Abstract
Electrical impedance tomography (EIT) is a method of internal imaging that requires fast, reliable reconstruction algorithms, especially as wearable technology becomes more popular. We introduce consecutive improvements to the historic Wexler scheme. Firstly, we express it in a form suitable for the finite element method. Furthermore, the algorithm is overrelaxed. The step size is variable and depends on the change in the gradient of the objective function, which improves convergence. The new, simpler electrode model further enhances image quality. We also propose a method for finding the initial conductivity. Those changes bring the scheme on par with standard algorithms such as NOSER and Gauss-Newton with total variational regularization. Moreover, we demonstrate its ability to scale with large-mesh tests.
Keywords: impedance tomography, inverse problems, optimization, finite element method
Subject terms: Biomedical engineering, Applied physics
Introduction
Electrical impedance tomography (EIT) is a non-destructive and non-invasive method of internal imaging. The examined body is stimulated by low-amplitude currents, and the resulting voltage is measured at electrodes placed on its surface. The acquired data are used to solve the inverse problem, which consists of determining the conductivity distribution within the body and knowing its response to electrical stimulation. The Wexler algorithm1,2 was one of the earliest methods in electrical impedance tomography (EIT), contributing significantly to the field’s development. However, its slow convergence rate and poor image quality led to its replacement by more suitable algorithms3–5. Real-time monitoring of intensive care patients6 and analysis of neonatal pulmonary function7 are cutting-edge topics of interest. Newer studies on the medical uses of EIT also include monitoring the brain8,9 or early screening of the liver and lungs10,11. Moreover, EIT has recently expanded beyond its traditional uses into wearable technologies12.
State-of-the-art EIT imaging methods can be categorized into several groups. Building on the advancements highlighted previously, one group comprises model-based iterative solvers with advanced regularizations that extend beyond classical Tikhonov or total-variation (TV) approaches4,13, including non-convex adaptive TV regularizations14 and second-order schemes15. Another group consists of direct (non-iterative) methods, with D-bar5 being the most developed. The integration of artificial intelligence, such as deep learning16, physics-informed neural networks (PINNs)17, and generative networks18, has also become prominent in EIT frameworks. Many recent studies employ artificial intelligence to enhance images reconstructed by model-based methods.
A clear trend in recent research is the enhancement and adaptation of classic EIT methods. For example, one-step linearized methods have been improved by incorporating anatomical information into the NOSER3,19 and GREIT20 algorithms. Iterative Gauss-Newton schemes have benefited from overrelaxation techniques13. They have also used post-processing with a convolutional neural network (CNN)21. Direct methods, such as D-bar and the Calderón method, have also advanced through deep learning22–25. These developments indicate ongoing opportunities for improving classic algorithms.
While recent EIT research has advanced powerful absolute reconstructions via Gauss–Newton with edge-preserving priors and a wide range of deep-learning hybrids, these approaches are computationally heavy and/or training-dependent, which limits adoption in real-time and wearable scenarios. On the other hand, widely used linear or difference schemes (e.g., NOSER/GREIT) are fast but provide only difference imaging capability. Moreover, as illustrated26–28, the trade-off between the complete electrode model (CEM) and the simplicity of the shunt model means that practical pipelines face modeling decisions with few documented middle-ground solutions suitable for real-time use. Finally, classic update rules such as Wexler’s have received little attention in modern step-size control and robust initialization, despite emerging evidence that principled acceleration and initialization can materially improve convergence.
To address these challenges, this paper proposes an enhanced Wexler scheme based on the finite element method. The new simple electrode model improves image reconstruction. The algorithm achieves a higher convergence rate by incorporating overrelaxation and a variable step length that adapts to changes in the objective function gradient. Additionally, a method for determining the initial conductivity is presented. The method will be tested and compared with the linearized one-step algorithm (NOSER)3 and the Gauss-Newton TV algorithm4.
Wexler algorithm
The Wexler algorithm is described in great detail in works1 and2. A brief overview of the algorithm is provided below. Consider the body 
with a conductivity distribution 
. On the boundary of the domain 
, there are N electrodes, and their edges are denoted by 
, where 
(see Fig. 1). During the measurements, the electrodes play different roles depending on the so-called projection angle k, which corresponds to one of the N possible choices of current electrodes located on the boundaries 
and 
. The driving electrodes act as sources of constant current density 
. Figure 1 presents the electrode configuration corresponding to 
. We also select a reference electrode connected to the boundary 
, with respect to which the voltages are measured on the remaining electrodes located on the boundaries 
, where 
.
Fig. 1.
Illustration (a) shows the arrangement of 
electrodes, with the boundary segments 
where 
marking their contact with the imaging domain 
. Illustration (b) presents the placement of the driving electrodes (boundaries 
and 
), the voltage measurement electrodes (boundaries 
, where 
), and the reference electrode 
for the measurement projection 
.
The EIT forward problem is governed by the partial differential equation of the structure, similar to the Laplace equation:
 |
1 |
where 
is the conductivity distribution and 
is the electric potential. For boundaries connected to current electrodes and the reference electrode, appropriate Neumann and Dirichlet conditions must be considered:
 |
2 |
To find the conductivity distribution based on the voltages measured on the electrodes, we introduce two systems A and B with the corresponding conductivity distributions 
and 
. These systems have different boundary conditions for each projection angle k:
 |
3 |
 |
4 |
is an element of the matrix that contains measured voltages on the boundaries 
for the projection angle k (Fig. 2). In this paper, we use data from an open dataset29,30. In the Methodology section, we provide a further description.
Fig. 2.
Systems A and B displayed in graphical form on the exemplary 8 electrode measurement set with adjacent current drive. Red and blue were used to mark current electrodes with positive and negative signs, respectively. Gray colour marks reference electrode.
Solving Eqs. (3) and (4) for conductivity distributions 
and 
we get electric potential distributions 
and 
. Ideal solution where we find conductivity distributions such as 
should lead to identical electric potential distributions in system B, with Dirichlet conditions set to measured voltages, and in system A without such constraints. The following quantity should then vanish:
 |
5 |
Unfortunately, while in EIT, we can measure electric potentials on the boundaries, but the conductivity of the studied body remains unknown. However, we can use the formula (5) to construct the iterative scheme. We treat R as the functional of the conductivity distribution 
. The potentials 
were calculated assuming that 
. Optimization of the functional with respect to 
leads to correction of the conductivity distribution. Assuming that domain 
is composed of 
elements 
and inside each element conductivity is constant and equal to 
:
 |
6 |
In order to minimize R, we compare derivatives to zero:
 |
7 |
We thus obtain the relationship between the conductivities 
and 
 |
8 |
which after substitutions
 |
9 |
leads to an iterative scheme:
 |
10 |
For the purpose of our article, we implement this algorithm using the finite element method (FEM). For conductivity calculations, we use the space of polynomials of degree 0. Potentials, on the other hand, are calculated using the space of shape functions that are polynomials of degree 1. The scheme (10) will be named Wexler in the later chapters.
Improvement of the algorithm
In the iterative scheme (10) for the Wexler algorithm, potentials and conductivities are approximated in finite element spaces: P1 (linear functions within each finite element) and P0 (constant functions within each finite element). In this section, we will unify this approximation, using space P1 for both 
and 
. We will introduce two methods to find the dependence of 
on 
:
- The first method is to express conductivity in the form of a combination of basis functions
of the P1 space in the functional (5).
Thus, we obtain:
Minimizing R with respect to the nodal values of
11 
:
where
, we get system of equations:
where matrix
12 
and 
are given with formulas: 
13 - The second method consists of direct minimization of the functional (5):
which leads to the equation:
14
Using the finite element method, we subject the coefficients on the right side of
15 
and 
to projection onto the P1 space. Performing appropriate averaging:
We get the following relation:
16
Another modification we will make is the way we bind 
to the values determined in each iteration. Unlike (9), we will make a substitution that will allow additional control of the convergence of the algorithm:
 |
17 |
where 
, while the symbols M1 and M2 refer to the proposed methods for calculating the relationship between 
and 
in the Eqs. (12) and (16). The notation (M1/M2) indicates that the formula can be applied to both methods; if the index is singular, e.g., M1, it means that the formula belongs purely to method one. Electric potentials will be calculated using the current iteration of conductivity distribution 
; therefore, we substitute:
 |
18 |
Eliminating the 
factor, we get the following iterative formulas:
 |
19 |
 |
20 |
Let us reformulate (19) and (20) as:
 |
21 |
where 
and
 |
The scheme (21) is a quasi-gradient method that could be subjected to overrelaxation by choosing 
, which can accelerate convergence. Constant steps, especially larger than unity, may lead to instability of the algorithm. Therefore, we propose a variable step size based on changes in the gradient of the goal function. Considering work on the gradient method with step adaptation31, we focus on the gradient directions of the consecutive iterations. Scalar product between last iteration gradient 
and current one 
is given as:
 |
22 |
Alignment of the gradients can be expressed as 
:
 |
Its value informs about stability between iterations and helps in choosing the step size:
- function is in a region of stable convergence, step size can be large,
- gradients are becoming perpendicular, iteration approaches the minimum of a function, step size should be reduced accordingly,
- gradients point in opposite directions, possible overshooting, step size should be reduced greatly.
New value of step size can be found through mapping of cosine value range to step size range 
:
 |
23 |
Empirically, we found the initial value 
and its range: 
and 
. It results in faster convergence and offers great stability. In the following, the improved algorithm is denoted as Wexler Gradient Direction Overrelaxation (WGDO).
Electrode model
A simplified electrode model is introduced to replace the shunt model. Each electrode contains two layers: the contact layer responsible for providing information on contact between the electrode and the studied body, and the conductor layer representing the conductivity of the electrode, as shown in Fig. 3. For brevity, reconstructions with the new electrode model will be labeled (NE).
Fig. 3.
Electrode models marked on the FEM grid: a) shunt electrode, b) new electrode, green colour marks area of contact conductivity, red - high electrode conductivity.
Finding initial conductivity
Information about background conductivity is not always available. Starting from an arbitrary, uniform value may lead to algorithm instability or slow convergence. We shall present a method for finding the body’s mean conductivity, which can serve as an initial point. Treating 
as constant inside the region, we are able to minimize the function from (5), similarly to how it was done for each element in the algorithm:
 |
24 |
where n is the current iteration. The default initial conductivity is set to the value of the background conductivity found in the paper29, the results will be compared to those achieved with those found using (24) (IC).
Methodology
An open dataset29,30 was used as the data source for the tests. We chose an adjacent electrode configuration (Fig. 4). However, without the use of Lagrange multipliers, Dirichlet boundary conditions on the potential field were imposed with respect to the reference electrode. To calculate the current density, we use the dimensions of the electrode submitted in the29 as height: 7.5 cm and width: 2.5 cm, and also the RMS (root mean square) of the current: 
mA. As a result, we get:
 |
25 |
The tank used in the experiment had a radius of 14 cm. The background conductivity measured by researchers in29 was 0.3 
.
Fig. 4.
Adjacent current injection pattern with reference electrode. The red and blue bars on the electrodes indicate positive and negative currents through the electrodes, respectively. Gray bar indicates reference electrode.
For the initial conductivity inside the tank, we chose a slightly lower value of 
. We set the new electrode conductivity to a high value: the contact area conductivity was set to 
and the electrode conductivity to 
. Those values were reached empirically. However, in the section on the initial value of conductivity for the Wexler algorithm, we discuss the method for obtaining the starting conductivity inside the tank.
Open data set29,30 provides many variants of experimental data utilizing plastic and metallic inclusions. We chose a few of them to compare the image reconstruction abilities of our modified version of the Wexler algorithm against its standard version, as well as more established methods such as NOSER and Gauss-Newton with total variational regularization (G-N TV). Each target photo was used to create simplified masks (Fig. 5) that contain information about conductivity. Using them as ground truth and as geometric information, quantitative analysis was possible.
Fig. 5.
Exemplary target photos with corresponding graphics. Gray scale in the range of [0, 1] describes the conductivity of the inclusions. Plastic objects as non-conductive were set to 0, background - 0.2, and metallic inclusions to 1. Exemplary photos of the phantoms with their graphical representations: a) case_3_4, b) case_5_1, c) case_6_4.
All of the methods were implemented using FreeFEM++32, an open-source partial differential equation solver based on the finite element method. Reconstructed images were analyzed utilizing python and its libraries such as numpy, scikit-image, and matplotlib.
Due to the methods being used for absolute (WGDO and G-N TV) and difference (NOSER) imaging, we scale our results with respect to the background conductivity and map it to the range [0, 1]. Given the known background conductivity 
, we define the dimensionless conductivity contrast
 |
26 |
If the reconstruction 
has a small scale bias, we calibrate it to the background by a single gain 
(where 
is the circular tank mask) and set 
.
For image comparisons, we use the Structural Similarity index (SSIM)33 via the scikit-image implementation34,35. SSIM is computed only inside 
. To limit outliers while preserving the sign of contrast, we map C to [0, 1] using tanh-based compression:
 |
27 |
where:
 |
28 |
typically with 
and 
. The second metric of our comparisons is the peak signal-to-noise ratio (PSNR). It quantifies the discrepancy between a reference image I and a reconstructed image image K through the mean squared error (MSE). For an image with dynamic range 
(e.g., 255 for 8-bit), PSNR in decibels is
 |
29 |
PSNR implementation from scikit-image was used. In addition, for comparison between WGDO and the standard Wexler algorithm, we use the error value (6) against the number of iterations. In all tests, NOSER is set to one iteration, and the G-N TV algorithm iterates until convergence with 5 inner iterations. Algorithm tests:
new electrode model compared with the shunt electrode for the standard Wexler algorithm,
WGDO against Wexler, both with the new electrode model, tested to the desired convergence,
WGDO, NOSER, and GN-TV comparison,
WGDO - NOSER comparison in terms of scalability, G-N TV is excluded here because NOSER is known to have faster frame times,
initial conductivity calculation tests for WGDO.
The algorithm scalability test will be performed by measuring the calculation time as a function of the mesh size (Fig. 6). The computation was done using an 8-core Ryzen 7 5700x CPU.
Fig. 6.
Finite element meshes used by the reconstruction algorithms, nv is the number of vertices and nt is the number of finite elements. Mesh 2 is used by the shunt electrode model; other meshes support the NE model.
The proposed method (24) of finding initial conductivity was compared to the results achieved by setting its value to a known constant from29.
Discussion of the results
New electrode model
The new electrode model provides noticeable improvements over the shunt model. Qualitatively reconstructed images present more structural similarity to the phantoms (Fig. 7). New electrode reconstructions present better contrast and also eliminate boundary noise. SSIM and PSNR metrics (table 1) show that the upgrade is measurable. Although quite modest due to noisy data and the small number of iterations, the value of PSNR increases by 
in case 4_4, and SSIM also confirms qualitative observations increasing by 
.
Fig. 7.
Photos of the experimental setups: Case_3_4, Case_4_4, Case_5_1 with metallic and plastic inclusions and corresponding reconstructions comparing electrode models in the first column29,30. New electrode reconstructions are signed NE. A new electrode model was computed on the Mesh 3, shunt model on Mesh 2 (Fig. 6).
Table 1.
Comparison between shunt electrode model and new electrode model (NE) for standard Wexler algorithm in test cases: Case_3_4, Case_4_4, Case_5_1.
| Wexler | Wexler(NE) | |
|---|---|---|
| SSIM case_3_4 | 0.58 | 0.73 |
| PSNR case_3_4 | 15.99 | 17.02 |
| SSIM case_4_4 | 0.60 | 0.82 |
| PSNR case_4_4 | 15.50 | 18.80 |
| SSIM case_5_1 | 0.53 | 0.68 |
| PSNR case_5_1 | 12.94 | 15.04 |
Figure 8 shows the evolution of error values as a function of the number of iterations. It can be observed that the NE model enables the Wexler algorithm to achieve lower errors. In addition, error curves of the NE model show a steeper descent in initial iterations.
Fig. 8.
Errors (6) in function of iterations for Wexler algorithm comparing two electrode models for test cases Case_3_4, Case_4_4, Case_5_1 from open data set29. The maximal number of iterations was set to 30.
Comparison of WGDO with the standard Wexler algorithm
The stopping condition used in this subsection relies on the change of error function values (6) between iterations:
 |
30 |
where 
is a small constant chosen arbitrarily. In the case of (Fig. 9) 
was chosen. This condition prevents excess iterations when the change of the error value R between iterations becomes insignificant.
Fig. 9.
Comparison between standard Wexler algorithm and WGDO using the following experimental setups data: Case_3_6, Case_4_4, Case_5_1. Both algorithms use a new electrode model. The number of iterations was determined by the convergence condition. Mesh 3 (Fig. 6) was used.
It can be seen (Fig. 10) that the WGDO algorithm often requires half as many iterations as the standard Wexler. In addition, the algorithm exhibited a steeper descent and achieved lower error values. Analysis of SSIM metrics shows that WGDO reconstructions are more similar to the target. The index values were improved from 
to [0.72, 0.84, 0.86]. PSNR displays reduced noise, improving from 
dB to 
dB (table 2).
Fig. 10.
Error values (6) in function of iterations for standard Wexler algorithm and WGDO with data of the test cases: Case_3_6, Case_4_4, Case_5_1 from open data set29. The maximal number of iterations was determined by the stopping condition.
Table 2.
SSIM and PSNR table comparison between standard Wexler algorithm and WGDO using the following experimental setups data: Case_3_6, Case_4_4, Case_5_1, with NE model.
| Wexler(NE) | WGDO(NE) | |
|---|---|---|
| SSIM case_3_6 | 0.79 | 0.84 |
| PSNR case_3_6 | 17.28 | 17.59 |
| SSIM case_4_4 | 0.81 | 0.86 |
| PSNR case_4_4 | 19.85 | 20.47 |
| SSIM case_5_1 | 0.65 | 0.72 |
| PSNR case_5_1 | 15.33 | 15.87 |
WGDO compared to NOSER and G-N TV
WGDO, NOSER, and G-N TV achieve comparable metric scores (3). However, they differ in contrast, edge sharpness, and background noise (Fig. 11). In case 3_6, NOSER presents the strongest image metrics (SSIM 0.85; PSNR 18.38), while WGDO and G-N TV reach similar SSIM ( 0.84) with lower PSNR (17.30). In case 4_4, WGDO leads SSIM (0.86), and PSNR is effectively tied between NOSER and WGDO (NOSER 20.55, WGDO 20.35) with a lower value for G-N TV (19.87). In the more challenging case 5_1, G-N TV provides the highest PSNR (17.16) and the best SSIM (0.75) while WGDO is close behind (0.74), NOSER is smoother but blurrier (SSIM 0.69; PSNR 15.86). Qualitatively, WGDO most consistently preserves edges and localization, but leaves a small residual background. NOSER suppresses noise but diffuses peaks as scene complexity increases. G-N TV offers clean, piecewise-constant maps that can lower peak contrast in easier scenes.
Fig. 11.
Comparison between WGDO, NOSER, and G-N TV, using NE model in reconstructions. Experimental setups: Case_3_6, Case_4_4, Case_5_1 from open data set29 were used to benchmark algorithms. WGDO was stopped after 30 iterations. Mesh 3 (Fig. 6) was used.
Initial conductivity
Qualitatively reconstructed images do not differ significantly (Fig. 13). The image metrics (4) indicate slightly worse image quality for images reconstructed with approximated initial conductivity (IC). Error functions (Fig. 12) show that the curves descend from the higher initial values. However, after 30 iterations, the error curves of all tested cases nearly converge. It shows that the proposed approximation may be useful when the initial conductivity is unknown.
Fig. 13.
Comparison between WGDO with starting from the conductivity set to the experimental value, with one computed with the proposed method (IC), both cases use the NE model. We use experimental setups: Case_2_2, Case_4_4, Case_5_1 from open data set29 in the tests. Computations on the Mesh 3 (Fig. 6).
Fig. 12.
Values of R (6) in function of iterations for WGDO with given conductivity and found by proposed method (IC) with data of the test cases: Case_2_2, Case_4_4, Case_5_1 from open data set29.
Scalability
Scalability was evaluated on meshes with 464, 647, 1,168, 1,511, 1,646, and 2,837 vertices. Figure 14 reports the average runtimes for experimental data from29,30 (cases 2_2, 2_6, 3_4, 3_6, 4_4, 5_1, and 6_3). WGDO, as well as NOSER, performed one iteration for each case. NOSER time excludes Jacobian assembly. In real-time reconstruction, even large conductivity changes can be handled efficiently by initializing each solve with the result of the previous iteration, which substantially improves performance. Frame time essentially becomes the time of a single WGDO iteration. What it means is that for the use cases where the Jacobian needs recomputing (changes to the geometry resulting in the mesh change, boundary conditions change), WGDO provides a faster frame rate due to comparable time frames when the Jacobian is not being recomputed for the NOSER algorithm.
Fig. 14.
Average computation times for experimental data from29,30 (cases: 2_2, 2_6, 3_4, 3_6, 4_4, 5_1, 6_3) in function of the mesh size. WGDO and NOSER performed one iteration. Tests were performed utilizing an 8-core Ryzen 7 5700x CPU. All meshes besides Mesh 2 (Fig. 6) were used.
Conclusions
The proposed modifications effectively address the limitations of the Wexler algorithm, thereby improving performance. The quality of the reconstructions was significantly improved. We notice it in Fig. 7 and the image quantitative metrics table 1: the proposed two-layer electrode enhances PSNR by up to 21
and SSIM by up to 32
. The scheme achieves lower error values and converges faster (Fig. 10) due to the introduced overrelaxation. The new scheme is in line with the G-N TV algorithm in terms of image quality (Fig. 11, table 3), often offering better PSNR, and in terms of iteration speed with NOSER (Fig. 14). We also introduced a practical method for estimating an appropriate initial conductivity (24). Image reconstructions with approximated initial conductivity match (Fig. 13, table 4) those with the known initial conductivity value. Error curves confirm that the convergence is not hindered by approximated initial conductivity (Fig. 12). The success of enhancing the Wexler algorithm creates an opportunity to further investigate the convergence acceleration procedures in the context of other EIT algorithms. By improving the Wexler algorithm, we increase the diversity and robustness of available EIT reconstruction techniques, which are essential for practical applications. Future work may revolve around using WGDO as a physics-based model for deep learning neural networks.
Table 3.
SSIM and PSNR comparison table for WGDO, NOSER, and G-N TV algorithms. Experimental setups: Case_3_6, Case_4_4, Case_5_1 from open data set29.
| WGDO(NE) | NOSER(NE) | G-N TV(NE) | |
|---|---|---|---|
| SSIM case_3_6 | 0.84 | 0.85 | 0.84 |
| PSNR case_3_6 | 17.52 | 18.38 | 17.30 |
| SSIM case_4_4 | 0.86 | 0.85 | 0.85 |
| PSNR case_4_4 | 20.35 | 20.55 | 19.87 |
| SSIM case_5_1 | 0.74 | 0.69 | 0.75 |
| PSNR case_5_1 | 16.00 | 15.86 | 17.16 |
Table 4.
SSIM and PSNR table. Comparison between WGDO with the conductivity set to the experimental value and one computed with the proposed method IC; both cases use the NE model. Experimental setups: Case_2_2, Case_4_4, Case_5_1 from open data set29.
| WGDO(NE) | WGDO(NEIC) | |
|---|---|---|
| SSIM case_2_2 | 0.79 | 0.77 |
| PSNR case_3_6 | 16.90 | 16.90 |
| SSIM case_4_4 | 0.86 | 0.83 |
| PSNR case_4_4 | 20.35 | 19.11 |
| SSIM case_5_1 | 0.74 | 0.73 |
| PSNR case_5_1 | 16.00 | 15.88 |
Author contributions
M.J. programmed algorithms and visualized results, M.J. and C.J.W. worked theoretically on algorithm improvement, and M.J. and C.J.W. wrote the manuscript.
Funding
No Funding.
Data availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
Declarations
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Data Availability Statement
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.