## 4 Proposed Scheme for Self-Embedding
### Watermark Extraction and Reconstructing Ambtc
The quality of the image was measured by the PSNR defined as

\[\text{PSNR}=10 log_{10}\frac{255^{2}}{\text{MSE}} \tag{19}\]

PSNR is calculated as \(10 log\) (signal power/noise power), and signal power and noise power are calculated using peak power. The MSE used for PSNR calculation is the difference in average intensity between the marked image and the reference image, and a low MSE value can be evaluated as good image quality. In other words, the MSE is the mean of the squares of the errors \((p_{i}-p_{i}^{\prime})^{2}\), where \(p\) and \(p^{\prime}\) are reference and distorted images, respectively. The MSE is calculated as follows:

\[\text{MSE}(p,p^{\prime})=\frac{1}{N}\sum_{i=1}^{N}(p_{i}-p_{i}^{\prime})^{2}. \tag{20}\]

Here, the allowable pixel intensity is \(255^{2}\). SSIM is a formula that measures the similarity between the original image and the displayed image. It consists of luminance, contrast and structure, and it measures the quality of an image. The range of the SSIM value is limited between 0 and 1, and if the value is close to 1, the image is similar to the cover image. The computation of SSIM is as follows:

\[SSIM(x,y)=\frac{(2\mu_{x}\mu_{y}+c_{1})(2\sigma_{xy}+c_{2})}{(\mu_{x}^{2}\mu_{ y}^{2}+c_{1})(\sigma_{x}^{2}+\sigma_{y}^{2}+c_{2})^{\prime}} \tag{21}\]

where \(\mu_{x}\) and \(\mu_{y}\) denote values of cover image \(x\) and the marked image \(y\), \(\sigma_{x}\) and \(\sigma_{y}\) are standard deviation values of the cover image and the marked image, while \(\sigma_{x,y}\) denotes the covariance of both two images. \(c_{1}\) and \(c_{2}\) are constants to stabilize the the division. Figure 5 shows a cropping attack on the marked image with limited ratios and the results of recovering the tampered images using the proposed method. The ratio was limited to 10% to 45%. For the cropping attack, the visual difference between the original image and the restored images as applying the method proposed in Section 4 (see Figure 5b) was very similar. Objective evaluations such as PSNR and SSIM of Figure 5 can be found in Tables 1 and 2. The merit of our proposed method is that it manages PSNR (Table 1) and SSIM (Table 2) about marked images and recovered images reasonably. In the case of Lena image, when the ratio of cropping attacks is 5% and 45%, the difference between the two PSNRs is only 3.4573 dB. While, in the case of the Barbara image, the difference was highest among the comparison PSNRs in Table 1, i.e., it was 6.3909 dB. In Table 2, the reason for introducing SSIM to measure image quality is that SSIM was high in the case of Baboon images with low PSNR (in Table 1), and subjective evaluation of image quality like the human visual system is possible, unlike PSNR measurement results. Overall, it means that SSIM was more accurate than PSNR in subjective terms. Therefore, two measurements are required for complementarity. As a result, it showed very good performance compared to other existing methods. Figure 4: Test images used in our experiments: (**a**) Lena, (**b**) Pepper, (**c**) Airplane, (**d**) Boat, (**e**) Goldhill, (**f**) Couple, (**g**) Baboon and (**h**) Zelda. Figure 6 shows the quality of the restored image as PSNR when the cropping attack ratio was applied from 5% to 45% for the marked image. As the attack rate increased, PSNR