# Edge-Guided Dual-Stream U-Net for Secure Image Steganography
## 4 Experiments
### Evaluation Metrics  
In this study, we utilized several evaluation metrics to quantitatively assess the performance of the steganography algorithm and the quality of the stego images. These metrics include the error rate, peak signal-to-noise ratio (PSNR) [43], structural similarity index (SSIM) [44], and feature similarity indexing method (FSIM) [45]. The error rate measures the difference between the original message and the message recovered from the stego image. A lower error rate indicates higher accuracy in message recovery by the steganography algorithm. It is defined as follows:  
\[\epsilon=\frac{1}{m}\parallel M^{\prime}-M\parallel_{0} \tag{19}\]  
where \(M\) represents the original message, \(M^{\prime}\) is the recovered message, and \(m\) denotes the total number of bits in the message. The error rate (\(\epsilon\)) is inversely equivalent to decoding accuracy, where \(Accuracy=1-\epsilon\). For example, an error rate of 0.02 (2%) indicates a decoding accuracy of 98%. The peak signal-to-noise ratio (PSNR) [43] is used to objectively evaluate the quality of an image after steganography, compression, or other processing, by calculating the pixel differences at corresponding positions in two images. A higher PSNR value indicates less image degradation, suggesting that the image has undergone minimal distortion. \[\text{PSNR}=10\cdot\log_{10}\left(\frac{MAX_{I}^{2}}{\text{MSE}}\right) \tag{20}\]  
where \(MAX_{I}\) is the maximum possible value of image pixels, and \(MSE\) represents the mean squared error, defined as follows:  
\[MSE=\frac{1}{N}\sum_{i=1}^{N}(S_{i}-X_{i})^{2} \tag{21}\]  
where \(N\) is the total number of pixels in the image, and \(S_{i}\) and \(X_{i}\) are the pixel values of the stego image and the cover image, respectively. The structural similarity index (SSIM) [44] is a critical metric for assessing the similarity between two images. Unlike PSNR, SSIM focuses not only on pixel-level differences but also on capturing structural changes within the image. It is defined as follows:  
\[\text{SSIM}(X,S)=\frac{(2\mu_{X}\mu_{S}+C_{1})(2\sigma_{XS}+C_{2})}{\left(\mu_ {X}^{2}+\mu_{S}^{2}+C_{1}\right)\left(\sigma_{XS}^{2}+\sigma_{S}^{2}+C_{2} \right)} \tag{22}\]  
where \(\mu_{X}\) and \(\mu_{S}\) represent the means of the original and stego images, respectively; \(\sigma_{XS}^{2}\) and \(\sigma_{S}^{2}\) are their respective variances; and \(\sigma_{XS}\) is their covariance. Constants \(C_{1}=(K_{1}L)^{2}\) and \(C_{2}=(K_{2}L)^{2}\) are used for computational stability, where \(L\) is the dynamic range of the image, typically set to 255, and \(K_{1}=0.011\), \(K_{2}=0.03\). The Feature Similarity Index (FSIM) is a novel image quality assessment metric designed to measure the similarity between two images. Unlike traditional metrics such as Peak Signal-to-Noise Ratio (PSNR) or Structural Similarity Index (SSIM), FSIM assesses perceptual image quality by analyzing two complementary low-level features--Phase Congruency (PC) and Gradient Magnitude (GM). Since our method integrates both edge and global information within the shared expansion path of the dual-stream U-Net, we also consider FSIM as an evaluation metric. FSIM is defined as follows:  
\[FSIM(f_{1},f_{2})=\frac{\sum_{i=0}S_{i}(x)\cdot PC_{m}(x)}{\sum_{x\in\Omega} PC_{m}(x)} \tag{23}\]where \(f_{1}\) and \(f_{2}\) are the reference and distorted images, respectively. \(S_{L}(x)\) is the local similarity at pixel x, computed as the product of the similarities of PC and GM features:  
\[S_{L}(x)=S_{PC}(x)\cdot S_{GM}(x) \tag{24}\]  
Here, \(S_{PC}(x)\)_and_\(S_{GM}(x)\) are the similarity measures for PC and GM features, respectively. \(PC_{m}(x)=max(PC_{1}(x),PC_{2}(x))\) is the maximum PC value at pixel x between the two images, used as a weighting function to reflect the importance of each pixel in the overall similarity.