## 3 Proposed Approach
### Network Architecture
#### 3.2.3 Chaos Mapping Enhanced Image Steganography Network (CHASE)  
Information-hiding encoder-decoder networks proposed by researchers such as [PERSON] [15] and [PERSON] et al. [22] typically adopt an independent design for encoder and decoder networks. This approach results in the irreversibility of the secret information encoding and decoding processes, consequently diminishing the success rate of secret information restoration. Taking inspiration from the architecture of invertible neural networks like Dihm [25] and Lu [32], our proposed CHASE model integrates the encoding and decoding of secret information within the same network. Figure 5 illustrates our steganographic network (CHASE), and Table 1 provides an explanation of the notation used below. \begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Description** \\ \hline \(X_{h}\) & Cover image \\ \(X_{s}\) & Secret image \\ \(X_{c}\) & Stego image \\ \(Scr(x)/Scr^{-1}(x)\) & Image scrambling process/Image inverse scrambling process \\ \(F(x)/F^{-1}(x)\) & Hiding process/Reconstruction process \\ \(\underline{X}_{s}\) & Reconstructed secret image \\ \hline \(\overline{X_{s}}\) & Inverse Scrambling reconstructed secret image \\ \(\underline{X}_{h}\) & Reconstructed host image \\ \(r\) & Loss function generated during forward hiding process \\ \(z\) & Auxiliary variable used to assist in reconstructing images \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of notations used in this article. Figure 5: The framework of CHASE. During forward hiding, the Haar wavelet transforms the scrambled secret image and cover image. Invertible blocks then process the transformed data, generating the stego image and loss information. In backward reconstruction, auxiliary variable and the stego image pass through invertible blocks to reconstruct the cover image and scrambled secret image. The inverse scrambling of the latter yields the final secret image. In the hiding and reconstructing blocks, \(\rho\), \(\eta\), \(\varphi\), and \(\psi\) can be arbitrary functions, and we utilized Dense block [37] to represent them to enhance the image processing effect. In the forward hiding process of the network, \(X_{h}\) is transformed from the RGB encoding format to the YcrCb format, with the \(Y\) channel selected as the actual cover image. Subsequently, the \(Y\) channel image and \(X_{s}\) undergo separate transformations using the Haar wavelet, a discrete wavelet transform variant that is easy to implement. Hiding information in the frequency domain is better advantageous for preserving the visual quality of the image than in the spatial domain. Additionally, the Haar wavelet transform is orthogonal, ensuring the retention of image information, and enabling complete reconstruction of the original image during the inverse transformation. The two transformed images are input into the network. Following the invertible block sequence and, ultimately, the inverse wavelet transform (IWT), the \(Y\) channel image with intact information and the loss information \(\tau\) are obtained. The \(Y\) channel image at this stage undergoes further transformation to yield \(X_{c}\), which is utilized for actual transmission. In the backward process, the \(Y\) channel image is initially extracted from \(X_{c}\). The reconstructed Y channel image and the scrambled secret image \(\hat{X}_{s}\) then undergo the scrambling inverse process to ultimately obtain the reconstructed secret image \(\overline{\hat{X}_{s}}\). We designed the hiding and reconstructing processes as reciprocal problems, and this process can be expressed by the following formula:  
\[IWT(F(DWT(X_{h}),DWT(Src(X_{s}))))=(X_{c},r) \tag{4}\]  
\[IWT\Big{(}F^{-1}(DWT(X_{c}),z)\Big{)}=\Big{(}\hat{X}_{h},\mathit{Scr}^{-1} \big{(}\hat{X}_{s}\big{)}=\overline{\hat{X}_{s}}\Big{)} \tag{5}\]  
**Invertible Block. ** The designed hiding and reconstruction networks share identical sub-modules and network parameters, employing the affine coupling layer as the invertible block.