http://journal.uir.ac.id/index.php/JGEET 
 

 
 

E-ISSN: 2541-5794  
   P-ISSN: 2503-216X  

Journal of Geoscience,  
Engineering, Environment, and Technology 
Vol 8 No 1 2023 

 

Darmawan, D. et al./ JGEET Vol 8 No 1/2023   39 

RESEARCH ARTICLE 
 

Electrical Resistance Tomographic by Using Current Injection and 

Magnetic Field Induction 

Dudi Darmawan1*, Deddy Kurniadi2, Suprijanto2 
1 Engineering Physics, Universitas Telkom, Jl Telekomunikasi no 1, Jawa Barat, Indonesia 

2Instrmentasi dan Kontrol, Institut Teknologi Bandung, Jl. Ganesha 10, Jawa Barat, Indonesia 
 

* Corresponding author: dudiddw@telkomuniversity.ac.id 
Tel.:+081320500247 
Received: Sep 20, 2022; Accepted: Mar 17, 2023. 
DOI: 10.25299/jgeet.2023.8.1.10560  

 
Abstract 

A critical issue in electrical tomography is ill-posed problems due to low sensitivity. In the electric current injection method, the placement 
of the injection electrode on the object boundary can influence it. This condition causes the reconstruction result of parameter change far away 
from the boundary to be inferior in quality. Another excitation method is using magnetic field induction proposed to overcome these problems. 
Each reconstruction image was obtained using two methods with three types of parameter changes, that represented the edge and the cent er 
of the object position. Both reconstruction results are merged and further processed to enhance the quality of the image, bas ed on the average 
value of the resistivity of each element. The results show that the final image reconstruction has a smaller root mean square error (RMSE) than 
the electric current injection method. 
 
Keywords: III-posed; sensitivity; current injection;  magnetic field induction; reconstruction image 
 

 
 
1. Introduction  

An important issue in electrical impedance tomography is 
ill-posed problems (Gong et al., 2016) (López C. et al., 2015) 
(Harikumar, Prabu, and Raghavan, 2013) (Khan and Ling, 
2019). The things that cause the appearance of ill-posed are 
mismatch model, non-linear, low sensitivity, and the limited 
amount of data (information) (Alsaker, Hamilton, and 
Hauptmann, 2017) (Seppänen et al., 2009) (Chitturi and 
Farrukh, 2017). These factors have been the subject of 
attention from some of the research related to current 
injection tomography. 

The issue of sensitivity is influenced by several factors, 
including the large injection currents, the current position of 
the injection point, the position of the measurement 
electrodes, and electrode measurement conditions. These 
factors form a system of data collection on the current 
injection tomography. Therefore, a solution to overcome this 
problem is to find the appropriate configuration of data 
collection systems, such as adjacent, cross, opposite, multi-
reference, and adaptive. However, all of those data collection 
systems use excitation electrodes attached to the object 
boundaries. This condition causes the changes in the 
parameters in the middle of the object to be hard to be 
detected.  

Another method that can be used to overcome the low 
sensitivity is the use of another excitation method, i.e. the 
magnetic field. The induction of a magnetic field to the inside 
of the object is expected to overcome the low sensitivity. This 
is possible because the stimulation of the magnetic field is 
done to the entire object (Wang et al., 2018) (Feldkamp and 
Quirk, 2019) (Ma, Wei, and Soleimani, 2013) (Ma and 
Soleimani, 2017). In this way, the same sensitivity to the 
whole object can be reached so that the spatial resolution of 

the uniform resistivity distribution is obtained (Seppänen et 
al., 2009) (Chitturi and Farrukh, 2017). 

However, this raises another problem which is 
irregularities of the magnetic field given part. The solution to 
solve this issue is to find the optimal configuration of the 
induction system (Darmawan et al., 2015) (Wang et al., 2018) 
(Alsaker, Hamilton, and Hauptmann, 2017). The induction 
system includes the shape and dimensions of the coil inducer, 
induction number, position, and configuration of the 
induction. One form of alternative coil used is the rectangular 
coil that has been used for imaging using the eddy current 
method (Deddy Kurniadi, 2014) (Darmawan et al., 2015). 
Experiments were performed using a variety of frequency 
values and give good results. Therefore, the selection of a 
rectangular shape is attractive to apply to the case of 
tomography. This form is believed to provide the 
homogeneous distribution of the magnetic field, especially 
attached to the square-shaped object. 

Therefore, the incorporation of the current injection 
method for magnetic field induction method is attractive for 
development. The ability of the current injection method in 
detecting parameter changes in the edge can be accomplished 
by the method of magnetic field induction. Providing a 
magnetic field to the center of the object is expected to 
address the issue of sensitivity in the current injection 
method. 

This study was conducted to obtain the reconstructed 
image of each method. Furthermore, both image 
reconstruction results are combined and evaluated. This 
image fusion has been widely used in various image 
processing and merging methods (Rane, Kakde, and Jain, 
2017) (Zhao, Li, and Cheng, 1993).  

2. Methods 

http://journal.uir.ac.id/index.php/JGEET


 
40  Darmawan, D. et al./ JGEET Vol 8 No 1/2023 
 

2.1  Forward and Inverse Model of Current Injection 
Method 

In the current injection method, an electrical current is 
injected through some point in the object's surface. Currents 
will spread and cause electric potential distribution inside the 
object. The relationship between the electric potential (V), 
and resistivity (r) was formulated by the Laplace equation 
(Darmawan et al., 2016) (Ma and Soleimani, 2017) (Darbas et 
al., 2021). 

𝛻 ∙  
1

𝜌
𝛻𝑉 = 0   in    (1) 

With boundary conditions of potential and current density on 
the surface, 

𝑉 = 𝑉0     on  𝜕  (2) 

1

𝜌

𝜕V

𝜕𝑛
= 𝐽0  on  𝜕  (3)                                                            

In the current injection tomography, the forward model is 
the mapping function that states the value of the electric 
potential distribution as a function of the resistivity 
distribution, 𝐹(𝜌) →  𝑽|_. The electric potential function is 
obtained through the solution of the Laplace equation.  

In this study, a solution of forward models is obtained 
through the concept of current conservation. This concept 
states that the net amount of current in each element is equal 
to zero. In other words, the current coming into each element 
is equal to the current coming out of that element.  
 In the case of two dimensions, the concept of current 
conservation is obtained through the application of double 
integrals in equation (1). 

∬ (𝛻 ∙
1

𝜌
𝛻𝑉)

𝑆

  ∙ 𝑑𝑆 = 0  

(4)   

By using the divergence theorem, the surface integral along S 
on the left side of equation (4) turns into an integral along a 
closed path l surrounding one element. 

∮
1

𝜌
 𝛻𝑽 ∙  𝑑𝑙 = 0

𝑙
  

 (5) 
The left side of equation (5) states the sum of the current flux 
penetrated to the entire surface of the boundary surrounding 
the element.  
In 2 dimensions, numerical solutions to equation (5) produce 
equation (6). 
 

∑
1

𝜌
𝛻𝑽 .  𝑙 = 0 𝑙    

 (6) 
with the boundary condition being current flux density (Je, i) 
[15]. 

𝐽𝑒(𝑖) = {
𝐽, 𝑓𝑜𝑟  𝑖𝑛𝑗𝑒𝑐𝑡𝑒𝑑 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 

0, 𝑓𝑜𝑟 𝑛𝑜𝑡 𝑖𝑛𝑗𝑒𝑐𝑡𝑒𝑑 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 
 

This is a Neumann boundary condition. 
Furthermore, equation (6) is applied to all elements and 

produces some linear equations connecting the potential 
value of an element with the potential values of neighboring 
elements. The linear equations of potential values of all 
elements can be arranged in the matrix-vector form, such as 
equation (7). 

 

𝐺𝑁×𝑁.  𝑉𝑁×1 =  𝐶�̅�×1  

(7) 
with 
 G = admittance matric 
 �̅�  = potential vector 

 𝐶̅  = current source vector 
 N = number of elements 

Furthermore, obtaining the resistivity distribution from 
the boundary potential distribution observed is done by using 
the linearization method. A function that maps the boundary 
potential distribution back into the resistivity distribution, 

𝐹−1[𝑉] →  𝜎|  is known as the inverse model. Through the 
linearization method, it was found that changes in resistivity 
become proportional to the potential boundary changes, 
according to the equation (8). 

∆�̅� = 𝑆 ∆�̅�  
(8) 

with S as the sensitivity matrix.  
 
The Tikhonov regularization (Dingley and Soleimani, 2021) is 
used to get a solution so that equation (8) turns into equation 
(9). 

∆�̅� =  (𝑆 + 𝛼𝐼)−1∆�̅� 
(9) 

And finally, resistivity reconstruction results were obtained 
through 

𝜌𝑎𝑛𝑜𝑚𝑎𝑙𝑖 = 𝜌ℎ𝑜𝑚𝑜𝑔𝑒𝑛 + ∆�̅�   

(10) 
Reconstruction results are evaluated numerically using 

the parameters of root mean square (RMS) as equation (11), 
and it’s called Error. 

𝐸𝑟𝑟𝑜𝑟 =
1

𝑁
√(𝜌𝑟𝑒𝑘𝑜𝑛𝑠𝑡𝑟𝑢𝑘𝑠𝑖 − 𝜌𝑢𝑗𝑖 )

2
  (11) 

 
2.2 Forward and Inverse Model of Magnetic Field Method 

In the method of magnetic field induction, the 
magnetic field change is raised in the coil. The induced 
current will appear in the object. The relationship between 
the electric potential (V), resistivity (), and magnetic 
potential (A) is defined by equation (12), which is known as 
Poisson's equation [20] 

𝛻 ∙  
1

𝜌
𝛻𝑉 = −𝜔𝐴 𝛻

1

𝜌
                      (12)   

The solution of equation (12) is obtained by the same 
approach as in section 2.1.  

∬ (𝛻 ∙
1

𝜌
𝛻𝑉)

𝑆
  𝑑𝑆 = ∬ (−𝝎𝑨 𝜵

1

𝜌
)

𝑆
𝑑𝑆 (13) 

The Numerical solution of equation (13) is equation (14). 

∑
1

𝜌
𝛻𝑽 .  𝑙 = − 𝜔𝐴 .  𝛻

1

𝜌
 ∆𝑆  

𝑙 

 

(14) 
Furthermore, the completion of some linear equations of all 
elements is done using equation (7). 
 
2.3 Magnetic Field Simulation 

Calculation of the magnetic field 𝐴 by the rectangular coil 
is carried out as follows: 

Induction of a magnetic field at a point by the 𝑑𝑙 ⃗⃗ ⃗⃗ ⃗conductive 
segment which is electrified by I satisfies the Biot-Savart 
equation. 



 
Darmawan, D. et al./ JGEET Vol 8 No 1/2023 41 

 

𝐴 = ∫
𝜇𝑜𝐼

4𝜋

𝑑𝑙⃗⃗⃗⃗  𝑥 �̂� 

𝑟 2
 

(15) 
with 

𝐴  = Magnetic field 
𝜇0 = permeability  
𝐼   = current 

𝑑𝑙  ⃗⃗⃗⃗⃗⃗ ⃗ = coil segment 
𝑟  = distance to the observation point 

If defined  𝑟 = √𝑥2 + 𝑦2 + 𝑧2   and  𝐾 =
𝜇0

4𝜋
  then the 

magnitude of the magnetic field by the current conductive 
segment is formulated in equation (16). 

𝑑𝐴⃗⃗⃗⃗⃗⃗ =
𝜇𝑜𝐼

4𝜋

𝑑𝑙⃗⃗⃗⃗  ×  �̂� 

𝑟
= 𝐾𝐼

(𝑖 ̂𝑑𝑥 + 𝑗 ̂𝑑𝑦 + 𝑘 ̂𝑑𝑧)

𝑟
 ×  

𝑟

𝑟
  

(16) 
The magnitude of the magnetic potential by the conductor 
along L is obtained by integrating equation (16) so that the 
magnetic potential is obtained by a straight conductor in 
equation (17). 

𝐴 = 𝐾𝐼 ∫
(𝑖 ̂𝑑𝑥+𝑗 ̂𝑑𝑦+𝑘 ̂𝑑𝑧)

𝑟
 ×  

𝑟

𝑟
 

𝐿

0
     (17) 

Thus the magnetic potential by the four sides of the 
rectangular coil conductor is obtained by adding up the 
magnetic field potential by the four rectangular sides. 

𝐴 = ∑ 𝐾𝐼 ∫
(𝑖 ̂𝑑𝑥+𝑗 ̂𝑑𝑦+𝑘 ̂𝑑𝑧)

𝑟
 ×  

𝑟

𝑟
 

𝐿

0
4
1                                               (18) 

The following is a numerical calculation of the magnetic field 
potential by one side of the rectangular coil. Suppose the 
starting point of the line conductor is at coordinates (x1, y1) 
and the endpoint of the line conductor is at coordinates (x2, 
y2). Divide the length of the conduit into n-line segments. If 
the distance of the two points to the observation surface is the 
same, z1 = z2 = z. 

The length of each delivery segment is 

∆𝑥 =
𝑥2 − 𝑥1

𝑛
 

∆𝑦 =
𝑦2 − 𝑦1

𝑛
 

The distance of a certain conductive segment to the magnetic 
potential calculation point satisfies equation (19) 
 

𝑟 2 = (𝑥𝑝 − 𝑥1)
2

+ (𝑦𝑝 − 𝑦1)
2

+ (𝑧𝑝 − 𝑧1)
2

 

(19) 

The magnetic potential components of any segment satisfy 
equation (20). 

∆𝐴𝑥𝑖 =
𝐾𝐼(𝑧𝑝 − 𝑧) ∆𝑦

𝑟
                         

∆𝐴𝑦𝑖 =
−𝐾𝐼(𝑧𝑝 − 𝑧) ∆𝑥

𝑟
                         

∆𝐴𝑧𝑖 =
𝐾𝐼[(𝑦𝑝−𝑦)∆𝑥−(𝑥𝑝−𝑥)∆𝑦]

𝑟
  (20) 

The magnetic potential component at an observation point by 
all conveying segments satisfies equation (21). 

𝐴𝑥 = ∑ ∆𝐴𝑥𝑖

𝑛

𝑖=1

,   𝐴𝑦 = ∑ ∆𝐴𝑦𝑖

𝑛

𝑖=1

  , 𝐴𝑧 = ∑ ∆𝐴𝑧𝑖

𝑛

𝑖=1

  

(21) 
The magnitude of the resultant magnetic potential at a point 
is obtained by complying with equation (22). 

𝐴 = √𝐴𝑥
2

+ 𝐴𝑦
2

+ 𝐴𝑧
2

               (22) 

3. Results and Discussion 

3.1 Simulation Results of Forward and Inverse Model of 

Current Injection Method 

The simulation results of the forward model using the 
method of the adjacent, cross, and opposite injection with 16 
times the injection shown in figure 1. 
The simulation results for the best reconstruction of the 
current injection method with three different anomalies are 
shown in table 1. 

The simulation of the forward model of magnetic field 
induction is done 16 times. Simulation results which show the 
distribution of magnetic field and electric potential 
distribution produced, are shown in figure 2. 

 

(a) 

 
(b) 

 
(c) 

Fig. 1. Potential distribution as the solution of the forward model of current injection method  (a) adjacent  (b) cross  (c) opposite. 
 



 
42  Darmawan, D. et al./ JGEET Vol 8 No 1/2023 
 

3.2 Simulation Results of Forward and Inverse Model 
of Magnetic Field Induction Method 

The simulation of the forward model of magnetic field 
induction is done 16 times. Simulation results show the 
distribution of magnetic field and electric potential 
distribution produced, shown in figure 2. 

 
(a) 

 
(b) 

Fig. 2. (a) Magnetic field distribution and (b) Potential distribution as the forward model solution of magnetic field induction method. 

Table  1.  Reconstruction Results Of Resistivity Distribution By 
Current Injection Method 

Table 2. Reconstruction Results Of Resistivity Distribution By 
Magnetic Field Induction Method 

Anomaly 
Reconstruction results of 16 times 

Injection 
Image Error 

  

0,0060 

  

0,0080 

  

0,0090 

Average 0,0077 

Furthermore, in the same way as the current injection 
method, determining the resistivity distribution on the 
magnetic field induction method is performed using 
equation (6) in equation (10). The best reconstruction 
results obtained are shown in table 2. 

3.3 Merging of Reconstruction Result of Current 
Injection and Magnetic Field Induction 

Merging cell value of the reconstruction results between 
the current injection and magnetic field method is 
conducted using 4 ways, namely minimum, maximum, max-
min, and average value (Kumar et al., 2016) (Arai, 2020)  
(Noushad, 2017) (Wang, 2020). 
- Minimum value 
 This way takes the smallest value of the value of each 
element of both methods, thus formulated as 

𝜌 = 𝑀𝐼𝑁(𝜌(𝑘), 𝜌(𝑑)) 

- Maximum value 
This way takes the greatest value from the value of each 
element of both methods, thus formulated as 

𝜌 = 𝑀𝐴𝑋(𝜌(𝑘), 𝜌(𝑑)) 

 
- Max-Min value 
This way retrieves the value of the smallest value of both 
methods if both resistivity values are smaller than a 
specified value and takes the greatest value if both 
resistivity values are greater than that specified value. 

𝜌_ = {

𝑀𝐼𝑁(𝜌(𝑘), 𝜌(𝑑)), 𝜌(𝑘)  𝑎𝑛𝑑  𝜌(𝑑) < 𝛿

𝑀𝐴𝑋(𝜌(𝑘), 𝜌(𝑑)), 𝜌(𝑘) 𝑎𝑛𝑑 𝜌(𝑑) > 𝛿

𝐴𝑉𝐺(𝜌(𝑘), 𝜌(𝑑)) ,   𝑓𝑜𝑟 𝑜𝑡ℎ𝑒𝑟𝑠

 

 
- Average value 

This way takes the value of the average of the 
value of both methods, thus formulated as 

𝜌 = (𝜌(𝑘) + 𝜌(𝑑))/2 

with 
k index for injection 
d index for induction 

Anomaly 
Reconstruction results in 16 times 

Induction 
Image Error 

  

0,0071 

  

0,0102 

  

0,0086 

Average 0,0086 



 
Darmawan, D. et al./ JGEET Vol 8 No 1/2023 43 

 

 
By using these ways, the simulation results of the merging 
of the two methods are shown in table 3. 

4. Conclusion 

The results of the reconstruction of the two methods, 
which were carried out separately, confirmed the 
advantages of each method. Tomography using the current 
injection method gives better results in detecting anomalies 
at the edges. While the magnetic field induction method 
gives better results in detecting anomalies in the middle of 
the test object. 

Furthermore, merging the reconstructed image of the 
two methods increases in image quality. This is indicated by 
the merged image which is better than the reconstructed 
image of each method. Through the three types of 
anomalies tested, the average method produces better 
results. 

Acknowledgments 

This work is supported by Hibah Penelitian Pasca 
Doktor 2019, Ministry of Research, Technology and Higher 
Education of the Republic of Indonesia 

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