# Suppression of Multiple Reflection Interference Signals in GPR Images Caused by Rebar Using VAE-GAN
## 2 Materials and Methods  
### Overall Network Structure  
The overall network model consists of a generative network based on Variational Autoencoders (VAEs) and a discriminative network based on Generative Adversarial Networks (GANs). It comprises two domain-specific image encoders, \(E_{A}\) and \(E_{B}\), two domain-specific image generators, \(G_{A}\) and \(G_{B}\), and two domain-specific image adversarial discriminators, \(D_{A}\) and \(D_{B}\). The encoder-generator pair \(\{E_{A},G_{B}\}\) is used to learn the mapping between two GPR image domains, \(A\) to \(B\). The encoder \(E_{A}\) is responsible for mapping GPR images with rebar clutter into shared latent space \(z\), and the generator \(G_{B}\) maps \(z\) back to the output data space, generating images without rebar clutter. The discriminator \(D_{B}\) ensures the generation of more realistic images without rebar clutter. The primary objective of the model is to generate images without rebar clutter from GPR images with rebar clutter interference. The overall network structure is illustrated in Figure 1.  
The generative network includes two sets of encoders and generators, corresponding to two different image domains, domain \(A\), which contains GPR images with rebar clutter, and domain B, which contains GPR images without rebar clutter. \(E_{A}\) and \(G_{A}\) are the encoder and generator for domain \(A\), referred to as VAE\({}_{1}\). Similarly, \(E_{B}\) and \(G_{B}\) are the encoder and generator for domain \(B\), referred to as VAE\({}_{2}\). The generative network achieves the mapping relationship between the two VAE by a shared latent space. Specifically, the encoders are tasked with extracting high-level feature representations from the input images of both domains and encoding them into a latent space of the same probability distribution. This latent space can be viewed as a \"bridge\" containing key information  
Figure 1: The overall network structure. from both domains. Similarly, the generators decode from the latent space to reconstruct the feature representations of the input images. This design enables the two VAEs to jointly learn the shared feature representations between the two domains. In VAE\({}_{1}\), the encoder \(E_{A}\) can extract features of rebar signals and void signals from the input images, which are mapped into the shared latent space \(z\). On the side of generator \(G_{A}\), \(z\) is used to reconstruct the images. Similarly, in VAE\({}_{2}\), the encoder \(E_{B}\) can also learn the features of void signals from the input images, which are mapped into \(z\), allowing the generator \(G_{B}\) to reconstruct the images. This design allows the model to perform mapping translations between the two image domains \(A\) to \(B\) and \(B\) to \(A\). Images from the two domains can be mapped into the shared latent space \(z\) through encoders \(E_{A}\) and \(E_{B}\), respectively, and then mapped back to their respective domain's output data space through generators \(G_{A}\) and \(G_{B}\). Each domain's generator can produce two types of images: reconstructed images within their domain and translated images from the other domain. For a GPR image \(x\in A\) with rebar clutter, the encoder \(E_{A}\) first maps \(x\) into \(z\). Subsequently, both sets of generators decode \(z\). The primary objective of generator \(G_{A}\) is to reconstruct image \(\tilde{x}^{A\to A}\) with rebar clutter that closely resembles the original image \(x\), emphasizing high consistency with GPR images from the original domain \(A\). Generator \(G_{B}\), on the other hand, aims to generate a clear image \(\tilde{x}^{A\to B}\) without rebar clutter interference. This means that \(G_{B}\) can extract features related to rebar clutter from \(z\) to eliminate clutter interference in the original image, producing a clear image containing only void defects. The structure of the generator is depicted in Figure 2a. The encoder consists of three convolutional layers at the front end.