# Enhanced Dual Reversible Data Hiding Using Combined Approaches
## 4 Experimental Results
The PSNR is generally expressed by the following formula (Equation (14)):  
\[PSNR(I,I^{\prime})=10\cdot log_{10}\bigg{(}\frac{MAX^{2}}{\sqrt{MSE}}\bigg{)}, \tag{14}\]  
Here, MAX is the maximum pixel value of the image, which is typically 255 for an 8-bit image. The MSE (Mean Squared Error) represents the average squared error between the original image and the marked image, and it is calculated as follows:  
\[MSE=\frac{1}{N^{2}}\sum_{i=1}^{N}\sum_{j=1}^{N}[I_{i,j}-I_{i,j}]^{2}, \tag{15}\]  
The symbols \(I_{i,j}\) and \(I^{\prime}_{i,j}\) stand for the pixel values of the original grayscale image and the marked image at the respective position, respectively, and \(N\times N\) is the width and height of the original image. In data hiding techniques, the PSNR is used to evaluate the difference between the original image and the marked image; the higher the PSNR value, the less quality degradation and the more similar the marked image is to the original. Generally, a PSNR value of 30 dB or higher is considered to be nearly imperceptible to the human eye. When comparing the performance of various data hiding techniques, the PSNR serves as an important criterion. A higher PSNR value indicates that the data have been effectively hidden while maintaining the image quality. The PSNR objectively measures how well a data hiding technique preserves image quality. Another performance measure is the SSIM (Structural Similarity Index Measure), a formula (Equation (16)) that measures the similarity between the original image and the marked image. The SSIM is a metric used to evaluate the structural similarity between images, precisely assessing the visual quality between the original image and the one after data embedding. The SSIM mimics the way the human visual system perceives images, providing a more intuitive quality assessment. It calculates the similarity by combining three components: luminance, contrast, and structure. The formula for the SSIM is as follows:  
\[SSIM(I,I^{\prime})=\frac{(2\mu_{I}\mu_{I^{\prime}}+C_{1})(2\sigma_{I^{\prime}} +C_{2})}{(\mu_{I}^{2}+\mu_{I^{\prime}}^{2}+C_{1})(\sigma_{I}^{2}+\sigma_{I^{ \prime}}^{2}+C_{2})} \tag{16}\]  
* \(\mu_{I}\) and \(\mu_{I^{\prime}}\) are the mean values of the original image \(I\) and the marked image \(I^{\prime}\), respectively. * \(\sigma_{I}\) and \(\sigma_{I^{\prime}}\) represent the standard deviations of \(I\) and \(I^{\prime}\), respectively. * \(\sigma_{I}\) and \(\sigma_{I^{\prime}}\) is the covariance between \(I\) and \(I^{\prime}\). * \(C_{1}\) and \(C_{2}\) are small constants added to ensure numerical stability in the calculations. The SSIM value is expressed as a number between \(-1\) and \(1\), where a value closer to 1 indicates a higher structural similarity between two images. By utilizing the SSIM in DH techniques, even subtle structural differences between the original image and the marked image can be accurately assessed. The SSIM is useful for evaluating how well the primary structure and detailed information of an image are preserved during the DH process. In particular, it enables the assessment of high-frequency components (such as edges or textures in the image), allowing for the verification of whether there are significant visual changes after data embedding. In this study, SSIM, ER, and RS values were measured to evaluate the performance of data hiding. For the SSIM, it is generally considered that values above 0.98 indicate that differences are nearly imperceptible to the human eye. In our study, we achieved an excellent average SSIM value of 0.999 or higher. Therefore, setting the minimum acceptable SSIM threshold at 0.98 or higher is appropriate. Additionally, an ER of 1.0 bpp or higher is typically regarded as a high-capacity data hiding technique. In our study, we achieved a high ER value of 1.5 bpp. Hence, the minimum acceptable ER threshold in this study is set at 1.0 bpp or higher. Finally, RS analysis can be used to detect traces of data hiding. Previous studies have evaluated RS values within the 0.3 to 0.4 range as a safe level.