A novel framework for three-dimensional electrical impedance tomography reconstruction of maize ear via feature reconfiguration and residual networks

Forward and inverse problems in 3D EIT

The two main elements that define EIT are the forward problem and the inverse problem. In the forward problem, the potential distribution $\varphi$ across the field is calculated using the previously determined excitation current $\overrightarrow{I}$ and conductivity distribution $\sigma$, establishing the framework for the inverse problem and empirical measurement. The challenges in 3D electrical impedance imaging include a broad solution field range, complex geometry, and more. By constructing the complete electrode model (CEM) and finite element method (FEM), the three-dimensional measurement domain $\mathrm{\Omega }$ is discretized into N tetrahedral units. This turns a continuous, infinite degree of freedom problem into a discrete, finite degree of freedom problem; that is, the continuously varying conductivity ${\sigma }_0$ in the measurement domain is discretized into N elements, as shown in Eq. (1):

(1) $${\sigma }_{\Omega }\stackrel{\text{discretize}}{\to }{\sigma }_{ℝ}={\left\{{\sigma }_1,{\sigma }_2,\dots ,{\sigma }_{N-1},{\sigma }_N\right\}}_{\Omega }.$$

Solving the forward problem by Eq. (2):

(2) $$\varphi \left(x,y,z\right)=U\left({\sigma }_{\mathrm{R}},\overrightarrow{I}\right),\left(x,y,z\right)\in \mathrm{\Omega }$$where $U\left({\sigma }_{\mathrm{R}},\overrightarrow{I}\right)$ signifies the mapping relationship used to solve for the field potential $\varphi$, using the excitation current vector $\overrightarrow{I}$ and the discrete conductivity ${\sigma }_{\mathbb{R}}$.

Equations (3) and (4) concisely summarize the inverse problem, which aims to solve for the internal electrical impedance distribution ${\sigma }_{\mathbb{R}}$, given the known excitation current $\overrightarrow{I}$ and boundary measurement voltage ${\varphi }_L$.

(3) $$\mathrm{\Delta }\varphi \left(x,y,z\right)=J\mathrm{\Delta }{\sigma }_{\mathbb{R}}$$

(4) $${\sigma }_{\mathbb{R}}^{\mathrm{\prime }}\left(x,y,z\right)=F\left({\varphi }_l-\epsilon \right),l\in \left[1,L\right].$$

The amount of change $\mathrm{\Delta }{\sigma }_{\mathbb{R}}$ in the conductivity distribution ${\sigma }_{\mathbb{R}}$ is what causes the boundary voltage perturbation, which is represented by the letter $\mathrm{\Delta }\varphi \left(x,y,z\right)$. $F\left({\varphi }_l-\epsilon \right)$ represents the imaging methodology, J stands for the Jacobian matrix, also known as the sensitivity matrix, and ${\varphi }_l$ represents the boundary measurement voltage at a particular place. The hardware system adopts an adjacent excitation and measurement pattern, sequentially exciting and measuring in pairs. Figure 1 shows the first three excitations. Initially, electrodes 2 and 1 are excited, and the measured electrode pairs include [4,3], [5,4],… [48,47], [33,48]. A measurement cycle consists of 45 electrode pairs, excluding the excited pair. Then, electrodes 3 and 2 are excited to complete another measurement cycle; this is followed by the excitation of electrodes 4 and 3. The process continues until electrodes 1 and 16 are excited, resulting in 16 excitations for the first layer of electrodes. All three layers of electrodes use the same excitation method, totaling 48 excitations and 2,160 measurements. L = 2,160. System noise is represented by $\epsilon$.

Reconstruction of conductivity based on deep learning

The mathematical expression for EIT image reconstruction is depicted in Eq. (4). When given a known ${\varphi }_L$, t the endeavor to solve for the discrete variable ${\sigma }_{\mathbb{R}}$ to approximate ${\sigma }_{\mathrm{\Omega }}$ presents an ill-conditioned nonlinear inverse problem. CNNs, as data-driven, general-purpose, function-based approximators, utilize nonlinear activation functions at every network layer. This allows the network to encapsulate nonlinear relationships. Furthermore, the design of convolutional layers, characterized by a local receptive field and a weight-sharing architecture, ensures each neuron connects only to a small portion of the input data. Moreover, neurons within the same layer share identical weights. This structure equips CNNs with the capability to discern local and translation-invariant features in high-dimensional data, such as images. As a result, they are adept at managing nonlinear and intricate data. Utilizing their deep architecture and convolutional operations, CNNs can autonomously learn and distill salient features from data. In an effort to minimize predictive errors, these networks employ techniques like back-propagation and gradient descent for parameter optimization. Importantly, during the training process, the multilayered architecture of CNNs facilitates a seamless automated feature extraction pathway. This pathway transitions from raw data to low-level features and then to high-level composite features. Notably, this automated process significantly reduces the condition number required for computational solutions, thereby enhancing efficiency. Such pronounced efficiency is particularly effective in addressing challenges associated with ill-conditioned scenarios.

Additionally, regularization tactics, such as weight decay, dropout, and batch normalization, when implemented within CNNs, are instrumental in addressing ill-conditioned problems and simultaneously enhancing the model’s generalization proficiency. Equation (5) demonstrates the CNN approach to tackle this problem. Here, F represents the network topology, E and g denote the cost function and regularization term, respectively, and $\theta$ symbolizes the network parameter. This approach eliminates the necessity of calculating the sensitivity matrix J or manually determining components like the forward model, cost function, regularization, and optimizer (Aggarwal, Mani & Jacob, 2018; Ren et al., 2020; Zhang et al., 2022; Chen, Yang & Bagnaninchi, 2021). Deep learning-based inverse problem-solving frameworks effectively tap into the correlation between potential distribution information in the spatial domain, thus producing an optimal model with formidable generalization abilities within a recognized data spectrum.

(5) $$F_{\mathrm{L}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{n}\mathrm{e}\mathrm{d}}=\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}{\sum }_{m=1}^M\left(E\left\{\sigma ,F_{\theta }\left(\varphi \right)\right\}\right)+g\left(\theta \right).$$

In contrast to the two-dimensional EIT reconstruction task, $\sigma \left(x,y,z\right)$ encompasses not only the conductivity information ${\sigma }_{\left(x,y\right)}$ in the electrode layer plane but also integrates the conductivity information ${\sigma }_{\left(x,y\right)}\left(z\right)$ from the non-measurement plane. This is elucidated in Eq. (6), where z signifies the position of the measurement plane along the vertical axis. Drawing inspiration from Eq. (6), we adopt the representation of conductivity as depicted in Eq. (7) for this study.

(6) $${\sigma }_{\mathrm{\Omega }}={\oint }_{\mathrm{\Omega }}{\sigma }_{\left(x,y\right)}\left(z\right)dz$$

(7) $${\sigma }_{\mathrm{\Omega }}={∰}_{\mathrm{\Omega }}\sigma \left(x,y,z\right)dxdydz.$$

Following the fusion of Eqs. (1) and (7), Eq. (8) provides the mathematical representation of the approximate conductivity set after field discretization. N represents the number of cells after the finite element method (FEM) computation.

(8) $$\{\begin{array}{c}{∰}_{\mathrm{\Omega }}\sigma \left(x,y,z\right)dxdydz\approx {\sum }_{n=1}^N\left\{{\sigma }_1,{\sigma }_2,\dots ,{\sigma }_{N-1},{\sigma }_{N=95295}\right\}\hfill \\ \left\{{\sigma }_1,{\sigma }_2,\dots ,{\sigma }_{N-1},{\sigma }_{N=95295}\right\}\subseteq \mathrm{\Omega }\hfill \end{array}.$$

Single channel feature reconfiguration network for 3D EIT

In EIT imaging, a sensitivity matrix typically characterizes the relationship between the boundary-measured voltage changes and the internal electrical conductivity variations. Given that the potential distribution is not limited to a 2D plane, data collected from the boundary of the measurement domain presents challenges in reconstructing a comprehensive 3D image. In this work, an end-to-end approach is used to couple arbitrary nonlinear functions $F_{learned}:{\varphi }_L\to {\sigma }_{\mathbb{R}}$, thereby circumventing the need for sensitivity matrix computation. In fact, differences in tissue electric properties of measurement domain have an effect on distribution of electric field, leading to varied boundary voltage changes. While image reconstruction premised on minor boundary voltage alterations may be efficacious in 2D EIT reconstruction grounded in image segmentation theory, this method underperforms in 3D EIT, which is characterized by a larger number of reconstruction blind spots.

Therefore, we address the 3D EIT reconstruction problem from the perspective of data distribution, taking into account the subtle differences in boundary voltage. We then opt for residual mapping instead of the direct mapping typically used in traditional CNN approaches.

Principle of feature reconstruction

Based on the matrix design guidelines presented in Eq. (9), the measurements of the boundary voltage $V_b$ include two types of information: (1) univariate data derived from identical excitation electrodes but varying measurement electrodes (information across matrix rows), and (2) bivariate data derived from distinct excitation and measurement electrodes (information across matrix columns). In this study, we employ single-channel convolution to extract the amalgamated features, which are represented in the two types of information embedded in the reconstructed feature map, as illustrated in Fig. 2D. Unlike the one-dimensional convolution strategy employed by Li et al. (2020), our method places greater emphasis on the feature differences across distinct measurement objects. Additionally, sections c and d of Fig. 2 highlight the unique features derived from both column and row matrices, which themselves originate from variations in the excitation and measurement electrodes. Given the pronounced ill-posedness characteristic of 3D EIT, our method extracts features that are more differentiated compared to the one-dimensional sequence shown in Fig. 2A. Furthermore, the implicit information contained in the original data is also preserved to the greatest extent possible, avoiding the possibility of correlation patterns and data distribution being altered as a result of forced transformations from single-channel one-dimensional data to multi-channel two-dimensional data (Zhu et al., 2021). In our experiments, using $V_b$ for projection, and the feature map Fig. 2C is transposed according to Fig. 2D. The main purpose is to adapt to the computational rules of the convolutional sliding window, and to prioritise the computation of the row data with the most obvious feature differences. This data representation is more amenable to the “end-to-end” processing mode of CNNs compared to the original voltage sequences.

(9) $V_b\text{=}{\left[\begin{array}{l}{V}_{3,4}^{2,1}{V}_{4,5}^{2,1}\cdots V_{15,16}^{2,1}{V}_{17,18}^{2,1}{V}_{18,19}^{2,1}\cdots V_{32,17}^{2,1}{V}_{33,34}^{2,1}{V}_{34,35}^{2,1}\cdots V_{48,33}^{2,1}\\ V_{4,5}^{3,2}{V}_{5,6}^{3,2}\cdots V_{16,1}^{3,2}{V}_{18,19}^{3,2}{V}_{19,20}^{3,2}\cdots V_{17,18}^{3,2}{V}_{34,35}^{3,2}{V}_{35,36}^{3,2}\cdots V_{33,34}^{3,2}\\ \ddots \ddots \\ V_{1,2}^{16,15}{V}_{2,3}^{16,15}\cdots V_{13,14}^{16,15}{V}_{31,32}^{16,15}{V}_{32,17}^{16,15}\cdots V_{30,31}^{16,15}{V}_{47,48}^{16,15}{V}_{48,33}^{16,15}\cdots V_{46,47}^{16,15}\\ V_{1,2}^{18,17}{V}_{2,3}^{18,17}\cdots V_{16,1}^{18,17}{V}_{19,20}^{18,17}{V}_{20,21}^{18,17}\cdots V_{31,32}^{18,17}{V}_{33,34}^{18,17}{V}_{34,35}^{18,17}\cdots V_{48,33}^{18,17}\\ V_{2,3}^{19,18}{V}_{3,4}^{19,18}\cdots V_{1,2}^{19,18}{V}_{20,21}^{19,18}{V}_{21,22}^{19,18}\cdots V_{32,17}^{19,18}{V}_{34,35}^{19,18}{V}_{35,36}^{19,18}\cdots V_{33,34}^{19,18}\\ \ddots \ddots \\ V_{16,1}^{17,32}{V}_{1,2}^{17,32}\cdots V_{15,16}^{17,32}{V}_{18,19}^{17,32}{V}_{19,20}^{17,32}\cdots V_{30,31}^{17,32}{V}_{48,33}^{17,32}{V}_{33,34}^{17,32}\cdots V_{47,48}^{17,32}\\ V_{1,2}^{34,33}{V}_{2,3}^{34,33}\cdots V_{16,1}^{34,33}{V}_{17,18}^{34,33}{V}_{18,19}^{34,33}\cdots V_{32,17}^{34,33}{V}_{35,36}^{34,33}{V}_{36,37}^{34,33}\cdots V_{47,48}^{34,33}\\ V_{2,3}^{35,34}{V}_{3,4}^{35,34}\cdots V_{1,2}^{35,34}{V}_{18,19}^{35,34}{V}_{19,20}^{35,34}\cdots V_{17,18}^{35,34}{V}_{36,37}^{35,34}{V}_{37,38}^{35,34}\cdots V_{48,17}^{35,34}\\ \ddots \ddots \\ V_{16,1}^{33,48}{V}_{1,2}^{33,48}\cdots V_{15,16}^{33,48}{V}_{32,17}^{33,48}{V}_{17,18}^{33,48}\cdots V_{31,32}^{33,48}{V}_{34,35}^{33,48}{V}_{35,36}^{33,48}\cdots V_{46,47}^{33,48}\end{array}\right]}_{48\times 45}$

In $V_{out1,out2}^{in1,in2}$, $in1$ and $in2$ represent the excitation electrodes, while $out1$ and $out2$ correspond to the measurement electrodes.

3D EIT reconstruction network

Motivated by Eqs. (3) and (4), this study aims to model the nonlinear relationship between the conductivity distribution and the boundary voltage values, thereby proposing a solution for the 3D EIT inverse problem. This objective is further elaborated upon in Eqs. (6) and (7).

(10) ${\sigma }_{ℝ}\left(x,y,z\right)=F\left(V_b,I_n,R_{\left(w,b\right)}\right)$

(11) $$\{\begin{array}{l}{\varphi }_l=\left\{{\nu }_1,{\nu }_2,\cdots ,{\nu }_L\right\},L=2160,{\varphi }_l\in \mathrm{\partial }\mathrm{\Omega }\\ V_b\equiv {\varphi }_l\\ I_n=\left\{i_1,i_2,\cdots ,i_N\right\},N=48,I_n\in \mathrm{\partial }\mathrm{\Omega }\\ R_{\left(w,b\right)}=\left\{\left(w_1,b_1\right),\left(w_2,b_2\right),\cdots ,\left(w_m,b_m\right)\right\}.\end{array}$$

${\sigma }_{\mathbb{R}}$ represents the three-dimensional field conductivity distribution, while F symbolizes the neural network imaging algorithm. $V_b$ aligns with element ${\varphi }_l$ and solely undergoes reconstruction according to Eq. (9). $I_n$ denotes the boundary excitation current, and $R_{\left(w,b\right)}$ embodies the weight and bias parameters of the neuron.

The original data, represented as a one-dimensional voltage signal of 2,160 × 1, is transformed into a single-channel voltage data of 45 × 48 according to the rules stipulated by $V_b$, serving as the network input for reconstruction. The initial convolution layer is devised with one input channel and 64 output channels. To expedite the convolution calculation, the kernel size, stride, and padding are set to 7, 2, and 3, respectively. The data size post the initial convolution is 64 × 23 × 24. Following Maxpool, the data is fed into the ‘Bottleneck’, which houses two structure types—‘Bottleneck_1’ and ‘Bottleneck_2’. Both utilize the Relu nonlinear activation function for intermediate activation. Post average pooling, the data size is transformed to 2,048 × 1 × 1 and finally, 95,295 conductivity data is outputted via the fully connected layer (fc). To avoid excessive resource occupation and simultaneously ensuring the validity of the output data, a single layer fully connected mapping is employed for outputting the conductivity results after numerous experiments. This method aims for the fc output to adhere to the target conductivity trend as closely as possible, ultimately procuring an approximate value that fluctuates within a certain range. This embodies a regression analysis approach. Upon observing the reconstruction results, notable differences can be discerned between the reconstructed conductivity and the target conductivity. However, the error values of back propagation remain small and traditional methods of calculating the loss cannot effectively constrain the model to converge. To mitigate this, the error term is amplified in this article, enabling the convolutional model to identify larger error terms more easily and effectively converge the model. An overview of the method proposed in this article is depicted in Fig. 3.

Experimental instrument

The open-source software package “Electrical Impedance Chromatography and Diffuse Optical Chromatography Reconstruction Software” (EIDORS 3.10 with Netgen 5.3) (Adler, 2019) was employed for the generation of simulated experimental data. The actual data were sourced from the EIT system, designed and developed by our research team (Li et al., 2022), as represented in Fig. 4.

The 3D EIT boundary voltage acquisition used adjacent current excitation and adjacent voltage measurement, where one pair of adjacent electrodes was selected as the initial excitation end and the other two adjacent electrodes were selected as the measurement end. The system uses a three-layer electrode structure as shown in Fig. 5A. The excitation electrode uses cyclic excitation, while the measurement electrode performs periodic measurements in the order of Eq. (9). The conductivity distribution ${\sigma }_{\mathbb{R}}$ was calculated by FEM, and the voltage distribution $\varphi \left(x,y,z\right)$ in the field was obtained from Eq. (2), and 2,160 boundary voltages were collected according to the principle of adjacent excitation-adjacent measurement. Finally, the conductivity values ${\sigma }^{\mathrm{\prime }}\left(x,y,z\right)$ are reconstructed by the algorithm F. The basic principle is shown in Fig. 5.

Datasets

Measurement data of acrylic models

In this experiment, a NaCl-based solution with a conductivity of $350\mu S/cm$, and a fluctuation range of $\pm 3\mathrm{\%}$ was used as the background medium. The test subjects included a sphere, a cylinder, a cube, and a combination of all three, as depicted in Fig. 6.

Measurement data of complete maize ears

Freshly picked maize cobs were selected for the experiment, and the measurement areas of interest included the husk, kernels, and cob. The electrical properties of biological tissues vary greatly across different frequency ranges (Cao et al., 2019). According to the impedance spectrum-based frequency selection method by Li et al. (2023), the magnitude of the impedance change in maize cobs decreases after 8 kHz, and the impedance hardly changes any further in the frequency range greater than 80 kHz. Therefore, in this article, we employ a simulated, noise-free, ideal boundary voltage as a benchmark to evaluate the quality of the measured signal at excitation frequencies within 80 kHz, based on the similarity between the global and local distribution characteristics of the boundary voltage data points.

Maximum mean discrepancy (MMD)

Distributional differences are measured by calculating the distance between the kernel matrices of the two sample sets in the feature space, as shown in Eq. (16).

(16) $\text{MMD}=\sqrt{\frac{1}{n\left(n-1\right)}{\displaystyle {\sum }_{i\ne j}k}\left(x_i,x_j\right)+\frac{1}{m\left(m-1\right)}{\displaystyle {\sum }_{i\ne j}k}\left(y_i,y_j\right)-\frac{2}{nm}{\displaystyle {\sum }_{i,j}k}\left(x_i,y_j\right)}$where n is the size of the sample set X, m is the size of the sample set Y, $x_i$ and $y_i$ are the ith samples in the sample sets X and Y, respectively, and $k\left(x,y\right)$ represents the value of the kernel function (typically a radial basis kernel function is employed, such as a Gaussian kernel function).

Kernel density estimation (KDE)

The difference between the distributions of the two sample sets X and Yis compared by calculating the integral squared difference (ISE) as shown in Eq. (17).

(17) $$\{\begin{array}{l}{\widehat{f}}_X\left(x\right)=\frac{1}{n_X{h}_X}{\displaystyle {\sum }_{i=1}^{n_X}K\left(\frac{x-x_i}{h_X}\right)}\\ {\widehat{f}}_Y\left(y\right)=\frac{1}{n_Y{h}_Y}{\displaystyle {\sum }_{j=1}^{n_Y}K\left(\frac{y-y_j}{h_Y}\right)}\\ KD{E}_{ISE}\left({\widehat{f}}_X,{\widehat{f}}_Y\right)={\displaystyle \int {\left({\widehat{f}}_X\left(x\right)-{\widehat{f}}_Y\left(y\right)\right)}^2}dx\end{array}$$where x and y denote data points, $n_X$ and $n_Y$ denote the number of data points, K denotes the kernel function, and $h_X$ and $h_Y$ denote the bandwidth parameters.

Local maximum mean discrepancy (LMMD)

It can be used to measure the average of the MMDs of the sample sets X and Y in different local windows, thus reflecting the local distributional differences. The LMMD is calculated as follows:

(18) $$LMMD\left(X,Y\right)={\displaystyle \frac{1}{N}{\sum }_{i=1}^{N}MMD\left(X,Y\right)}$$where N is the number of windows.

k-nearest neighbor distribution complexity (KNNDC)

KNNDC quantifies the distributional complexity between two sample sets by calculating the average distance that each data point in sample set X needs to traverse to locate its k nearest neighbors within sample set Y. Similarly, the KNNDC denotes the average distance required by each data point in sample set Y to identify its k nearest neighbors. The formula is presented below:

(19) $$KNNDC\left(x,Y,k\right)={\displaystyle \frac{1}{k}{\sum }_{i=1}^{k}D\left(x,y_i\right)}$$where $y_i$ is the i-th nearest neighbor of x in the sample set Y, and $D\left(x,y_i\right)$ is the distance (e.g., Euclidean distance) between x and $y_i$.

Combining the results of the four indicators in Fig. 7, the similarity between the measured boundary voltage and the reference ideal voltage is high when the excitation frequency is in the range of 1–10 kHz. Therefore, the excitation frequencies selected for the experiment are 2,000, 4,000, 6,000 and 8,000 Hz.

Evaluation indicators

The algorithm’s performance is evaluated using root mean square error (RMSE), correlation coefficients (CC), structural similarity index (SSIM), and inverse problem solving time (IPST).

The RMSE is widely used to measure the deviation between predicted and true values:

(20) $$\mathrm{R}\mathrm{M}\mathrm{S}E\left(P,A\right)=\sqrt{{\displaystyle \frac{{\sum }_{i=1}^N{\left(P_i-A_i\right)}^2}{N}}}.$$

CC is commonly used as a statistical measure to quantify the degree of correlation between two variables. In this article, it is used to evaluate the correlation between two images, as described by the following equation:

(21) $$CC\left(P,A\right)=\frac{{\sigma }_{P_i{A}_i}}{\sqrt{{\sigma }_{P_i}^2\cdot {\sigma }_{A_i}^2}}.$$

SSIM is a measure of the degree of structural similarity between images:

(22) $$SSIM\left(P,A\right)={\displaystyle \frac{\left(2{\mu }_{P_i}{\mu }_{A_i}+c_1\right)\left(2{\sigma }_{P_i{A}_i}+c_2\right)}{\left({\mu }_{P_i}^2+{\mu }_{A_i}^2+c_1\right)\left({\sigma }_{P_i}^2+{\sigma }_{A_i}^2+c_2\right)}}$$where N denotes the number of samples, P denotes the reconstructed image, and A denotes the actual image. The variables $\mu$, ${\sigma }^2$, and ${\sigma }_{P_i{A}_i}$ represent the mean, variance, and covariance of $P_i$ and $A_i$, respectively. Constants $c_1$ and $c_2$ are also involved in the equation.

In 3D electrical impedance tomography, solving the inverse problem can be time-consuming due to the large volume of reconstructed data. Therefore, in this article, the inverse problem solving time (IPST) is used as a measure to evaluate the computational efficiency of the algorithm.

Computational platform and experimental details

The computer used for the experiments had an Intel Core i5-10400F 2.90 GHz hexa-core CPU, an NVIDIA GeForce RTX 3060 (12 GB) GPU, and 32 GB of RAM. Model simulation was performed using MATLAB R2022b on a Windows 10 OS. Network training was performed in the PyTorch framework, with an integrated development environment (IDE) of PyCharm and a compiled language of Python. Training data generation and imaging were based on the open platform EIDORS 3.10.

The model training process does not use a pre-trained model. The epoch is set to 50, the batch size is 32, and the loss function is mean square error loss (MSE Loss). The adaptive moment estimation (Adam) optimizer is used, along with a dynamic learning rate adjustment strategy. The initial learning rate is set to 0.0001.

Simulation experiment

The algorithm’s reconstruction performance is evaluated through experiments on simulated single and multi-object scenarios. The results are compared with the classical GN and cGN algorithms, as well as the widely used UNet. The ability of the algorithm to accurately reconstruct the shape and spatial position of the measured objects is assessed qualitatively through 3D visualization. Additionally, the reconstruction accuracy is assessed quantitatively using metrics such as RMSE, CC, SSIM, and IPST. The experiments demonstrate that the proposed method effectively reconstructs the size and position of both single and multiple targets. RFNetEIT achieves optimal reconstruction quality in terms of shape constraints and position localization for each measured object. The evaluation metrics, including RMSE, CC, SSIM, and IPST, indicate excellent results, further validating the effectiveness of the proposed method. Table 3 presents a comparison of the reconstruction results under four different levels of noise levels. The results show notable differences in the quality of reconstruction between the conventional algorithms (GN and cGN) and the deep learning algorithms (UNet and Ours) for Objects 1 to 4 under different noise levels. Particularly, the discrepancy in reconstruction results is most significant between the 30 dB noise condition and the noiseless ideal condition for the same object. Taking Object 1 (sphere) as an example, the three-dimensional slices obtained from the GN and exhibit evident differences in reconstruction, with noticeable changes in morphology as noise increases.

Table 3:

Reconstruction results at different SNRs.

		Object 1	Object 2	Object 3	Object 4
None	GN
	cGN
	UNet
	Ours
50 dB	GN
	cGN
	UNet
	Ours
40 dB	GN
	cGN
	UNet
	Ours
30 dB	GN
	cGN
	UNet
	Ours

DOI: 10.7717/peerj-cs.1944/table-3

Note:

Objects 1–4 represent different simulation models, namely, sphere, cube, cylinder, and three acrylic models. ‘None,’ ‘50 dB,’ ‘40 dB,’ and ‘30 dB’ denote the signal-to-noise ratios of the simulation experiments. At each signal-to-noise ratio, qualitative comparisons are made between GN, cGN, UNet, and our method, including the reconstruction results for the four simulation models from Object 1 to 4, respectively. The imaging results for each model are presented through three-dimensional views and three-dimensional sliced views, with conductivity levels displayed using colorbar.

Specifically, under the 30 dB signal-to-noise ratio (S/N) condition, both GN and cGN can roughly reconstruct the morphology of Object 1 (sphere) and Object 2 (cylinder). However, when it comes to Object 3 (cube) and Object 4 (multiple objects), the GN algorithm shows vague conductivity changes in the field, while cGN is limited in reconstructing them. On the other hand, UNet can capture changes in the shape and number of objects, but it struggles with the accurate reconstruction of the edges of Object 3 and Object 4. The proposed method in this article demonstrates superior reconstruction quality for objects with varying shapes and quantities. The reconstructed objects exhibit uniform conductivity distributions with minimal artifacts, as observed from the slice results. Among the four sets of experiments, Object 4 presents the greatest reconstruction challenge. The GN can only approximate the conductivity range, while the cGN struggles to effectively reconstruct the visualized conductivity. The UNet incorrectly reconstructs four cylinders in the noiseless and SNR = 40 dB experiments, but produces clearer conductivity distributions at SNR = 50 and 30 dB, albeit with poor reconstruction quality for the spheres. In contrast, the method proposed in this article achieves high-quality reconstruction for objects with diverse shapes.

Table 4 shows the IPST comparison results of the four methods in the Object 1–4 experiments. The computation time range of the inverse problem for traditional methods such as GN and cGN is roughly 60–200 s, which mainly depends on the CPU and memory usage. While the time required by UNet and RFNetEIT is within 0.1 s, although the computation time of RFNetEIT is slightly higher than that of UNet, the results of the metrics such as RMSE, CC, and SSIM are better. In Table 5, the RMSE for RFNetEIT is 0.0029 with a 95% confidence interval of (0.00274, 0.00306), while for UNet it is 0.0052 with a confidence interval of (0.00508, 0.00532). For CC, RFNetEIT yields a mean of 0.9504 with a confidence interval of (0.9394, 0.9614), compared to UNet’s 0.846 (0.836, 0.856). The SSIM value for RFNetEIT is 0.9923 with a confidence interval of (0.9909, 0.9937), whereas UNet presents 0.9769 (0.9757, 0.9781). As the SNR decreases, significant differences are presented between RFNetEIT and the comparison methods (experimental results for RMSE, CC, and SSIM are in the Supplemental Material). Among these, the contrast is most pronounced at 30 dB. Using the comparison experiment for Object 2 as an example, the range of RMSE for the method presented in this article is primarily between 0.002 and 0.003, the range of CC is approximately between 0.93 and 0.98, and the SSIM is mostly 0.99 or higher. Based on data from 100 samples, the fluctuations in the method presented in this article are minimized, indicating strong robustness. Additionally, the confidence intervals for the Object 4 experiments are shown in Table 6. Specifically, the confidence intervals for this article’s method are [0.274,0.306]_*0.01 for RMSE, [0.9394, 0.9614] for CC, and [0.9909, 0.9937] for SSIM, respectively.

Table 5:

Evaluation indicators for the 30 dB SNR simulation experiment.

		Object 1	Object 2	Object 3	Object 4
30 dB	RMSE
	CC
	SSIM

DOI: 10.7717/peerj-cs.1944/table-5

Note:

Objects 1–4 represent experiments with four types of test objects, and the displayed results are from experiments conducted under a 30 dB signal-to-noise ratio condition. RMSE, CC, and SSIM represent three evaluation metrics. The results of the comparative experiments are represented by curves in four different colors. Black represents GN, green corresponds to cGN, blue represents UNet, and red represents our proposed method, RFNetEIT. The number of experimental samples is 100.

Table 6 presents the confidence intervals for the different methods in the Object 4 experiment. For the proposed method, the confidence intervals are [0.274, 0.306]_*0.01 for RMSE, [0.9394, 0.9614] for CC, and [0.9909, 0.9937] for SSIM. These intervals are significantly better than those of UNet, which are [0.508, 0.532]*0.01 for RMSE, [0.836, 0.856] for CC, and [0.9757, 0.9781] for SSIM. For the confidence intervals of each indicator at different signal-to-noise ratios, please refer to the Supplemental Material.

Experiments with acrylic models

A solution of $350\mu S/cm$ and an acrylic model serve as the experimental objects. A training model, augmented with Gaussian noise based on numerical simulation experiments, is employed as a priori information to reconstruct the conductivity distribution of the acrylic in the measured experiments. Comparative experiments are conducted using the methodology described in “Simulation Experiment”. Given that it is not feasible to quantify the experimental objects and test fields, the quantitative metrics employed in the simulation experiments are not directly applicable to these real-world tests. Accordingly, this section evaluates the results of the experiments through qualitative analysis. The specific objects tested are listed in Table 7.

Table 7:

Image reconstruction results based on measured data.

Excitation frequencies	Method	Object 1	Object 2	Object 3	Object 4
1,000 Hz	GN
	cGN
	UNet
	Ours
10,000 Hz	GN
	cGN
	UNet
	Ours
50,000 Hz	GN
	cGN
	UNet
	Ours

DOI: 10.7717/peerj-cs.1944/table-7

Note:

The excitation frequencies chosen for the actual measurement experiments were 1, 10 and 50 KHz. The comparison methods at each frequency are GN, cGN, UNet, and Ours. The experimental subjects are Objects 1–4. The imaging results of each method at different frequencies and on different objects are displayed through three-dimensional views and three-dimensional slice views, and the conductivity levels are shown using a colorbar.

Experiments were conducted using different stimulation frequencies (1, 10, and 50 kHz) with acrylic models of various shapes (sphere, cylinder, rectangle, and multi-target) placed in the container as reconstruction targets. The results of all reconstructions are shown in Table 7. The GN is able to capture coarse conductivity variations in the measurement area, although the imaging quality is influenced by the excitation frequency. However, the cGN has difficulty in accurately representing conductivity variations at all excitation frequencies. On the other hand, the UNet can reconstruct the image, but the position and shape of the reconstructed object are biased. For single target objects (Object 1–Object 3), the object reconstructed using RFNetEIT exhibits the best position and shape. However, for Object 4, both the GN and cGN struggle to reconstruct clearer targets due to the complexity of the target and the presence of hardware system noise, while the UNet produces incorrect reconstruction results. Three slices of each reconstruction result are shown in Table 7. A comparison of the slices reveals that the GN is not suited for reconstruct a uniform conductivity distribution, the UNet generates a significant artifact in the target conductivity, while RFNetEIT achieves a more uniform conductivity distribution in 3D space.

Reconstructing the electrical conductivity distribution of maize ears

In the experiments, currents were excited at frequencies ranging from 2 to 8 kHz. Three layers of electrodes were mounted on the maize ear. The electrode positions remained stationary during measurements, as depicted in Fig. 8A. “Simulation Experiment” shows that the UNet lacks robustness in 3D imaging, while “Experiments with Acrylic Models” discusses how the cGN is less effective for accurately reconstructing the 3D conductivity distribution. To evaluate the performance of our proposed method, we combined these findings with current research on the 3D conductivity reconstruction of maize ears. The comparative algorithms we selected include the interpolation imaging method by Li et al. (2022) and the GN algorithm.

Figure 7A displays the complete maize ear measurement apparatus. Since the maize ear has a three-layer structure, the main part of the ear is selected as the imaging target, as shown in Fig. 8B. Figure 8C represents the conductivity model using transverse and longitudinal slices to better illustrate the distribution. Figure 8D contrasts the absolute reconstruction results, and Fig. 8E displays the reconstructed frequency difference image obtained through interpolation. In Li’s experiment (Li et al., 2022), three layers of electrodes were also used, but the electrodes on different layers operated independently. A separate set of voltage data was excited and measured for each layer, resulting in a total of 208 × 3 voltage measurements. Compared to the absolute reconstruction results in Fig. 8D, Li’s method only demonstrates visual three-dimensional imaging. The conductivity between the layers of electrodes is estimated using interpolation methods, as the true value cannot be calculated using EIT theory. In Fig. 8D, the GN can reconstruct variations in conductivity distribution, but the differences are significant at varying excitation frequencies. In contrast, the method proposed in this article provides the best reconstruction results, and these results are stable across different frequencies. Figure 9 displays the conductivity reconstruction results for five maize ears using a uniform colorbar, revealing differences in conductivity distribution among different maize ears.

Comparison of reconstruction quality in numerical simulation experiments

GN and cGN algorithms struggle to accurately reconstruct the morphology and position of the simulated object due to challenges in determining suitable parameter values for constructing Jacobi matrices with good generalization and robustness. Additionally, the impact of system noise is a significant factor contributing to the poor or even non-imaging quality. The UNet network’s performance is not adequate to achieve a stable model due to the high number of reconstructed 3D conductivities and the complexity of the training dataset, resulting in significant fluctuations in the imaging results. In contrast, the method proposed in this article, incorporating a residual structure, exhibits improved feature extraction capability and network stability. As a result, it can more accurately approximate the distribution pattern of the real conductivity of the object under measurement, leading to superior imaging quality and evaluation metrics compared to the comparison method.

The reconstruction difficulty is in the order of: multi-target > cube > cylinder > sphere. The reconstruction difficulty of multi-target is the highest, followed by rectangular. One possible reason is that the rectangular structure is more complex compared to the sphere and cylinder simulation models, with a surface shape that is rectangular and the presence of vertical transitions on adjacent surfaces. Moreover, the field containing multiple objects to be measured exhibits a greater variation in conductivity, resulting in the highest level of reconstruction difficulty.

Reconstruction quality comparison of acrylic measurement experiments

The limited ability of GN and cGN to accurately reconstruct the conductivity distribution of the acrylic model can be attributed to the influence of environmental noise and hardware system noise. The difficulties faced by cGN in imaging are primarily due to the higher impact of such noise compared to the Gaussian noise added in the simulation experiments, resulting in decreased stability compared to the GN algorithm. Furthermore, variations in the excitation frequency can lead to changes in the boundary voltage, further contributing to differences in the reconstruction results.

For single-target reconstruction, the experimental and simulation results of this article are close to each other, and the deviation from the shape size and spatial position of the real object to be measured is minimal. However, in multi-target reconstruction, the accuracy of the reconstructed shape of the measured object needs further improvement due to the influence of system noise and model complexity. UNet is prone to overfitting due to the limited feature extraction capability of the model, and the training process is prone to produce incorrect reconstruction results, which is more obvious in multi-target reconstruction.

Comparison of reconstructed quality of maize ear experiments

Impedance distribution within the maize ear correlates with its moisture content. Factors such as humidity, temperature, and maize maturity significantly influence impedance. Typically, regions with higher moisture content exhibit lower impedance. Thus, internal impedance variations in the maize ear could serve as an indirect method to evaluate moisture levels. However, challenges arise due to the non-destructive nature of the acquisition devices and impedance differences across electrodes at various maize ear positions, complicating conductivity reconstruction.

The study by Li et al. (2022) used three layers of independently operating electrodes to generate 2D EIT images, which were then interpolated by an interpolation algorithm to visualize 3D conductivity distributions, but this method fills in the missing conductivity values by calculating linear interpolation between the known points, and is not the true conductivity distribution of the maize ear. While the imaging effect of the GN is greatly affected by the excitation frequency, it is a challenge to determine the optimal excitation frequency in practical applications.

In contrast, the absolute imaging results of the method in this article can demonstrate relatively complete 3D structural information. The reconstruction results from RFNetEIT can clearly distinguish between husk, kernels, and cob. Furthermore, the conductivity distribution within each part shows gradient changes, which aligns with the actual conditions. However, due to the low signal-to-noise ratio in the actual boundary voltage measurements of the maize ear and the intricate physiological attributes and structure of the maize ear, obtaining authentic training samples becomes a challenge. This has led to the current study only being able to observe the 3D conductivity distribution within the maize ear without accurately reconstructing the conductivity values for specific sections.