# Cyclic Consistent Image Style Transformation: From Model to System
## 4 Method  
### Overview  
The proposed C3 GAN model is built upon the foundational principles of Generative Adversarial Networks (GANs). In a traditional GAN, two networks, the generator \(G\) and the discriminator \(D\) compete in a zero-sum game. The generator \(G\) aims to generate data samples that are indistinguishable from real data, while the discriminator \(D\) strives to distinguish between real and generated data. The overall objective function for a basic GAN can be expressed as follows:  
\[\min_{G}\max_{D}\mathbb{E}_{x\sim p_{data}(x)}[\log D(x)]+\mathbb{E}_{z\sim p_{ z(z)}}[\log(1-D(G(z)))]\]  
However, one limitation of traditional GANs is their difficulty in maintaining training stability and generating diverse outputs. To address these challenges, we propose the C3 GAN model, which introduces cyclic consistency to enhance both stability and diversity. The schematic diagram of the C3 GAN model is shown in Figure 2. The model comprises four neural networks: two generator networks and two discriminator networks. Specifically, one generator network is responsible for transforming images from domain A to domain B, while the other performs the inverse transformation. Each discriminator is trained to distinguish between real images and those generated by the corresponding generator. **Generator Networks:** The C3 GAN model employs two generator networks, denoted as \(G_{A}\) and \(G_{B}\), for bidirectional transformation between domains \(A\) and \(B\). Both generators utilize a convolutional neural network (CNN) architecture. For an input image \(x_{A}\) from domain \(A\), \(G_{A}\) generates a corresponding image \(x_{B}=G_{A}(x_{A})\) in domain \(B\). Conversely, \(G_{A}\) maps an image \(x_{B}\) from domain \(B\) back to domain \(A\), producing \(x_{A}^{\prime}=G_{B}(x_{B})\). The structure of these networks includes Batch Normalization to stabilize training and ReLU activation functions for non-linearity. **Discriminator Networks:** The model also incorporates two discriminator networks, denoted as \(D_{A}\) and \(D_{B}\), which evaluate the realism of images in domains \(A\) and \(B\), respectively. \(D_{A}\) is trained to distinguish between real images \(x_{A}\) and generated images \(G_{B}(x_{B})\), while \(D_{B}\) discriminates between \(x_{B}\) and \(G_{A}(x_{A})\). The discriminator networks are also implemented using CNN architectures, with LeakyReLU activations to handle negative inputs more effectively. **Cyclic Consistency Loss:** A critical feature of C3 GAN is the introduction of cyclic consistency loss, which ensures that the transformations are reversible and consistent. This loss is defined as the \(L_{1}\) distance between the original images and the images reconstructed after a complete cycle of transformations. The training process of the C3 GAN model, illustrated in Figure 3, begins with two inputs: image sets from two distinct domains that are labeled as domain \(A\) and domain \(B\). These image sets are composed of \(256\times 256\) pixel RGB images, ensuring that both domains share the same number of channels and resolution. The model outputs two sets of images--one set  
Figure 2: The schematic diagram of the C3 GAN model. transforming images from domain \(A\) to domain \(B\), and the other performing the reverse--with the outputs retaining the original resolution and channel dimensions. The C3 GAN architecture is built upon four key components: two generators (\(G_{A}\) and \(G_{B}\)) and two discriminators (\(D_{A}\) and \(D_{B}\)). The generators are designed to perform the image-to-image translation tasks, where \(G_{A}\) converts images from domain \(A\) to domain \(B\), and \(G_{B}\) performs the inverse operation. The discriminators \(D_{A}\) and \(D_{B}\) evaluate the authenticity of images in their respective domains, determining whether the images are real or generated. Each of these networks is constructed from multiple convolutional layers, interspersed with normalization layers to stabilize training and activation layers to introduce non-linearity.