EBOiwthCM
AR
6.10E-04
8.24E-05
1.43E-03
0.0827
0.0693
0.2426
0.1988
0.0685
0.2917
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Table 3. Reconstruction errors in % of the final solution generated by the proposed
heuristic algorithms for all cases.
Algorithm
Case 1
Case 2
Case 3
FA
5.28
6.82
16.56
NBA
0.152
2.28
21.91
GA-MPC
0.054
15.11
13.07
LSHADE-SPACMA
0.060
5.84
7.90
LSHADE-cnEpSin
0.061
9.63
14.95
EBOwithCMAR
0.058
6.43
12.00
Table 4. Reconstruction errors in % of each parameter in the final solution generated by the proposed heuristic algorithms
for Cases 1 and 2.
The results shown in black bold characters indicate the best values. For Case 1, all the heuristic algorithms studied
were very successful in the recovery of the conductivity values of the inclusions in Ω. For Case 2, all the heuristic
algorithms studied well retrieved the center of the ellipse (see Table 4). The difference in performance between them
lies in the estimation of both the conductivity and the geometric parameters. EBOwithCMAR performed the best while
getting the least mean error and relatively small average cost. LSHADE-SPACMA got the smallest standard deviation
and average cost. As expected, because of the low-resolution property of EIT, the estimate for the conductivity value
inside the inclusion is not as accurate as of the approximation of the geometry. Nevertheless, NBA provided an
excellent approximation of the conductivity (error ) while FA, LSHADE-SPACMA, and EBOwithCMAR were
still able to obtain good conductivity value estimates (see Table 4). GA-MPC yielded the least accurate conductivity
estimate which justifies the relative error given in Table 3. This means that GA-MPC finds it hard to balance its
exploration and exploitation when approximating both the geometry of the inclusion and the conductivity inside it.
NBA and the three most recent algorithms – namely, LSHADE-SPACMA, EBOwithCMAR, and LSHADE-cnEpSin
– presented impressive reconstructions (see Figure 2); FA is less efficient. Lastly, NBA offered the best relative error
of the final solution.
For Case 3, LSHADE-SPACMA showed the best performance with the least mean error and average cost while GA-
MPC is the most consistent among the algorithm because it obtained the least standard deviation of the reconstruction
error. The recovered images of LSHADE-SPACMA and LSHADE-cnEpSin are the closest ones to the original image.
GA-MPC and EBOwithCMAR were also able to reconstruct the two inclusions. NBA only obtained a not too bad
Algorithm
Case 1 – thorax
Case 2 – one elliptical inclusion
FA
1.19E-02
5.34E-02
5.28
2.68
2.06
17.61
41.40
6.37
NBA
3.69E-03
1.42E-03
1.94
0.68
0.41
7.34
2.15
6.92
GA-MPC
3.09E-03
2.51E-04
15.50
1.26
0.43
4.76
2.43
4.07
LSHADE-SPACMA
3.18E-03
3.43E-04
5.92
1.46
0.07
4.00
8.65
0.34
LSHADE-cnEpSin
3.18E-03
3.62E-04
9.86
1.20
0.22
2.03
3.72
3.95
EBOiwthCMAR
3.19E-03
3.06E-04
6.55
1.13
0.14
0.09
4.56
4.21
Table 5. Reconstruction errors in % of each parameter in the final solution generated by the proposed heuristic algorithms for Case 3.
Algorithm
Case 3 – two elliptic al inclusions
FA
21.41
11.99
26.56
22.77
70.79
3.81
1.23
86.77
112.4
3.50
23.06
1.63
NBA
24.73
6.25
19.51
28.82
114.9
5.93
3.66
81.20
88.59
27.28
45.07
2.54
GA-MPC
19.18
2.18
1.37
13.65
13.32
0.45
7.90
2.59
4.56
8.40
0.51
0.05
LSHADE-SPACMA
12.59
2.05
0.49
5.32
6.05
0.25
2.52
0.57
1.19
2.11
1.96
0.77
LSHADE-cnEpSin
23.51
0.62
0.76
9.28
8.18
0.27
5.82
1.66
0.44
1.34
1.52
8.08
EBOiwthCMAR
18.14
7.45
4.32
10.57
16.59
0.37
1.27
12.57
11.70
8.86
16.47
5.82
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location of the bigger inclusion but failed to reconstruct the smaller ellipse. FA gave a poor reconstruction of both
inclusions. For the estimation of conductivity values inside the inclusions, LSHADE-SPACMA achieved a good
balance in the estimation of the conductivities, EBOwithCMAR yielded the most accurate estimate for the
conductivity in the smaller inclusion, and LSHADE-SPACMA got the best conductivity estimate for the bigger
inclusion. NBA is the least accurate in the estimation of the conductivity inside the bigger inclusion, while GA-MPC
has the least accurate conductivity value estimate for the smaller ellipse.
Moreover, GA-MPC gave more accurate approximations for both the geometry of the inclusion and its conductivity
than the standard GA used in (Kim et al. 2006; Ribeiro et al. 2014). Improvements of DE – namely, LSHADE-
SPACMA and LSHADE-cnEpSin – also performed better than the DE used in the study of Rashid et al. (2010). Lastly,
to compare our results to that of the deterministic method, we applied a quasi-Newton iterative method [Broyden-
Fletcher-Goldfarb-Shanno (BFGS) algorithm] to Cases 1 and 2 with a random initial guess. We use the Matlab built-
in command function fminunc to implement the quasi-Newton iterative algorithm. In Case 1, the reconstruction error
is 6.17−04, the standard deviation is 2.21−04, and the average cost function is 1.4−03. This is almost the
same result that we got from the six heuristic algorithms. But for Case 2, BFGS yielded a mean error of 151.6, standard
deviation 361.8, and average cost function 5.97, which are all huge comparing to the results of the six heuristic
algorithms.
Metaheuristic algorithms can be computationally costly since they require many cost function evaluations. Because
we made the stopping criterion based on the maximum number of function evaluations alone, the run time of all the
algorithms are approximately equal. For this reason, we only provide the time for each domain case. For Case 1, it
takes approximately 3 for one function evaluation to be done and the number of function evaluations is set to 1000.
Meanwhile, for Case 2, we have 3.7 for one function evaluation and the number of function evaluations is 6000.
Lastly for Case 3, we get 4.3 for one function evaluation with 12000 total number of function evaluations. To expect
driving 3D computational simulations is challenging. Metaheuristic algorithms are time-consuming and improvemen ts
are needed to get 3D results.
CONCLUSION
In this paper, we have presented a conclusive study of the applicability of several heuristic approaches for EIT image
reconstruction. Up to our knowledge, this is the first time that such a comparative study (between FA, NBA, GA-
MPC, LSHADE-SPACMA, LSHADE-cnEpSin, and EBOwithCMAR) is addressed for EIT. Numerical simulations,
given a fixed number of cost function evaluations and default heuristic algorithm parameters, showed that the more
recent algorithms – namely LSHADE-SPACMA, LSHADE-cnEpSin, and EBOwithCMAR – obtained the best results
in terms of accuracy, repeatability, and average cost. This indicates the continuous improvement in metaheuristic
techniques, reinforcing their potential to solve other similar problems. FA did not fare as well the other algorithms,
especially in retrieving two disjoint inclusions (Case 3), because the maximum number of evaluations and the
population size for each iteration might not be enough to have a balance between exploitation and exploration. This
might be also the case for NBA and GA-MPC.
Although NBA was successful in the image reconstruction of one defect (Cases 1 and 2), it failed in the configuration
with two defects. Modifications and improvement in these three algorithms can be further studied to obtain more
competitive results. Since FA, NBA, and GA-MPC populations for each iteration is fixed, we can recommend
adjusting the population size so that there will be a balance between exploitation and exploration. Also, the maximum
number of evaluations can be increased. Because of their population size linear reduction and self-adapting parameters
system, the most recent algorithms LSHADE-SPACMA, LSHADE-cnEpSin, and EBOwithCMAR are more
consistent and accurate. The different numerical results attest to their efficiency.
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Figure 1. Estimation of conductivity values inside the fixed geometries of heart and lungs of all proposed heuristic
algorithms for Case 1. The conductivity of the lungs and the heart are, respectively, = .− and
=. .
− in the original image.
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In the thorax domain, where only the conductivities of the inclusions are unknown, the six heuristic algorithms
provided excellent results. In other cases, the recovery of conductivity needs more improvement. Possible future works
include using a regularization term in the cost functional to deal with the ill-posedness of the problem or applying
parameter tuning techniques to better fit the algorithm to the EIT inverse problem. A sensitivity analysis for the CEM
forward problem could also be used to get a priori information on the dependence of the measured boundary voltages
on the conductivity and the characteristics of inclusions. This is part of ongoing work.
ACKNOWLEDGEMENTS
The authors acknowledge the financial support of the Natural Sciences Research Institute, University of the
Philippines Diliman under the research grant MAT-16-1-01.
Figure 2. Image reconstructions for Case 2. The conductivity
inside the ellipse of the original image is σe
= 6.7
mS∙cm–1.
Figure 3. Image reconstructions for Case 3. The conductivities are
σ1 = 6.1 mS∙cm–1 (big ellipse) and σ2 = 6.7 mS∙cm–1
(small ellipse) in the original image.
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NOTES ON APPENDICES
The complete appendices section of the study is accessible at http://philjournsci.dost.gov.ph
REFERENCES
ASTALA K, PÄIVÄRINTA L. 2006. Calderon’s inverse conductivity problem in the plane. Annals of Mathematics
163: 265–299.
AWAD N, ALI M, LIANG J, QU B, SUGANTHAN P. 2016a. Problem definitions and evaluation criteria for the
CEC 2017 special session and competition on single objective bound constrained real-parameter numerical
optimization [Technical Report]. Nanyang Technological University, Singapore.
AWAD N, ALI M, SUGANTHAN P, REYNOLDS R. 2016b. An ensemble sinusoidal parameter adaptation
incorporated with L-SHADE for solving CEC 2014 benchmark problems. In: Proceeding of the 2016 IEEE
Congress on Evolutionary Computation (CEC); IEEE. p. 2958–2965.
AWAD N, ALI M, SUGANTHAN P. 2017. Ensemble sinusoidal differential covariance matrix adaptation with
Euclidean neighborhood for solving CEC 2017 benchmark problems. In: Proceedings of the 2017 IEEE Congress
on Evolutionary Computation (CEC); IEEE. p. 372–379.
BARBOSA V et al. 2018. Image reconstruction of electrical impedance tomography using fish school search and
differential evolution. Critical Developments and Applications of Swarm Intelligence. p. 301–338.
BARBOSA V, RIBEIRO R, SILVA A, ROCHA V, FREITAS R, FEITOSA AR, DE SOUZA RE, DOS SANTOS W.
2017. Reconstruction of electrical impedance tomography images using chaotic ring-topology particle swarm
optimization and non-blind search. International Journal of Swarm Intelligence Research 8(2): 17–33.
BORCEA L. 2011. Electrical impedance tomography. Inverse Problems 18: R99–R136.
CAJAYON RC, LUCILO JA, PILAR-ARCEO CPC, MENDOZA ER. 2020. Comparison of two nature-inspired
algorithms for parameter estimation of S-system models. Philippine Journal of Science 149(1): 63–78.
CALDERÓN A. 1980. On an inverse boundary value problem. In: Seminar on Numerical Analysis and its
Applications to Continuum Physics; Río de Janeiro, Brazil. p. 65–73.
CHENEY M, ISAACSON D, NEWELL J. 1999. Electrical impedance tomography. SIAM Review 41: 85–101.
COLTON D, KRESS R. 1998. Inverse acoustic and electromagnetic scattering theory. Berlin: Springer-Verlag.
CRABB M. 2017. Convergence study of 2D forward problem of electrical impedance tomography with high order
finite elements. Inverse Problems in Science and Engineering. p. 1–26.
DAS S, MULLICK SS, SUGANTHAN P. 2016. Recent advances in differential evolution – an updated survey. Swarm
and Evolutionary Computation 27: 1–30.
DAS S, SUGANTHAN P. 2010. Problem definitions and evaluation criteria for CEC 2011 competition on testing
evolutionary algorithms on real world optimization problems [Technical Report]. Jadavpur University, India.
DOS SANTOS WP et al. 2018. Hybrid metaheuristics applied to image reconstruction for an electrical impedance
tomography prototype. In: Proceedings of the Hybrid Metaheuristics for Image Analysis; Springer. p. 209–251.
EGGLESTON M, SCHWABE R, ISAACSON D, COFFIN L. 1990. The application of electric current computed
tomography to defect imaging in metals. Review of Progress in Quantitative Nondestructive Evaluation 24: 455–
462.
ELSAYED S, SARKER R, ESSAM D. 2011. GA with a new multi-parent crossover for solving IEEE-CEC 2011
competition problems. In: Proceedings of the 2011 IEEE Congress on Evolutionary Computation (CEC); IEEE. p.
1034–1040.
Philippine Journal of Science
Vol. 149 No. 3-a, October 2020
Velasco et al.: Comparative Study of Heuristic
Algorithms for EIT
758
FEITOSA AR, RIBEIRO R, BARBOSA V, DE SOUZA RE, DOS SANTOS W. 2014a. Reconstruction of electrical
impedance tomography images using chaotic ring-topology particle swarm optimization and non-blind search. In:
Proceedings of the 2014 IEEE International Conference on Systems, Man, and Cybernetics (SMC); IEEE. p. 2618–
2623.
FEITOSA AR, RIBEIRO R, BARBOSA V, DE SOUZA RE, DOS SANTOS W. 2014b. Reconstruction of electrical
impedance tomography images using particle swarm optimization, genetic algorithms and non-blind search. In:
Proceedings of Biosignals and Robotics for Better and Safer Living (BRC) at the 5th ISSNIP-IEEE Biosignals and
Biorobotics Conference (2014); IEEE. p. 1–6.
FISTER I, FISTER JR. I, YANG XS, BREST J. 2013. A comprehensive review of firefly algorithms. Swarm and
Evolutionary Computation 13: 34–46.
HANSEN N. 2006. The CMA evolution strategy: a comparing review. In: Towards a new evolutionary computation.
Springer. p. 75–102.
HECHT F. 2012. New development in FreeFem++. Journal of Numerical Mathematics 20(3–4): 251–265. Retrieved
from https://freefem.org/
HERRERA CN, VALLEJO MF, DE MOURA F, AYA JC, LIMA R. 2007. Electrical impedance tomography
algorithm using simulated annealing as a search method. In: Proceedings of COBEM 2007 at the 19th International
Congress of Mechanical Engineering, Brazil.
HINTERMÜLLER M, LAURAIN A. 2008. Electrical impedance tomography: from topology to shape. Control and
Cybernetics 37(4): 913–933.
HOLDER D. 2000. Electrical Impedance Tomography: Methods, History and Applications. London: IOP Publishing.
HOLLAND J. 1975. Adaptation in Natural and Artificial Systems. Ann Arbor, MI: University of Michigan Press.
HUSSAIN K, SALLEH MM, CHENG S, SHI Y. 2019. Metaheuristic research: a comprehensive survey. Artificial
Intelligence Review 52(4): 2191–2233.
HYVÖNEN N, PIIROINEN P, SEISKARI O. 2012. Point measurements for a Neumann-to-Dirichlet map and the
Calderón problem in the plane. SIAM Journal on Mathematical Analysis 44: 3526–3536.
ISAACSON D, MUELLER J, SILTANEN S. 2003. Biomedical applications of electrical impedance tomography.
Physiological Measurement 24(2).
JORDANA J, GASULA M, PALLAS-ARENY R. 2001. Electrical resistance tomography to detect leaks from buried
pipes. Measurement Science and Technology 12(8): 1061–1068.
KAIPIO J, KOLEHMAINEN V, SOMERSALO E, VAUHKONEN M. 2000. Statistical inversion and Monte Carlo
sampling methods in electrical impedance tomography. Inverse Problems 16: 1487–1522.
KIM HC, BOO CJ, KANG MJ. 2006. Image reconstruction using genetic algorithm in electrical impedance
tomography. In: Neural Information Processing. King I, Wang J, Chan LW, Wang D eds. Lecture Notes in
Computer Science, Vol. 4234. Springer.
KUMAR A, MISRA R, SINGH D. 2015. Butterfly optimizer. In: Proceedings of the 2015 IEEE Workshop on
Computational Intelligence: Theories, Applications and Future Directions (WCI); IEEE. p. 1–6.
KUMAR A, MISRA R, SINGH D. 2017. Improving the local search capability of effective butterfly optimizer using
covariance matrix adapted retreat phase. In: Proceedings of the 2017 IEEE Congress on Evolutionary Computation
(CEC); IEEE. p. 1835–1842.
LI Y, RAO L, HE R, XU G, WU Q, GE M, YAN W. 2003. Image reconstruction of EIT using differential evolution
algorithm. In: Proceedings of the 25th Annual International Conference of the IEEE EMBS; IEEE. p. 1011–1014.
Philippine Journal of Science
Vol. 149 No. 3-a, October 2020
Velasco et al.: Comparative Study of Heuristic
Algorithms for EIT
759
MANGNALL Y, BAXTER A, AVILL R, BIRD N, BROWN B, BARBER D, SEAGAR A, JOHNSON A, READ N.
2003. Applied potential tomography: a new noninvasive technique for assessing gastric function. Clinical Physics
and Physiological Measurement 8: 119–129.
MARINGER DG. 2005. Portfolio management with heuristic optimization, 1st ed. Springer.
MARTINS T, CAMARGO E, LIMA R, AMATO M, TSUZUKI M. 2012. Image reconstruction using interval
simulated annealing in electrical impedance tomography. IEEE Transactions on Bio-medical Engineering 59:
1861–1870.
MENDOZA R, KEELING S. 2016. A two-phase segmentation approach to the impedance tomography problem.
Inverse Problems 33: 015001.
MENDOZA R, LOPE JE. 2012. Reconstructing images in electrical impedance tomography using hybrid genetic
algorithms. Science Diliman 24(2): 50–66.
MENG XB, GAO X, LIU Y, ZHANG H. 2015. A novel bat algorithm with habitat selection and Doppler Effect in
echoes for optimization. Expert Systems with Applications 42: 6350–6364.
METTLER F. 2012. Medical effects and risks of exposure to ionising radiation. Journal of Radiological Protection
32: N9–N13.
MIAO L, MA Y, WANG J. 2014. ROI-based image reconstruction of electrical impedance tomography used to detect
regional conductivity variation. IEEE Transactions on Instrumentation and Measurement 63: 2903–2910.
MOHAMED A, HADI A, FATTOUH A, JAMBI K. 2017. LSHADE with semi-parameter adaptation hybrid with
CMA-ES for solving CEC 2017 benchmark problems. In: Proceedings of the 2017 IEEE Congress on Evolutionary
Computation (CEC); IEEE. p. 145–152.
MOLINA D, LATORRE A, HERRERA F. 2018. An insight into bio-inspired and evolutionary algorithms for global
optimization: Review, analysis, and lessons learnt over a decade of competitions. Cognitive Computation 10: 517–
544.
NISSINEN A, KOLEHMAINEN V, KAIPIO J. 2011. Compensation of modelling errors due to unknown domain
boundary in electrical impedance tomography. IEEE Transactions on Medical Imaging 30: 231–242.
PARKER R. 1984. The inverse problem of resistivity sounding. Geophysics 49: 2143–2158.
PRICE K, STORN R, LAMPINEN J. 2005. Differential Evolution: A Practical Approach to Global Optimization.
Berlin: Springer.
RAO S. 2005. The Finite Element Method in Engineering, 4th ed. Oxford: Elsevier Butterworth-Heinemann.
RASHID A, KHAMBAMPATI AK, KIM BS, LIU D, KIM S, KIM KY. 2010. A differential evolution based approach
to estimate the shape and size of complex shaped anomalies using EIT measurements. In: Proceedings of the Grid
and Distributed Computing, Control and Automation, Vol. 121. Berlin: Springer.
RIBEIRO R, FEITOSA A, SOUZA R, DOS SANTOS W. 2014. Reconstruction of electrical impedance tomography
images using genetic algorithms and non-blind search. In: Proceedings of the 11th International Symposium on
Biomedical Imaging. p. 153–156.
SIARRY P. 2016. Metaheuristics. Springer International Publishing.
SOMERSALO E, CHENEY M, ISAACSON D. 1992. Existence and uniqueness for electrode models of electric
current computed tomography. SIAM Journal of Applied Mathematics 52: 1023–1040.
TANABE R, FUKUNAGA A. 2013. Success-history based parameter adaptation for differential evolution. In:
Proceedings of the 2013 IEEE Congress on Evolutionary Computation (CEC); IEEE. p. 71–78.
TANABE R, FUKUNAGA A. 2014. Improving the search performance of SHADE using linear population size
reduction. In: Proceedings of the 2014 IEEE Congress on Evolutionary Computation (CEC); IEEE. p. 1658–1665.
Philippine Journal of Science
Vol. 149 No. 3-a, October 2020
Velasco et al.: Comparative Study of Heuristic
Algorithms for EIT
760
TAVARES RS, MARTINS TC, TSUZUKI MSG. 2012. Electrical impedance tomography reconstruction through
simulated annealing using a new outside-in heuristic and GPU parallelization. In: Journal of Physics: Conference
Series Vol. 407(012015); IOP Publishing. p. 1–15.
TIAN N, LAI CH, PERICLEOUS K, SUN J, XU W. 2012. Contraction-expansion coefficient learning in quantum-
behaved particle swarm optimization. In: Proceedings of the 2011 Tenth International Symposium on Distributed
Computing and Applications to Business, Engineering and Science (DCABES); IEEE.
UHLMANN G. 2009. Electrical impedance tomography and Calderón’s problem. Inverse Problems 25: 123011.
VENKATRATNAM C, NAGI F. 2017. Spatial resolution in electrical impedance tomography: a topical review.
Journal of Electrical Bioimpedance 8: 66.
YANG XS. 2008. Nature-Inspired Metaheuristic Algorithms. Luniver Press.
YANG XS. 2010. A new metaheuristic bat-inspired algorithm. In: Cruz C, González JR, Krasnogor N, Pelta DA,
Terrazas G eds. Nature Inspired Cooperative Strategies for Optimization ( NISCO 2010); Vol. 284 of Studies in
Computational Intelligence; Berlin: Springer. p. 65–74.
YORKEY T, WEBSTER J, TOMPKINS W. 1987. Comparing reconstruction algorithms for electrical impedance
tomography. IEEE Transactions on Biomedical Engineering 34: 843–852.
ZHANG J, SANDERSON A. 2009. JADE: adaptive differential evolution with optional external archive. IEEE
Transactions on evolutionary computation 13(5): 945–958.
ZUO Y, GUO Z. 2003. A review of electrical impedance techniques for breast cancer detection. Medical Engineering
& Physics 5: 79–90.
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Vol. 149 No. 3-a, October 2020
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Algorithms for EIT
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APPENDICES
Appendix I. Firefly algorithm.
The flashing light produced by fireflies in a tropical summer sky are known to have two fundamental functions –
namely, to attract mating partners and to attract potential prey. In FA, these flashes can be formulated in such a way
that they are linked to the objective function to be optimized. It is a metaheuristic algorithm designed by Xin-She
Yang in 2007 (Yang 2008). One can find many applications of this algorithm in the literature [see, for example, Fister
et al. (2013) and Cajayon et al. (2020)].
An initial population of virtual fireflies is randomly generated. In each time step, the light intensity of each firefly is
compared pairwisely. In the standard FA, light intensity is determined by the objective function. If a firefly has a
greater light intensity than firefly i, the latter will fly towards the former. Note, however, that the movement of firefly
is determined by three terms: its current position, attraction to the brighter firefly , and a random walk. While the
light intensity is referred to as an absolute measure of emitted light by the firefly, the attractiveness is a relative
measure of the light that should be seen in the eyes of the beholders and judged by other fireflies (Fister et al. 2013).
Attractiveness is affected by the distance between firefly and firefly , attractiveness at , and the degree of
absorption of light in the air. The light intensities of the fireflies are then updated given the new positions. The fireflies’
positions (solutions) are ranked and the current best solution is updated. Detailed discussion of this firefly-inspired
algorithm can be found in the study of Fister et al. (2013), while the pseudo-code is given below.
Extensive simulations shown by Fister et al. (2013) were carried out to compare the performance of FA with particle
swarm optimization and GA. Results showed that FA finds the global minima more efficiently and with higher success
rate.
Firefly Algorithm Pseudo-code
Input: Objective function
for dimensions, number of fireflies ,
, light intensity
is determined by
Output: cost function
at optimal
1: Generate initial population of fireflies
2: Initial evaluation of all fireflies.
3: while do
4: Increment .
5: for to do
6: for to do
7: if
then
8: Move firefly towards with
, where
is the Cartesian distance between
two fireflies
and
and
is a vector of random numbers.
9: end if
10: Evaluate new solutions and update light intensity.
11: end for
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12: end for
13: Reduce .
14: Rank the fireflies and find the current global best solution.
15: end while
Appendix II. Novel bat algorithm.
NBA is a metaheuristic method proposed by Xian-Bing Meng et al. (Meng et al. 2015). NBA is one of the variants of
the basic bat algorithm (BA) developed by Xin-She Yang in 2010 (Yang 2010) based on the echolocation behavior of
bats. One of the bat species, known as microbats, are famous for using echolocation extensively. These bats emit a
very loud sound pulse and listen for the echo that bounces back from the surrounding objects. They use this
echolocation behavior to detect prey, avoid obstacles, and locate their roosting crevices in the dark (Yang 2010). The
original BA, however, did not take into account the capacity for Doppler shift compensation of these bats. Their ability
to locate surrounding objects or targets is attributed not only to their advanced capability of echolocation but also to
their self-adaptive compensation for Doppler Effect in echoes. Moreover, the original BA did not consider the fact
that bats hunt in a wide range of habitats. For these reasons, we employed NBA instead of the basic BA.
In NBA, all virtual bats, depicted by their positions and velocities, search for food in an dimensional space.
Starting with a randomly generated population of bats, each bat is subjected to a selection of habitat/s where it will
forage. This habitat selection is a stochastic decision such that if a uniform random number in is smaller than the
selection threshold, bat will forage in a wide range of habitats; otherwise, it would hunt in limited habitats. If a
randomly generated number is bigger that bat ’s pulse emission rate, a local search is performed by making the bat
fly randomly around a certain neighborhood of the current best position (solution). If bat ’s new position is closer to
the food than the current best, the rate of its pulse emissions is increased while the loudness is decreased. Finally, after
looping through all the bats, the bats are ranked according to their proximity to the food, which is represented by their
objective function values. If the best solution does not improve after a certain time steps, the loudness and pulse
emission rates of the bats are re-initialized. A detailed explanation of NBA can be found in the study of Meng et al.
(2015), while the pseudo-code is given below.
In the study (Meng et al. 2015), the performance of NBA was tested under twenty optimization problems and four
real-world engineering designs. Simulations showed that NBA is effective, efficient, stable, and superior over some
well-known algorithms such as the original BA, particle swarm optimization, flower pollination algorithm, and even
DE.
Novel Bat Algorithm Pseudo-code
Input: Objective function for dimensions, number of bats ,
maximum number of iterations
,
Output: cost function at optimal
1: Generate initial population of bats
and velocities
,
2: Initial evaluation of all bats.
3: Rank the bats with as the best global position and its velocity is
.
4: while do
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5: if
, where
then
6: Generate new solutions with
7: else
8: Generate new solutions with
,
,
where is a uniform random vector, is the smallest constant in the
computer, , and is the speed in the air.
9: end if
10: if then
11: Generate a local solution around the selected best solution using
where mean
, and rand is a Gaussian distribution with
mean 0 and standard deviation , and mean
is the average loudness of all bats
at time step . Note that is used to ensure that.
12: end if
13: Evaluate new solutions.
14: Update solutions, the loudness, and pulse emission rate using
,
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15: Rank the solutions and find the current best
.
16: if does not improve in time step then
Re-initialize the loudness and set temporary pulse rates which is a uniform
number between
17: end if
18: Increment .
19: end while
Appendix III. Genetic algorithm with multi-parent crossover.
First proposed by John Holland in 1975 (Holland 1975), GA is originally based on the Darwinian principle of
evolution. There have been a number of developments in GA theory and it is still a growing area. GA is one of the
most popular heuristic algorithms and has been applied to different problems in science and engineering. A typical
design for a classical GA would be as follows. Starting with a randomly generated population, GA carries out a process
of fitness-based selection and recombination to produce a successor population – the next generation. In the fitness-
based selection, the more fit members – called parents – of the population are selected. The selected members are then
recombined to form members of the successor population. Recombination has two components: crossover operator
and mutation operator. The crossover operator represents the combination of vector entries of a pair of parents to
produce children. The mutation operator, on the other hand, refers to making random changes to a single parent. The
new population is then carried over to the next generation. A widely used evolution scheme is elitism, where the best
one or two individuals from the current population are carried over to the next generation unaltered to guarantee that
the solution quality obtained by the algorithm will not decrease from one generation to the next.
In this study, GA-MPC – a variant of GA that proposes a new crossover method – and randomized operation in lieu
of mutation, is considered (Elsayed et al. 2011). Unlike GA’s original formulation, GA-MPC creates an archive pool
where the best individuals are stored and a selection pool with size , chosen randomly, is reserved for successful
individuals from the tournament selection. Individuals in the selection pool are used for performing crossover, where
three parents generate three offsprings: two are designed for exploitation; the other for exploration. After which, a
randomized operator is performed with probability to escape any local minimum. Individuals in the archive pool are
then merged with all of the offsprings, where the worst individuals are removed from the population. The surviving
population are then carried over to the next generation. A detailed explanation of GA-MPC can be found in the study
of Elsayed et al. (2011), while the pseudo-code is given below.
In the study (Elsayed et al. 2011), GA-MPC was tested using the real-world numerical optimization problems of the
IEEE CEC 2011 Real-World Numerical Optimization Special Session (Das and Suganthan 2010), and ranked first
among fourteen participating algorithms (Molina et al. 2018).
GA-MPC Pseudo-code
Input: Objective function for dimensions,
Output: cost function at optimal
1: Generate initial random population of size with
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where is a random vector with values in .
2: while do
3: Rank all the individuals in the population by their cost function value and choose
the best individuals to form the archive pool .
4: Apply a tournament selection with size and fill the selection pool.
5: Generate a random number in .
6: for each three consecutive individuals in the selection pool do
7: if one of the selected individual is the same to another then
8: Replace one by a randomly-selected individual in the selection pool.
9: end if
10: if then
11: Rank these three individuals .
12: Calculate .
13: Generate three offspring from the three parents with
14: end if
15: for each offspring do
16: Generate a random number in .
17: if then
18: Mutate the offspring by
, where is an individual from the
archive pool and
19: end if
20: end for
21: end for
22: if there is a duplicate individual then
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23: Replace the duplicate with
where
24: end if
25: end while
Appendix IV. LSHADE-SPACMA algorithm.
DE is another popular population-based algorithm similar to GA. Currently, there is a huge progress in the study of
improvements of DE and its diverse applications (Das et al. 2016). LSHADE, a DE-based algorithm (Price et al.
2005), is a proposed improvement to SHADE (Tanabe and Fukunaga 2013) that implements a linear population size
reduction scheme to focus on exploitation as the optimization process progresses (Tanabe and Fukunaga 2014).
Similar to SHADE, it uses the current-to- best/1 mutation strategy initially proposed in JADE (Zhang and Sanderson
2009), and a binomial crossover method (Tanabe and Fukunaga 2013, 2014). LSHADE also retains the improvements
proposed in SHADE for the adaptation of the scaling factor and the crossover rate using function distributions
(Tanabe and Fukunaga 2013, 2014).
Covariance Matrix Adaptation – Evolutionary Strategy (CMA-ES), on the other hand, is another adaptive algorithm
that adapts the multi-variate normal distribution (Hansen 2006). As an evolution-inspired algorithm, CMA-ES steps
are very similar to that of DE (Price et al. 2005) and GA (Holland 1975). It starts with an initial “population” of search
points sampled from the initial multi-variate normal distribution followed by selection and recombination to update
the mean, step size control to update the evolution path, and CMA (Hansen 2006).
In this study, LSHADE-SPACMA is considered. It is an improved version of LSHADE (Tanabe and Fukunaga 2014)
that uses a semi-parameter adaptation method for the scaling factor and crossover rate , and a hybridization
framework with a modified version of CMA-ES (Hansen 2006; Mohamed et al. 2017). As opposed to complete- or
self-adaptation, LSHADE-SPACMA uses semi-adaptation. The adaptation process depends on the number of function
evaluations carried out so far, i.e. until the algorithm reaches half of the defined maximum number of function
evaluations, it will only focus on adapting the crossover rate , while the scaling factor is generated randomly
using a uniform distribution within the range of . In the second half of the adaptation process, the scaling
factor is adapted. Furthermore, the algorithm uses a modified CMA-ES where a crossover operation is added after
the offspring generation (sampling of new points) step. The hybridization is done by allocating subpopulations
between LSHADE and modified CMA-ES to produce donor vectors, where the allocation throughout the optimization
process varies depending on the performance of each algorithm. A detailed discussion of LSHADE-SPACMA can be
found in the study of Mohamed et al. (2017), while a simplified pseudo-code is given below.
In the study (Mohamed et al. 2017), LSHADE-SPACMA was evaluated using the set of problems presented in IEEE
CEC 2017 Real-Parameter Special Session bound constrained case (Awad et al. 2016a), where it ranked fourth out of
the twelve participating algorithms (Molina et al. 2018).
Simplified LSHADE-SPACMA Pseudo-code
Input: Objective function for dimensions, memories and
Output: cost function at optimal
1: Generate initial random population with size .
2: Set values of memories and to .
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3: Initialize CMA parameters.
4: while < do
5: Semi-parameter adaptation (SPA) for scaling factor and crossover rate .
During the first part of SPA, the adaptation is concentrated on and for the
second part the focus of the adaptation is on .
6: Split the population into two.
7: Generate donor vectors using LSHADE or modified CMA-ES.
8: Concatenate resulting vectors from LSHADE and modified CMA-ES.
9: Generate trial vectors and mutate with
.
10: Evaluate fitness of trial vectors.
11: Implement selection strategy.
12: Update population allocated to LSHADE and CMA-ES according to the relative
performance of the two methods.
13: Store successful parameters to memory of size .
14: Update archive with rate
.
15: Update memory memory
during the first part of SPA and
with during the
second part. Update also
.
16: Implement linear population size reduction with min pop size
.
17: Sort individuals and retain them based on the new population size.
18: Update CMA-ES parameters.
19: end while
Appendix V. LSHADE-cnEpSin algorithm.
As the name suggests, LSHADE-cnEpSin is another DE-based algorithm that is similar to LSHADE (Tanabe and
Fukunaga 2014) and its variants. While it shares certain characteristics with LSHADE (Tanabe and Fukunaga 2014)
or LSHADE-SPACMA (Mohamed et al. 2017) such as linear population size reduction or its use of current-to- best/1
mutation strategy, LSHADE-cnEpSin adds an ensemble sinusoidal approach to adapt the values of the scaling factor
non-adaptive sinusoidal decreasing adjustment and adaptive history-based sinusoidal increasing adjustment (Awad
et al. 2016b, 2017). In the former, a decreasing sine-based formula is used where the wave-like configuration dampens
as the optimization process progresses; in the latter, an increasing adaptive sine-based formula using Cauchy
distributions with mean taken from an external memory (which stores successful mean frequencies) is considered. The
choice of which sinusoidal approach to use is based on previous performance. In particular, a learning period is first
implemented for a certain number of generations before the respective probabilities for each sine-based formula is
updated (Awad et al. 2017). These sinusoidal approaches are only activated in the first half of the optimization process,
while the usual formulation of SHADE (Tanabe and Fukunaga 2013) in adapting the scaling factor , using Cauchy
distributions, is used in the second half (Awad et al. 2017).
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LSHADE-cnEpSin, although sharing the same adaptation process for the crossover rate with LSHADE (Tanabe
and Fukunaga 2014), adds another crossover operator using covariance matrix learning with Euclidean neighborhood
(Awad et al. 2017). This process is done by marking the best individual, and computing the Euclidean distance
between the best and every other individual in the population. A number of best individuals in terms of Euclidean
distance are then used to generate the covariance matrix, which are then used to update the target and trial vectors
(Awad et al. 2017). A detailed discussion of LSHADE-cnEpSin can be found in the study of Awad et al. (2017), while
a simplified pseudo-code is given below.
In the study (Awad et al. 2017), LSHADE-cnEpSin was evaluated using the set of problems presented in IEEE CEC
2017 Real-Parameter Special Session bound constrained case (Awad et al. 2016a), where it ranked third out of the
twelve participating algorithms (Molina et al. 2018).
Simplified LSHADE-cnEpSin Pseudo-code
Input: Objective function
for dimensions, memories
and
Output: cost function
at optimal
1: Generate initial random population with size
.
2: Initialize
3: Set values of memories
and
to .
4: Initialize covariance matrix settings.
5: while do
6: if number of generation is
then
7: Implement sinusoidal configuration to adapt using .
8: else
9: Use Cauchy distribution to adapt .
10: end if
11: Adapt using normal distribution.
12: for to population size do
13: Generate mutant vectors with
.
14: Apply covariance matrix learning or binomial crossover to generate trial vectors.
15: Store successful and .
16: end for
17: Update memory
of size , and archive with rate
.
18: Implement linear population size reduction with min pop size
.
19: Sort individuals and retain them based on the new population size.
20: end while
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Appendix VI. EBOwithCMAR algorithm.
Effective butterfly optimizer (EBO), proposed in 2015, is inspired by the mate locating behavior of male butterflies
(Kumar et al. 2015). Two mate locating behaviors are used for the modification of the population: perching for
exploitation and patrolling for exploration. The algorithm starts with an initial population of male butterflies, which
are then divided into two subpopulations: main and auxiliary butterflies. The main population of butterflies start
perching and only change to patrolling if their positions are not updated by the former strategy.
In this study, an improvement to EBO named EBOwithCMAR is considered. Unlike EBO that uses two
subpopulations, EBOwithCMAR uses three: two are for EBO (Kumar et al. 2015) and one for covariance matrix
adapted retreat (Hansen 2006; Kumar et al. 2017). The additional subpopulation is meant to improve the exploitation
capability of the algorithm. Moreover, EBOwithCMAR improves the perching and patrolling strategies by adjusting
the crisscross and towards-best modification, and adding a crossover operator (Kumar et al. 2017). It also uses an
adaptive strategy for the scaling factor and the crossover rate and a linear population size reduction – the same
schemes used in LSHADE (Tanabe and Fukunaga 2014). A data sharing scheme is also added where the better
performing algorithm shares information about the solution to the other algorithm (Kumar et al. 2017).
Each cycle (iteration) begins with a “learning period” where both perching and patrolling schemes are given equal
probability. Probabilities are held fixed until half a cycle is reached, which are then updated. Once a full cycle is
reached, the algorithm implements a data sharing scheme where the better performing algorithm between EBO and
CMAR is determined. If EBO is the better algorithm, the population dedicated to CMAR is replaced by a random
element from the main population of EBO. On the other hand, if CMAR is the better algorithm, the worst individual
in the main population of EBO is replaced by the best individual from the population dedicated to CMAR. After the
data sharing scheme, parameter values are reset and the probabilities for perching and patrolling are returned to their
initial values. EBOwithCMAR further enhances the exploitation capability of EBO by employing sequential quadratic
programming at the later phases of the optimization process. A detailed discussion of EBOwithCMAR can be found
in the study of Kumar et al. (2017), while a simplified pseudo-code is given below.
In the study (Kumar et al. 2017), EBOwithCMAR was evaluated using the set of problems presented in IEEE CEC
2017 Real-Parameter Special Session bound constrained case (Awad et al. 2016a), where it ranked first out of the
twelve participating algorithms (Molina et al. 2018).
Simplified EBOwithCMAR Pseudo-code
Input: Objective function for dimensions, population sizes and ,
and , and ,, , , and cycle
Output: cost function at optimal
1: Generate initial random population with size .
2: Set for EBO and for CMAR to 1. Set other parameters.
3: Randomly assign members of the main population to three subpopulations , and
for EBO and for CMAR. The max and min population size for are and
, respectively, and for , and is the pop size for .
4: while do
5: if number of cycle is then
6: Update and .
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7: end if
8: if number of cycle is then
9: Implement data sharing.
10: Update parameters for EBO using and CMAR with .
11: Reset
,
and number of cycles.
12: end if
13: if
then
14: Apply EBO.
15: Implement linear population size reduction and reallocate subpopulations.
16: end if
17: if rand
then
18: Apply CMAR.
19: end if
20: if rand
and then
21: Apply SEQ.
22: if best solution is improved then
23: Set
to .
24: Update
and
.
25: else
26: Set
to .
27: end if
28: end if
29: Sort individuals and update allocations between subpopulations.
30: end while
Appendix VII. Parameter setting for heuristic methods.
The parameter settings used in all the algorithms are set to values in their original formulation. In particular, the default
values in the main papers cited in Section 3 for each algorithm are chosen, unless certain configurations need to be
made to address issues, such as algorithmic complexity. The lack of parameter tuning to fit the EIT problem in
conducting the numerical simulations is intended to ensure that the study does not favor any algorithm, which allows
for a thorough comparison of the original methods.
Parameters for FA were set as follows
, and, where
is the initial randomization
parameter, is the light absorption coefficient, and is the base attraction coefficient. The population size in each
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generation for FA varies for each case: Case 1 has
fireflies, Case 2 has
, and Case 3 has
. Each firefly would
require the computation of the cost functional (18), which is expensive and accounts for the major running time of the
algorithm.
0
0.9
1
0.6
0
0.9
1.5
0.1
1
0.9
2
10
0.9
In the table above, the parameter settings for NBA are enumerated, and denote the rate, frequency, and loudness
of sound pulses emitted by bats, respectively denotes the probability of habitat selection, while denotes the
compensation rate for Doppler effect in echoes. The constants and are used to update and. The
parametersand are re-initialized if the best solution does not change after time steps. Meanwhile, the
contraction-expansion coefficientand the inertia weight are adjusted according to the parameter control method
described by Tian et al. (2012), i.e. using
where is the current time step, and for
while for. While
is defined to be the maximum number of time steps in the study of Tian et al. (2012),
in this work, we set its value to be the IEEE arithmetic representation for positive infinity.
Parameters for GA-MPC were set as follows: population size, “crossover factor”
crossover rate, tournament selection size generated randomly between 2 and 3, and archive pool
size(half the population size) (Elsayed et al. 2011). LSHADE-SPACMA and LSHADE-cnEpSin share
several parameter values: initial population sizeand minimum population size as both
algorithms implement a linear population size reduction, initial values for at , factor that controls the
greediness of the mutation strategy archive rate, and memory sizeused in
storing adapted parameters (Awad et al. 2017; Mohamed et al. 2017). Parameter settings specific to LSHADE-
SPACMA are as follows: initial value for used in the hybridization framework, and learning rate
used in updating the probability for hybridization (Mohamed et al. 2017); while the parameter specific to
LSHADE-cnEpSin used for the non-adaptive sinusoidal decreasing adjustmentis set to(Awad et al. 2017).
EBOwithCMAR uses the following parameter values: for EBO, it uses the same initial population size
, minimum population size for with LSHADE-SPACMA and LSHADE-cnEpSin (Awad et
al. 2017; Mohamed et al. 2017), , and memory size; while for
CMAR
, number of evaluations constituting a cycle
, and local search
update probability
(Kumar et al. 2017).
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