## 2 Background and Related Work
### Singular Value Decomposition  
Given a matrix \(X\) of size \(m\times n\), the singular value decomposition of \(X\) results in three matrices (\(U,S,\) and \(V\)). \(U\) and \(V\) are orthogonal matrices with size \(m\times m\), whereas \(S\) is a diagonal matrix containing values called \"the singular values\". Equation (1) shows the SVD results. \[X=U*S*V^{T} \tag{1}\]  
To prove the importance of the diagonal matrix \(S\), an image is split into blocks sized \(4\times 4\), and the svd is applied on each. As shown in Equation (2), the singular matrix for each block contains four singular values : \(S_{1}\dots S_{4}\). Due to the huge difference between the first value and all other values, we can conclude that \(S_{1}\) has the most significant impact on  
Figure 1: (**a**) An MPEG video sequence (**b**) MPEG video sequence structure. the image. The values of the singular values for some blocks of the standard images are shown in Table 1.  
\[B_{i}(X) =U\cdot S\cdot V^{T}\] \[=U\cdot \begin{bmatrix}S_{1}&0&0&0\\ 0&S_{2}&0&0\\ 0&0&S_{3}&0\\ 0&0&0&S_{4}\end{bmatrix} \cdot V^{T} \tag{2}\]  
To prove the importance of the first singular value (S1) for each block in comparison with all other singular values, multiple standard images are used (shown in Figure 2), and each image is split into blocks sized \(4\times 4\). Singular value decomposition is applied on each block, and the last three singular values (\(S_{2}\), \(S_{3}\), and \(S_{4}\)) are set to zeros. Then, the images are reconstructed, and the results are illustrated in Figure 3. The results prove that even with only a quarter of the singular values, we are able to reconstruct the images with acceptable quality. This compels us to conclude that if this quarter is encrypted, it will result in the encryption of the whole image, while the amount of data to be encrypted is reduced to a quarter. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Standard** **Image**} & \multirow{2}{*}{**Block** **Number**} & \multicolumn{5}{c}{**Value**} \\ \cline{3-6}  & & **S1** & **S2** & **S3** & **S4** \\ \hline \multirow{3}{*}{cameraman} & \(1\) & 628.5056 & 16.6462 & 7.8219 & 2.7480 \\ \cline{2-6}  & 33 & 474.5510 & 24.6315 & 8.2719 & 0.4573 \\ \hline \multirow{3}{*}{tank} & 40 & 522.7868 & 14.9429 & 3.7784 & 1.1981 \\ \cline{2-6}  & 200 & 527.5800 & 13.3748 & 10.0325 & 0.9129 \\ \hline \multirow{3}{*}{lena} & 4000 & 530.3523 & 36.7818 & 8.0956 & 3.1669 \\ \cline{2-6}  & 2150 & 455.2935 & 14.5609 & 6.7711 & 3.6052 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of the singular matrix for some blocks of the standard images. Figure 2: Original standard images. (**a**) aerial; (**b**) lena; (**c**) cameraman; (**d**) jet; (**e**) lake; (**f**) tank. The structural similarity (SSIM) is a metric that measures the similarity between two images [34]. It is used in many fields such as watermarking and compression to measure the effects of distortion or compression on images. The closer the value of SSIM is to \"1\", the higher the similarity of the images. SSIM is defined by Equation (3). \[SSIM(x,y)=\frac{(2\mu_{x}\mu_{y}+C_{1})(2\sigma_{xy}+C_{2})}{(\mu_{x}^{2}+\mu_{y }^{2}+C_{1})(\sigma_{x}^{2}+\sigma_{y}^{2}+C_{2})} \tag{3}\]  
where:  
* \(\mu_{x}\) and \(\mu_{y}\) represent the averages of \(x\) and \(y\), respectively. * \(\sigma_{x}^{2}\) and \(\sigma_{y}^{2}\) represent the variances of \(x\) and \(y\), respectively. * \(\sigma_{xy}\) represents the covariance of \(x\) and \(y\). * \(C_{1}=(k_{1}L)^{2}\) and \(C_{2}=(k_{2}L)^{2}\) are two variables used to stabilize a division with a weak denominator. * \(L\) is the dynamic range of the pixel values. * \(k_{1}=0.01\) and \(k_{2}=0.03\). Table 2 shows the SSIM values between the original images and the images reconstructed using only one-quarter of the singular values. The results prove that the most important information of image blocks is contained within the first singular value.