## 3 Proposed Work
### Image Restoration Phase
Applying Equation (11) to all the non-overlapping \(2\times 2\) blocks of \(V^{\prime}\) will yield, say, \(V^{0}\). Similarly, we obtain the remaining three image blocks of \(V^{\prime}\), say, \(V^{1},V^{2}\), and \(V^{3}\)on applying Equations (12) through (14). We perform decryption through bit-wise XOR operation of the four different image blocks of \(V^{\prime}\) and the corresponding block of pseudo-random matrix \(M^{\prime}\); i.e., if the block \(V^{\prime}\) is the \(K\)th block of \(E^{\prime}\), we choose the corresponding \(K\)th block of \(M^{\prime}\) to decrypt all the four image blocks \(\{V^{0},V^{1},V^{2},\) and \(V^{3}\}\). After decryption, we obtain new blocks \(V^{0}_{D},V^{1}_{D},V^{2}_{D}\), and \(V^{3}_{D}\). \[V^{0}_{D} =V^{0}\oplus M^{\prime K}. \tag{15}\] \[V^{1}_{D} =V^{1}\oplus M^{\prime K}.\] (16) \[V^{2}_{D} =V^{2}\oplus M^{\prime K}.\] (17) \[V^{3}_{D} =V^{3}\oplus M^{\prime K}. \tag{18}\]  
The correlation between pixels in the decrypted blocks \(\{V^{0}_{D},V^{1}_{D},V^{2}_{D},V^{3}_{D}\}\) paves the way for recovering the correct block. This is defined via a smoothness function. The algorithm for computing the multi-directional smoothness measure of a block is given in Algorithm 3. The input to Algorithm 3 is a decrypted image block \(V^{i}_{D}\) of size \(H\times H\) pixels, where \(0\leq i<4\). The computation of the multi-directional smoothness measure considers every overlapping \(2\times 2\) block of the decrypted block \(V^{i}_{D}\). Consider \(Q\) as one of the \(2\times 2\) blocks from a decrypted block \(V^{i}_{D}\); i.e., let  
\[Q= \begin{bmatrix}T_{11}&T_{12}\\ T_{21}&T_{22}\end{bmatrix} \;\;then,\]  
The smoothness measure \(SM\) of the \(2\times 2\) block \(Q\) is computed as follows:  
\[SM_{2\times 2}=\sum_{n=1}^{6}D_{n}, \tag{19}\]  
where  
\[ \begin{array}{l}D_{1}{=}\left|T_{11}-T_{12}\right|\cdot\\ D_{2}{=}\left|T_{21}-T_{22}\right|\cdot\\ D_{3}{=}\left|T_{11}-T_{21}\right|\cdot\\ D_{4}{=}\left|T_{12}-T_{22}\right|\cdot\\ D_{5}{=}\left|T_{11}-T_{22}\right|\cdot\\ D_{6}{=}\left|T_{12}-T_{21}\right|\cdot\end{array} \]  
The smoothness measure of every overlapping \(2\times 2\) block of \(V^{i}_{D}\) will be calculated. The sum of all these values gives the final smoothness measure. The smoothness measure is an indication of the strength of pixels correlation within a block. We expect a weak correlation among the pixels after encryption. Hence, the block \(V^{i}_{D}\) with minimum smoothness value is the restored image block. Note that an incorrect block from \(V^{i}_{D}\) will have a random pixel distribution, resulting in a higher smoothness value. The recovered block, through the smoothness measurement, is placed at the corresponding position of the restored image. Consider \(SM^{0},SM^{1},SM^{2}\), and \(SM^{3}\) to be the smoothness measure computed from \(V^{0}_{D},V^{1}_{D},V^{2}_{D},V^{2}_{D},\) and \(V^{3}_{D}\), respectively. If \(SM^{m}\) gives the minimum measurement from \(SM^{0},SM^{1}\), \(SM^{2},SM^{3}\) where \(m\in\{0,1,2,3\}\) then the restored image block and the extracted hidden bits are determined as follows:  
* Case 1: If \(m==0\) then the 00 will be extracted message bits and \(V^{0}_{D}\) will be the restored block. * Case 2: If \(m==1\) then the 01 will be extracted message bits and \(V^{1}_{D}\) will be the restored block. * Case 3: If \(m==2\) then the 10 will be extracted message bits and \(V^{2}_{D}\) will be the restored block. * Case 4: If \(m==3\) then the 11 will be extracted message bits and \(V^{3}_{D}\) will be the restored block. I.e., the correct block identified via the smoothness measurement calculation helps in recovering the hidden data in that block. It is obvious that the data hidden are 00 if the recovered block is \(V_{D}^{0}\). This is because the decrypted block \(V_{D}^{0}=V^{0}\oplus M^{K}\) and \(V^{0}\) is generated by applying Equation (11), which corresponds to the operation employed for embedding 00.