# Verifiable (2, \(n\)) Image Secret Sharing Scheme Using Sudoku Matrix
## 4 Experimental Results  
In the experiment, two secret images of size 256 \(\times\) 256 pixels, as shown in Figure 7a,b, and four cover images of size 512 \(\times\) 512 pixels, as shown in Figure 7c-f, were used. During the retrieval and authentication procedure, if the cover images were real, without evidence of tampering, we could reconstruct the full secret images completely, as shown in Figure 8a,b. To measure the visual quality of the cover images and reconstructed secret image, PSNR (peak signal-to-noise ratio) was used with Equation (12), where MSE (mean-squared error) was calculated with Equation (13), in which \(I(i,j)\) and \(K(i,j)\) were the values of the original pixel and that of the stego pixel located at location (\(i\),\(j\)), and \(m\) and \(n\) were the size (height and width) of the image. Generally, the higher the PSNR value is, the better the visual quality will be. Furthermore, we evaluated the visual quality between the cover image and the shared image using the structural similarity (SSIM) of Equation (14), where \(\mu_{x}\) and \(\mu_{y}\) are the averages of \(x\) and \(y\); \(\sigma_{x}\) and \(\sigma_{y}\) are the standard deviation values of \(x\) and \(y\); and values \(c_{1}\) and \(c_{2}\) are two constant values, which are set to 1s. If the SSIM value equals 1, it means that the two images are identical without any distortion. Visual quality comparisons of the shadow images shown in Table 1 demonstrate high visual quality. It is difficult to recognize the differences between cover images and share images. \[PSNR=10\log_{10}\frac{255^{2}}{MSE} \tag{12}\]  
\[MSE=\frac{1}{mn}\sum_{i=1}^{m}\sum_{j=1}^{n}[I(i,j)-K(i,j)]^{2} \tag{13}\]  
\[SSIM(x,y)=\frac{(2\mu_{x}\mu_{y}+c_{1})(\sigma_{x}y+c_{2})}{(\mu_{x}^{2}+\mu_ {y}^{2}+c_{1})(\sigma_{x}^{2}+\sigma_{y}^{2}+c_{2})} \tag{14}\] \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Secret Images**} & \multicolumn{2}{c}{**Lena**} & \multicolumn{2}{c}{**[PERSON]**} & \multicolumn{2}{c}{**Pepper**} & \multicolumn{2}{c}{**Zelda**} \\ \cline{2-9}  & **PSNR** & **SSIM** & **PSNR** & **SSIM** & **PSNR** & **SSIM** & **PSNR** & **SSIM** \\ \hline Bird & 49.88 & 0.993 & 49.88 & 0.998 & 49.88 & 0.993 & 49.89 & 0.990 \\ \hline Jet(F16) & 49.88 & 0.994 & 49.89 & 0.998 & 49.88 & 0.993 & 49.88 & 0.990 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Visual quality of shadow images when embedding different secret images. Figure 7: _Cont._We also compared several additional features listed in Table 2, which shows that our proposed scheme achieved high performance and met all requirements. \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline
**Features** & **Gao et al.’s Scheme**[5] & **Li et al.’s Scheme**[8] & **Liu et al.’s Scheme**[13] & **Liu et al.’s Scheme**[14] & **Gao et al.’s Scheme**[15] & **Proposed Scheme** \\ \hline Meaningful shares & Yes & Yes & Yes & Yes & Yes & Yes \\ Reversibility & Yes & Yes & Yes & - & Yes \\ Different cover images & - & - & Yes & Yes & Yes \\ (\(\mu\))-SIS & (2,2)-SIS & (3,3)-SIS & (2,2)-SIS & (2,3)-SIS & (2, \(\mu\))-SIS \\ Fault tolerance & Yes & - & - & - & Yes & Yes \\ Average PSNR & 32.96 & 49.07 & 48.72 & 41.71 & 40.41 & 49.88 \\ Embedding capacity (bits) & 786,432 & 624,215 & 524,288 & 785,525 & 1,310,720 & 786,432 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison to other methods. Figure 8: Reconstructed secret images. (**a**) Recovered image “Bird” (PSNR = \(\infty\)). (**b**) Recovered image “Jet(F16)” (PSNR = \(\infty\)). Figure 7: Secret images and cover images. (**a**) Secret image “Bird”. (**b**) Secret image “Jet(F16)”. (**c**) Cover image “Lena”. (**d**) Cover image “Baboon”. (**e**) Cover image “Pepper”. (**f**) Cover image “Zelda”. There were three fake logos, as shown in Figure 9a-c, which were replaced with the content of stego images. If a stego image had been modified (tampered with), as shown in Figure 9d (128 \(\times\) 128 in size), the secret image could not be completely reconstructed, as shown in Figure 9f.