## 5 Experimental Study
### Analysis of Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity Index (SSIM)  
PSNR is a factor of the maximum possible value of a pixel to the error that distorts the original quality of the image. It is expressed as follows:  
\[PSNR=10\log_{10}\left(\frac{I_{max}^{2}}{MSE}\right)\ \ \mathrm{dB}, \tag{21}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Name of the Image** & **Block Length** & **Embedding Rate** & **BER** \\ \hline \multirow{4}{*}{Airplane} & \(4\times 4\) & 0.12500 & 0.00617 \\ \cline{2-4}  & \(8\times 8\) & 0.03125 & 0 \\ \cline{2-4}  & \(16\times 16\) & 0.00781 & 0 \\ \cline{2-4}  & \(32\times 32\) & 0.00195 & 0 \\ \cline{2-4}  & \(64\times 64\) & 0.00049 & 0 \\ \hline \multirow{4}{*}{Baboon} & \(4\times 4\) & 0.12500 & 0.01849 \\ \cline{2-4}  & \(8\times 8\) & 0.03125 & 0.00024 \\ \cline{2-4}  & \(16\times 16\) & 0.00781 & 0 \\ \cline{2-4}  & \(32\times 32\) & 0.00195 & 0 \\ \cline{2-4}  & \(64\times 64\) & 0.00049 & 0 \\ \hline \multirow{4}{*}{Boat} & \(4\times 4\) & 0.12500 & 0.00604 \\ \cline{2-4}  & \(8\times 8\) & 0.03125 & 0 \\ \cline{2-4}  & \(16\times 16\) & 0.00781 & 0 \\ \cline{2-4}  & \(32\times 32\) & 0.00195 & 0 \\ \cline{2-4}  & \(64\times 64\) & 0.00049 & 0 \\ \hline \multirow{4}{*}{Peppers} & \(4\times 4\) & 0.12500 & 0.00275 \\ \cline{2-4}  & \(8\times 8\) & 0.03125 & 0 \\ \cline{1-1} \cline{2-4}  & \(16\times 16\) & 0.00781 & 0 \\ \cline{1-1} \cline{2-4}  & \(32\times 32\) & 0.00195 & 0 \\ \cline{1-1} \cline{2-4}  & \(64\times 64\) & 0.00049 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Embedding rate and BER obtained while using different block sizes. where \(I_{max}\) indicates the maximum possible value of an image pixel and \(MSE\) is the mean squared error, defined as follows:  
\[MSE=\left(\frac{1}{XY}\sum_{a=1}^{X}\sum_{b=1}^{Y}|I(a,b)-R(a,b)|^{2}\right), \tag{22}\]  
where the reference image is \(I\) and the restored image is \(R\). The \(MPSA\) scheme is expected to restore the correct cover image losslessly, and hence MSE = 0. This implies that PSNR should give \(\infty\) for a good approach. The SSIM gives the structural similarity index, another parameter for effective quality estimation of the restored image. Here, the structural quality degradation of the restored image gets quantified based on the luminance masking and contrast masking factors. SSIM should be 1 for the correctly recovered original image. The results of PSNR and SSIM values are also tabulated in Table 2.  
The PSNR values returned from the \(MPSA\) scheme on well-known images, along with the SSIM outputs, are indicated in Table 2. All three images, except _baboon_, were restored without any loss of information. This yields a PSNR \(\infty\) and SSIM measure of 1 with a block size of \(8\times 8\) pixels and higher. As mentioned in the previous section, our scheme recovers the correct block via the smoothness measurement, which is a reflection of the correlation between the pixels in the block. A smaller block with edges or corners does not give significant scope for a good smoothness measure. Hence, perfect recovery of a smaller block is difficult, as evident from the PSNR and SSIM values for \(4\times 4\)-pixel block sizes in Table 2.