## 3 Proposed Method
### Extracting the Message and Restoring the Original Image  
At this stage, the receiver uses the instructions of the additional information to extract the secret message embedded in the stego image \(I^{\prime}\) according to the reverse operation method of the embedding procedure to recover the original image \(I\). Like the recurrent robust reversible data hiding (triple-RDH) method in this study, we distinguish the stego image (\(I^{\prime}\)) into embeddable and un-embeddable areas: the pixel values of the first column, first row, last column, and last row of the stego image are the un-embeddable part. For each embeddable stego pixel \(y_{i}\), the quotient operation is \(y_{i}^{q}=y_{i}\:div\:2^{n}\). As mentioned above, since we need to reverse the hidden method to extract the secret message and recover the original pixel value, we need to extract the message bits from the white pixels with the recursive parameter \(t_{w}\) first until all the secret messages hidden in the white pixels are retrieved and the white pixel value is recovered, then the recursive parameter \(t_{g}\) is used to start message extraction and recovery of gray pixel value. In the following extraction equation, we take the recursive parameters \(t_{w}=2\) and \(t_{g}=2\) as an illustration. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Recursive Parameter \(t_{w}\)** & **Original Pixel** & **Predictor** & \(e\) & \(S\) & **Stego Pixel** \\ \hline \(t_{w}=1\) & \(x_{2,3}=20\) & \(p_{1}=19\) & \(e^{out}=1\) & \(s_{5}=1\) & \(x_{2,3}^{(out)}=21\) \\  & \(x_{2,3}^{(out)}=21\) & \(p_{2}=22\) & \(e^{back}=-1\) & \(s_{6}=1\) & \(x_{2,3}^{(back)}=20\) \\ \(t_{w}=2\) & \(x_{2,3}^{(back)}=20\) & \(p_{1}=19\) & \(e^{out}=1\) & \(s_{7}=1\) & \(x_{2,3}^{(out)}=21\) \\  & \(x_{2,3}^{(out)}=21\) & \(p_{2}=22\) & \(e^{back}=-1\) & \(s_{8}=0\) & \(x_{2,3}^{(back)}=21\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: White pixel value and the changes of various values in the embedding procedure. The receiver will use the following Equations (13) and (14) in the information extraction and recovery procedure of the pixel value. When \(t_{w}=2\), the white pixel value will be processed first. \[x_{c}^{(hack)}=\left\{ \begin{array}{ll}y_{i}^{g}+1,if\;e^{back}<-2\\ y_{i}^{g}+1,if\;e^{back}=-2\\ y_{i}^{g}\;\;\;\;\;\;,if\;e^{back}=-1\end{array} \right.\;\;and\;s_{4}=\left\{ \begin{array}{ll}none,if\;e^{back}<-2\\ 1,if\;e^{back}=-2\\ 0,if\;e^{back}=-1\\ none,if\;e^{back}>-1\end{array} \right. \tag{13}\]  
\[x_{c}^{(out)}=\left\{ \begin{array}{ll}x_{c}^{(back)}&,if\;e^{out}<1\\ x_{c}^{(back)}&,if\;e^{out}=1\\ x_{c}^{(back)}-1,if\;e^{out}=2\\ x_{c}^{(back)}-1,if\;e^{out}>2\end{array} \right.\;and\;s_{3}=\left\{ \begin{array}{ll}none,if\;e^{out}<1\\ 0,if\;e^{out}=1\\ 1,if\;e^{out}=2\\ none,if\;e^{out}>2\end{array} \right. \tag{14}\]  
Then, when \(t_{w}=1\), the message extraction and recovery procedure of pixel value use the following Equations (15) and (16) to recovery white pixel value and extract the secret message until all the white pixel values are restored and the secret message is extracted. \[x_{c}^{(back)}=\left\{ \begin{array}{ll}x_{c}^{(out)}+1,if\;e^{back}<-2\\ x_{c}^{(out)}+1,if\;e^{back}=-2\\ x_{c}^{(out)}&,if\;e^{back}=-1\end{array} \right.\;and\;s_{2}=\left\{ \begin{array}{ll}none,if\;e^{back}<-2\\ 1,if\;e^{back}=-2\\ 0,if\;e^{back}=-1\\ none,if\;e^{back}>-1\end{array} \right. \tag{15}\]  
\[x_{c}=\left\{ \begin{array}{ll}x_{c}^{(back)}&,if\;e^{out}<1\\ x_{c}^{(back)}&,if\;e^{out}=1\\ x_{c}^{(back)}-1,if\;e^{out}=2\\ x_{c}^{(back)}-1,if\;e^{out}>2\end{array} \right.\;and\;s_{1}=\left\{ \begin{array}{ll}none,if\;e^{out}<0\\ 0,if\;e^{out}=0\\ 1,if\;e^{out}=1\\ none,if\;e^{out}>1\end{array} \right.