## 3 Proposed Approach
### Network Architecture
#### 3.2.3 Chaos Mapping Enhanced Image Steganography Network (CHASE)
In contrast to the additive coupling layer, the affine coupling layer adeptly captures intricate relationships in input information and effectively manages high-dimensional data distribution. The input accepted at the beginning of the reversible block is \((X_{h},X_{s})\), which can also be denoted as \((X_{h}^{1},X_{s}^{1})\) to indicate the first input. Throughout the forward hiding process, multiple invertible blocks process information. For the \(i\)-th invertible block, its input \((X_{h}^{i},X_{s}^{i})\) and output \((X_{h}^{i+1},X_{s}^{i+1})\) can be represented by the following formula:  
\[\left\{ \begin{aligned} X_{h}^{i+1}=e\big{(}\alpha\odot\psi\big{(}X_ {s}^{i}\big{)}\big{)}\odot X_{h}^{i}\oplus\varphi\big{(}X_{s}^{i}\big{)}\\ X_{s}^{i+1}=e\Big{(}\alpha\odot\rho\Big{(}X_{h}^{i+1}\Big{)} \Big{)}\odot X_{s}^{i}\oplus\eta\Big{(}X_{h}^{i+1}\Big{)}\end{aligned} \right. \tag{6}\]  
where \(\rho(\cdot)\), \(\eta(\cdot)\), \(\varphi(\cdot)\), and \(\psi(\cdot)\) can represent arbitrary functions. Here, \(e(\cdot)\) signifies a Sigmoid function, and the coefficient \(\alpha\) acts as a regularization factor to mitigate numerical instability arising from the \(e(\cdot)\) function. Following the last forward hiding block, the outcomes undergo IWT to obtain the stego image and loss information, respectively. For the backward reconstruction module, the information flow direction is inverted. In the i-th module, the output \((\hat{X}_{h}^{i},\hat{X}_{s}^{i})\) from the \(i+1\)-th module serves as input \((\hat{X}_{h}^{i+1},\hat{X}_{s}^{i+1})\), as expressed by the following equation:  
\[\left\{ \begin{aligned} \hat{X}_{s}^{i}=\Big{(}\hat{X}_{s}^{i+1} \odot\eta\Big{(}\hat{X}_{h}^{i+1}\Big{)}\odot\exp\Big{(}-\alpha\odot\rho\Big{(} \hat{X}_{h}^{i+1}\Big{)}\Big{)}\Big{)}\\ \hat{X}_{h}^{i}=\Big{(}\hat{X}_{h}^{i+1}\odot\varphi\big{(}\hat{X}_ {s}^{i}\big{)}\Big{)}\odot\exp\big{(}-\alpha\odot\psi\big{(}\hat{X}_{s}^{i} \big{)}\big{)})\end{aligned} \right. \tag{7}\]  
Initially, the inverse extraction model takes the stego image and auxiliary information as input. At the final stage, executing IWT and image inverse permutation yields the reconstructed secret image. **The Loss information r and auxiliary variable z. ** In the forward hiding process, the loss of information in the cover and secret image results in the generation of loss information r in the output. For successful secret information reconstruction in the backward reconstruction module, both the loss information r and the stego image must be used as input. In actual transmission processes, transmitting loss information is impractical, as it may make it easier for an attacker to deduce the original secret information or result in information leakage. Moreover, transmitting loss information poses a risk of information leakage. While some steganography methods employing reversible neural networks utilize auxiliary variables sampled from standard Gaussian distributions during the reverse extraction process, this approach may compromise the accuracy of secret image extraction. The success of deep learning-based image steganography is attributed to the frequency differences between the cover image and the secret image [38]. To enhance secret image extraction accuracy, we leveraged the high-frequency sub-bands of the Stego image as auxiliary variables, containing partial information of the secret image to better assist in the reverse extraction operation. **Generative Adversarial Networks. ** The introduction of the auxiliary variable z facilitates secret information extraction but may compromise the completeness of the secret information, leading to a reduction in the quality of the secret image. To enhance the fidelity of the reconstructed secret image, we incorporated a Generative Adversarial Network (GAN) [39]. The GAN framework comprises a generator responsible for image generation and a discriminator tasked with evaluating the difference between the generated and real images.