# Electromagnetic Imaging for Buried Conductors Using Deep Convolutional Neural Networks
## 3 Deep Convolutional Neural Network (DCNN) and Inverse Problem  
DCNN is a DL method in machine learning, as shown in Figure 2. By imitating the mathematical model of the biological nervous system, multiple calculations and trainings of different levels and structures are performed to seek the optimal and most effective DL model. In contract to the past, where rules must be pre-defined by human beings for machine learning, DCNN only requires the finetuning of the training data and parameters in the designed neural network. The machine is capable of learning by itself to discriminate the features and then calculate the best result. The key to develop DCNN is to continuously provide high-quality data and perform computational training to produce better DCNN models. In order to accelerate the training process, advanced hardware platforms, such as the GPU (graphics processing unit), and improved software have been introduced. DCNN training can be regarded as a \"heterogeneous\" process that transfers the measured scattered field into a preliminary contrast of the object domain. The input in the DCNN framework consists of the real and imaginary parts of the objective contrast. The convolutional layer is a special neural layer used to extract features in the image. It converts each part of the image into a smaller feature vector by sliding a small matrix called a convolutional kernel over the image. The main advantage of the convolutional layer is that it can capture local patterns and textures in the image and can classify objects in the image regardless of their location. The activation layer is a layer that performs non-linear transformation of the feature vectors extracted from the convolutional layer to increase the representational power of the neural network. It usually uses the ReLU (rectified linear unit) function, which sets the negative values to 0 and keeps the positive values. This choice is made because ReLU can effectively prevent the gradient disappearance problem while maintaining computational efficiency. Finally, these feature vectors are fed into a fully-connected layer for the final classification or regression operation. This layer usually consists of multiple neurons, each of which is connected to some feature vectors in the preceding convolutional and activation layers. Through this combination, the neural network can learn to make associations between the extracted features and the desired output classes or values for reconstructing the image. We also try to use the most popular U-Net to reconstruct our images. As U-Net requires a preliminary image, the direct sampling method is utilized. However, in our experience, the initial preliminary image obtained solely from the upper scattered field, i.e., rather than from the full space of U-Net, would be bad. This is why we propose DCNN to reconstruct the buried conductor in this research. We employed the measured scattered fields \(E_{s}\) as the input data and put it into the DCNN module for training. The \(M_{i}\)\(\times\)\(M_{r}\)\(\times\) 2 matrix comprises the input data transformed from the scattered fields using \(E_{s}\). \(M_{i}\) represents the number of incident fields, \(M_{r}\) represents the number of receivers, and the real and imaginary parts of \(E_{s}\) will be split into two channels, as shown in Figure 2. Briefly speaking, comparing with the past conventional methods, using \(E_{s}\) as the input value for the DCNN can cope with the EMIS problem efficiently. The mean square error (MSE) is used as a loss function for training the DCNN model; it is the mean of the square differences between our target and predicted values and is defined as follows:  
\[Loss=0.5\times\sum\
olimits_{m=1}^{WHC}(\widetilde{y}_{n}-y_{n})^{2}, \tag{14}\]  
where \(W\), \(H\), and \(C\) denote the width, height, and number of channels of the output, respectively, and \(n\) presents the transmit number indexing, linearly, into each element of \(\widetilde{y}_{n}\) and \(y_{n}\).