# Electromagnetic Imaging for Buried Conductors Using Deep Convolutional Neural Networks
## 4 Numerical Result  
In this section, a simulation environment in half-space (Figure 1) is designed to numerically analyze the inverse scattering problem of perfect conductors. The frequency of the incident wave is 3 GHz. The buried depth, \(A\), is set to a wavelength (0.1 m). The background substance in Region 1 is air with \(\varepsilon_{1}=\varepsilon_{0}\) and \(\sigma_{1}=0\); while the background substance in Region 2 is the soil with \(\varepsilon_{2}=2.56\), \(\varepsilon_{0}\) and \(\sigma_{2}=0\). TM-polarized plane waves are emitted to illuminate the scatterers. As much as 5% Gaussian noises are added to the environment, and 32 receivers from \(\theta=195^{\"}\) to \(350^{\"}\), with radius 3 m, and 32 transmitters from \(\varnothing_{1}=-80^{\"}\) to \(80^{\"}\) with \(5^{\"}\) intervals are deployed. In short, a total of 32 receivers and 32 transmitters are placed in our simulated environment. We aim to utilize the received scattered field collected from various incident angles and input it to the DCNN to reconstruct the object. In our simulated environment, we place the scatterer in the 32 \(\times\) 32 pixel area. We assume that the scatterer has different sizes which can be located at any places within the measurement area. We simulate different shapes for various random positions. We then illuminate each profile with 32 different incident angles. The total number of dataset images is about 2000. They are split into two parts, in which 80% of the data is used for training and 20% for testing. The high computational performance of GPU parallelization is well suited for training because each training is independent during the process. In the training option, we choose adaptive moment estimation (Adam) because it is commonly used in practice. Intuitively, Adam is a fusion of the adaptive gradient algorithm (Adagrad) and momentum; it has the  
Figure 3: Flowchart of the experimental process. advantage of bias correction so as to limit the learning rate for each iteration, allowing the parameters to be updated more smoothly. Meanwhile, we set the initial learning rate at \(10^{-5}\) and drop the learning rate by half at every 70 epochs. The momentum parameter is set at 0.99, the batch size is 32, and the max epoch is 200. We will shuffle the data upon each epoch trained in the DCNN training process. This is due to the strong fitting capability of DCNN. In other words, the order of the samples may be cached if the same combination of batches appears repeatedly, which could affect the generalization outcome. In order to calculate the performance of each scheme, we define the root mean square error (RMSE) formula as follows:  
\[RMSE=\frac{1}{M}\sum_{i=1}^{M}\|Y-Y^{i}\|_{F}/\|Y\|_{F}, \tag{16}\]  
where \(Y\) and \(Y^{i}\) denote the true shape and the reconstructed shape, respectively, \(M\) is the total number of tests, and \(F\) is the Frobenius norm. In order to compare the reconstruction results of each graph trained by the neural-like network, we define the structural similarity index measure (SSIM) as follows:  
\[SSIM=\frac{(2\mu_{\tilde{y}}\mu_{y}+C_{1})(2\sigma_{\tilde{y}\tilde{y}}+C_{2}) }{(\mu_{\tilde{y}}^{2}+\mu_{y}^{2}+C_{1})(\sigma_{\tilde{y}}^{2}+\sigma_{ \tilde{y}}^{2}+C_{2})}. \tag{17}\]  
For SSIM details, please refer to reference [7]. We reconstruct the images by using DCNN. As much as 5% noise is added to the scattered field to take care of the interference of external factors that may be encountered in reality and input the scattered field information into DCNN for training. Figure 4a,b shows the ground truth circular shape and the reconstructed shape by DCNN, respectively. It is seen that the reconstructed shape has a slight depression in the left half and a slight bulge in the upper half of the shape. In general, the reconstructed result is quite good. The RMSE and SSIM are 2.95% and 95.91%, respectively, as listed in Table 1.  
Figure 4: Reconstructed image of case 1: (a) ground truth; (b) DCNN.