A hybrid of iterative Gauss–Newton and one-dimensional convolutional neural network for high-resolution electrical impedance tomography

Electrical impedance tomography (EIT) is a nondestructive tomographic technique used to visualize the interior of materials. Compared to other tomography methods, such as x-ray computed tomography and magnetic resonance imaging, because no large magnets or radiation are required for EIT, the EIT apparatus has advantages in terms of its weight and size.¹ Therefore, it is expected to be applied as a nondestructive structural health-monitoring (SHM) method for existing intricate and complicated cementitious building materials.² In fact, there are reports on the visualization of foreign objects, such as steel bars, polyurethane blocks, and plastic plates in concrete;³ visualization of moisture transport in cementitious materials;^4–7 and visualization of cracks by EIT using self-sensing concrete coated with multi-walled carbon nanotubes.⁸ 

In EIT, the spatial impedance distribution in the sample can be visualized by solving the inverse problem using mathematical solution methods, such as one-step Gauss–Newton method, primal dual interior point method, and iterative Gauss–Newton (IGN) method. The governing equation of EIT can be expressed in the case of low frequency as $∇⋅σ∇ϕ=0$⁠, where σ is the conductivity and ϕ is the electric potential. Since EIT is an ill-posed problem, obtaining a unique solution is challenging. Without using mathematical approximations, it is impossible to solve ill-posed problems, which means that the solutions inevitably contain errors. In other words, the impedance distribution obtained using the EIT has some degree of inaccuracy.⁹ Therefore, improving the spatial resolution of the impedance distribution using mathematical methods is difficult.

Recently, to overcome this issue, the use of machine learning has been reported to improve the spatial impedance distribution in EIT.¹⁰ Machine learning-based methods using algorithms such as artificial neural networks (ANNs)^11,12 and convolutional neural networks (CNNs)^13–17 have been reported. Machine learning-based EIT has also been applied to SHM. Some reports have shown that EIT can visualize the moisture distribution in bricks and cement¹⁸ and classify the position of the rebar embedded in cement paste with an accuracy of 95.6%.¹⁹ However, the weakness of machine learning is the precise reconstruction of unseen data or data with few features.²⁰ Recent reports have described reconstruction methods that use reconstructed images as precursor data, such as the ANN with the image reconstructed by the one-step Gauss–Newton algorithm as training data²⁰ and the D-bar method and a CNN.²¹ 

In this study, to improve the spatial resolution of EIT, we developed a novel post-processing method that combines images reconstructed using different inverse problem-solving methods based on logical conjunctions. Comparative analyses were conducted between the traditional reconstruction approach and the newly developed method, utilizing both computational and experimental potential data. The findings conclusively demonstrated the enhanced efficacy of the newly developed method.

Figure 1(a) shows a schematic of the experimental setup used for the typical EIT measurements. N electrodes (16 in this case) were placed on the surface of the circular base material within which a circular foreign object was embedded. Fundamentally, when a current is injected between arbitrary adjacent electrodes, the potential differences between all adjacent electrodes, excluding the current-injected pair, are measured. During current injection, the typical distributions of the electric potential and electric field, which were calculated by the finite element method (FEM) using COMSOL Multiphysics 6.0, are shown in Fig. 1(b). When current was injected between electrodes 1 and 16, potentials were generated at each electrode. The potential difference between adjacent electrodes of N and (N + 1) can be expressed as $∅N→N+1=−∫NN+1E⋅ds$⁠. The foreign object affects the distribution of the electric field and thus affects the potential at each electrode. Figure 1(c) shows the potential difference V between the electrodes when a current was injected between electrodes 1 and 16. Figure 1(d) shows the potential difference measured when the current-injection electrode pairs were sequentially changed. This is referred to as the potential spectrum. When the number of electrodes is N, the number of data points in the potential spectrum is N(N − 3). Because 16 electrodes were used in this study, the number of data points in the potential spectrum was 208. Since a larger number of electrodes do not necessarily increase the resolution,²² 16 electrodes were employed to avoid the time and labor required for electrode formation and increased computational costs.

From the potential spectra, conductivity distribution images were obtained using three different solution methods. The first was the IGN method using MATLAB-based EIDORS.²³ This software is generally used to visualize the EIT.

The second is a machine learning-based method with a one-dimensional CNN (1D-CNN).²⁴ The training procedure for the 1D-CNN is shown follows: First, we made 8448 datasets obtained using COMSOL. We solved the forward problem using a simulation model with embedded foreign objects. The simulation model consisted of a base material of 89.5 mm in diameter with foreign objects of various diameters, as shown in Fig. 2(a). The conductivities of the base material and conductive foreign object were set to 0.1 S/m and 10⁹ S/m, respectively. The conductivity images are distinguished by relative color, with the base material, conductor, and insulator colored gray, black, and white, respectively. The obtained potential spectrum was normalized by dividing it with the minimum value. Then, we used TensorFlow2 (Python 3.6). The input layer had 208 voltages, and the output layer had 10 000 conductivity values corresponding to each position of the reconstructed image (100 × 100 pixels²). Four convolution, pooling, flattened, and dropout layers were used to learn the relationship between the input and output. The rectified linear unit function was used as the activation function. For the output layer, the activation function is a sigmoid function. The loss function and number of epochs were set to the MSE and 1000, respectively.

The third is the logical multiplication method proposed in this study, in which two conductivity images were obtained using two different reconstruction methods, namely, the IGN method and the 1D-CNN method, and the 256 grayscale of each pixel was inverted. Each pixel was normalized to a minimum value of −1 and a maximum value of 1. The grayscale data at the same position in the two reconstructed images were then logically multiplied by polar (a specific value of −1 or +1) with the grayscale data. Each pixel value is output from −1 to +1, as shown in Table I. Each pixel value was converted from 0 to 255 to produce a grayscale reconstructed image. We refer to this method as AND.

To compare the three reconstruction methods described above, we first used ideal potential data calculated from FEM simulations to obtain conductivity images reconstructed by each method. The simulation model consisted of a base material of 89.5 mm in diameter with foreign objects. The diameter of the foreign object was varied from 2 to 30 mm, and the center-to-center distance between the base material and foreign object was fixed at 15 mm, as shown in Fig. 2(a). The conductivities of the base material and conductive foreign object were set to 0.1 and 10⁹ S/m, respectively.

Then, using the potential data obtained from the experiment measurement, each method was compared similarly. Cement materials were used for the experiment. Samples were prepared using the following procedure. First, we fabricated a cement paste with a water-to-cement weight ratio of 0.5. The cement powder (Ordinary Portland Cement, TAIHEIYO CEMENT CORPORATION, Japan) and water were thoroughly mixed using a mortar mixer (MIC-362-0-01, Marui & Co., Ltd., Japan). Second, the cement paste was poured into cylindrical acrylic containers, of which inner diameter and height were 95.5 and 40 mm, respectively, with 16 electrodes made of stainless steel (SUS303) and a foreign object (S50C), as shown in Fig. 2(b). The diameter and length of the electrodes were 3 and 70 mm, respectively. Sixteen electrodes were placed at equally spaced locations around each circle. The distance between the center of the cylindrical container and the foreign object was ∼15 mm. The diameters of the foreign objects varied from 10 to 30 mm. Third, after leaving the samples at room temperature (20 °C) in air for one day, they were placed in water (20 °C) and left for seven days.

Figure 2(c) shows the top view of the cement paste sample used for the EIT measurements. An EIT system was developed for this study. An LCR-meter (IM3536, HIOKI E.E. CORPORATION, Japan) was used as the current source, and a digital multimeter (GDM-8261A, TEXIO TECHNOLOGY CORPORATION, Japan) was used as the voltmeter. The injection current was 5 mA, and the measurement frequency was 1 kHz. Since some researchers had previously measured electrical impedance properties of cement samples at a measurement frequency of 1 kHz,^11,12 we adopted the same frequency.

Using the potential spectrum obtained from the FEM simulation, we evaluated the effectiveness of the three methods: IGN, 1D-CNN, and AND. Figure 3(a) shows the conductivity distribution of the simulation model. The black, white, and gray areas in the images represent high, low, and standard conductivities, respectively. Figures 3(b)–3(d) show the reconstructed relative conductivity images derived by IGN, 1D-CNN, and AND methods as a function of size ratio p, respectively.

Using the IGN method, when p = 112.4 × 10⁻³, a circular foreign object was clearly visualized at the same position as in the model (black circle). However, with decreasing p, ripple-like artifacts appeared around the foreign objects. For p = 0.5 × 10⁻³, it was no longer possible to distinguish between artifacts and true data. However, using 1D-CNN and AND, the foreign object was clearly reproduced at all values of p. Upon close examination by visual inspection, for the 1D-CNN method, artifacts resembling haze or mistiness were observed in areas without foreign objects. However, such haze was not observed using the AND method.

For further analysis, we compared the contrast line profiles of the area indicated by the dotted line in Fig. 3(a). Figures 4(a)–4(c) show the line profiles of the images reconstructed using the IGN, 1D-CNN, and AND methods, respectively. The dotted line represents the model profile. Using the IGN method, the change in contrast at the foreign object boundary is gradual, indicating that the foreign object boundary is indistinct. As p decreased, clear peaks and dips associated with artifacts appeared in areas other than the location of the foreign object. When p is 0.5 × 10⁻³, the peak due to artifacts is large and it is difficult to identify foreign objects. For both the 1D-CNN and AND, the line profile closely followed that of the model when p was greater than or equal to 8.0 × 10⁻³. However, when p was less than 8.0 × 10⁻³, as p decreased, the peak position shifted to the center. For the 1D-CNN method, slight artifacts were observed in areas where there were no foreign objects.

Figures 4(d) and 4(e) show the NPE and NSE of the line profiles shown in Fig. 3(b)–3(d) as a function of p, respectively. Dotted lines indicate ideal values. The full width at half maximum (FWHM) of the foreign object peak was defined as the size φ, and the center coordinate of the FWHM was defined as the position r. As shown in Fig. 4(d), using the IGN method, as p decreased, multiple peaks due to artifacts appeared, while using both the 1D-CNN and AND methods, only one peak appeared, and the NPE decreased when p was less than 8.0 × 10⁻³. Comparing with the 1D-CNN and AND methods, below 8.0 × 10⁻³, NPE using the AND method was slightly better than that using the 1D-CNN method. As shown in Fig. 4(e), when the IGN method was p decreased, the size was overestimated. Using the 1D-CNN method, the error was reduced by more than 68% with p less than 78.0 × 10⁻³. AND reduced the error by more than 27% with p less than 8.0 × 10⁻³. Comparing with the 1D-CNN and AND methods, below 2.0 × 10⁻³, NSE using the AND method was slightly better than that using the 1D-CNN method. Therefore, when the size of the foreign object is small, the AND method has better NPE and NSE than the 1D-CNN method.

Based on the analysis using simulation data, the AND method was found to have advantages over other reconstruction methods. Next, we used the experimental potential data obtained from the actual cement samples for evaluating the effectiveness of the three methods. As in the previous analysis, three methods were used to convert the measured potential data into conductivity data: Fig. 5(a) illustrates the top views of the cement samples containing foreign objects with various diameters. Figures 5(b)–5(d) show the conductivity distribution images obtained using the IGN, 1D-CNN, and AND methods as a function of p. As mentioned earlier, using the IGN method, a foreign object was visualized in the same position as in mode. However, as p decreases, the artifacts increase. Using both the 1D-CNN and AND methods, no artifacts appeared for any p, and foreign objects were clearly visualized.

The reconstructed images and models were compared using the line profiles of the contrast at the center of the image. Figures 6(a)–6(c) show the line profiles of the images reconstructed using the IGN, 1D-CNN, and AND methods, respectively. Using the IGN method, as p decreased, the artifact increased, and at p = 12.5 × 10⁻³, a large low conductivity distribution appeared on the left side of the reconstructed image of the foreign object. 1D-CNN and AND methods follow the line profiles of the model.

Figures 6(d) and 6(e) show the NPE and NSE of the line profiles shown in Figs. 5(b)–5(d), respectively, as a functions of p. Dotted lines indicate ideal values. Using the IGN method, as p decreased, multiple peaks due to artifacts appeared. As shown in Fig. 6(d), using both the 1D-CNN and AND methods, there were no artifacts and only one peak was observed. As shown in Fig. 6(e), when the IGN method was p decreased, the size was overestimated. Using the 1D-CNN method, the error was reduced by more than 45% with p less than 49.9 × 10⁻³. In the range of p greater than 12.5 × 10⁻³, 1D-CNN and AND methods were almost the same. These trends align with those shown in Figs. 4(d) and 4(e). In this study, we used only foreign objects, where p is larger than or equal to 12.5 × 10⁻³. Consequently, the superiority of AND remains unclear. Future investigations using smaller foreign objects may highlight the advantages of the AND method.

Figure 7(c) shows the potential intensity ΔV as a function of p. When p were 112.4 × 10⁻³ and 12.5 × 10⁻³, ΔV were 62.1 and 8.5 mV, respectively. From 12.5 × 10⁻³ to 0.5 × 10⁻³, ΔV decreased and converged to about 2.5 mV. Within this p range, different ΔV was obtained for each p, and thus, visualization was possible even when p was small. The IGN method was not able to accurately distinguish small differences in ΔV, and artifacts appeared because it was not possible to estimate the solution. Because the 1D-CNN method solves the inverse problem using machine learning with differences in the potential spectrum as features, the resolution is considered to be higher than that of the mathematically solved IGN method. Therefore, even when p is small, a foreign object can be reconstructed separately from the base material. Figure 7(d) shows the change in potential spectrum ΔV as a function of position r when p is between 2.0 × 10⁻³ and 12.5 × 10⁻³. As position r decreased, ΔV decreased. p less than 4.5 × 10⁻³ showed almost no change in ΔV for r less than 15. The reconstructed images and line profiles for each r are shown in the supplementary material. This may explain why the position could not be visualized precisely. ∆V was insensitive to r but sensitive to p. If current injection patterns and potential measurement patterns that are sensitive to r and ΔV are selected, it is expected that the visualization will be sensitive to r as well as to p.

In this study, we developed a new post-processing AND method that combines conductivity images reconstructed using different inverse problem-solving methods (IGN and 1D-CNN) and conducted evaluations using a novel evaluation index. Using the potential spectra obtained from the simulations as input data, we reconstructed the conductivity distribution using the conventional IGN, 1D-CNN, and AND methods. Consequently, using the IGN method, with decreasing p, the size error was increased. The AND method proposed in this study exhibited smaller positional and size errors than the 1D-CNN method. Using these methods to convert the potential spectrum obtained from the EIT measurements of actual cement samples into a conductivity distribution, we demonstrated that both the 1D-CNN and AND methods are more accurate in terms of the position and size, compared to the IGN method. The proposed post-processing combines the benefits of each inverse problem-solving method. This is effective for the EIT in solving ill-posed problems. In addition, the proposed evaluation index can indicate the limits of the size ratio, enabling uniform evaluation.

In addition to post-processing, another way to improve resolution is to devise a measurement technique. One of the methods is to change the current injection pattern, for example, current injection via opposing electrodes or asymmetrically opposed electrodes. In principle, the spatial distribution of the electric field can be determined by the current injection pattern. By intentionally changing the spatial distribution of the electric field, the measured potential of each electrode can be clearly distinguished. Thus, the resolution for detecting the position and size of foreign particles can be improved. In the future, the effect of the current injection pattern on spatial resolution will be investigated.

The supplementary material contains conductivity mappings at different r.

The authors thank Mr. R. Ikeda for the sample preparation. T.F. and T.I. acknowledge the Asahi Glass Foundation for their financial support.

The authors declare no conflict of interest.

Keiya Minakawa: Data curation (equal); Formal analysis (equal); Investigation (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). Keigo Ohta: Data curation (equal); Formal analysis (equal); Investigation (equal); Software (equal); Validation (equal). Hiroaki Komatsu: Software (equal). Tomoko Fukuyama: Funding acquisition (equal); Methodology (equal); Resources (equal). Takashi Ikuno: Conceptualization (lead); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (lead); Resources (equal); Software (equal); Supervision (equal); Visualization (equal); Writing – review & editing (equal).

The datasets used and/or analyzed in the current study are available from the corresponding author upon reasonable request. The supplementary material provides other data.