## 3 The Proposed Local-Moment-Driven Robust Reversible Data-Hiding Scheme
### Embedding Method
#### 3.1.4 Embedding Algorithm
_Extraction and recovering example:_ The proposed extraction and recovery process is illustrated using an example in Figure 4. The example considers a partial stego-image (\(I\)) of 6 \(\times\) 5 pixels which is decomposed into two images (\(I^{\prime\ MSB}\) and \(I^{LSB}\)) by considering \(n=3\). Then, the \(I^{\prime\ MSB}\) image is represented in a checkboard pattern and divided into blocks of 3 \(\times\) 3 pixels in size, excluding border pixels (shown in blue). Next, the pixels shown in orange are used for extraction and recovery. For this, the pixels are segregated into two groups (\(F_{G}\), \(S_{G}\)) using Equation (10). Here, the pixels grouped in the first group (\(F_{G}\)) are tagged and background-colored green (for easy identification to readers) whereas the other pixels are tagged as \(S_{G}\) and background-colored red. Since the first group (\(F_{G}\)) has only two pixels (\(x^{\prime}_{1}\) and \(x^{\prime}_{2}\)) of values 18 and 20, respectively, two prediction errors (\(e_{1}^{\prime PEE}\) and \(e_{2}^{\prime PEE}\)) are calculated, such as \(e_{1}^{{}^{\prime}PEE}=x^{\prime}_{1}-[\mu_{1}]\) and \(e_{2}^{\prime PEE}=x^{\prime}_{2}-[\mu_{2}]\), where \([. ]\) gives a rounded-off value and the \(\mu_{1}\) and \(\mu_{2}\) represent the local moment of pixel \(x^{\prime}_{1}\) and \(x^{\prime}_{2}\), respectively. Thus, the calculated values of \([\mu_{1}]\) and \([\mu_{2}]\) are 18 and 18, respectively. Therefore, \(e_{1}^{\prime PEE}=\)0 and \(e_{2}^{\prime PEE}=\) 2 is obtained. Next, the error pair (0, 2) is contracted to (0, 1) by following Figure 2b and reversing the arrows of Figure 2b. Additionally, the hidden \begin{table}
\begin{tabular}{c c c c c c} \hline \(\mathbf{x_{1}^{\prime}-[\mu_{1}]}\) & \(\mathbf{<-2}\) & \(\mathbf{-2}\) & \(\mathbf{-1}\) & \(\mathbf{0}\) & \(\mathbf{1}\) & \(\mathbf{>1}\) \\ \hline \(x_{ij}\) & \(x^{\prime}_{1}+1\) & \(x^{\prime}_{1}+1\) & \(x^{\prime}_{1}\) & \(x^{\prime}_{1}\) & \(x^{\prime}_{1}-1\) & \(x^{\prime}_{1}-1\) \\ \hline \(b_{1}\) & - & \(\mathbf{0}\) & \(\mathbf{1}\) & \(\mathbf{0}\) & \(\mathbf{1}\) & - \\ \hline \end{tabular}
\end{table} 
Table 1: S-PEE extraction and recovery rules. information obtained from the contraction is '0'. Next, the pixels (\(x_{1}\) and \(x_{2}\)) are recovered. Therefore, (18, 19) are received as the values of the pixel pair. For extracting from \(S_{G}=(17,\ 18,\ 18)\), the count of pixels in the group is checked, as it is 3, meaning that the two prediction errors (\(e_{1}^{\prime IPVO}\) and \(e_{2}^{\prime IPVO}\)) are calculated, such as \(e_{1}^{\prime IPVO}=x_{s}-x_{t}\) and \(e_{2}^{\prime IPVO}\)=\(x_{u}-x_{v}\), where the value of \(x_{s}\), \(x_{t}\), \(x_{u}\), and \(x_{v}\) are 17, 18, 18, and 18, respectively. Therefore, \(e_{1}^{\prime IPVO}=-1\) and \(e_{2}^{\prime IPVO}=0\) are obtained. Next, the error pair (\(-1\), 0) is contracted to (0, 0) by following Figure 2a and reversing the arrows of Figure 2b. Additionally, the hidden information obtained from the contraction is '01'. Next, the pixels (\(x_{\pi(1)}\) and \(x_{\pi(N)}\)) are recovered. Therefore, (18, 18) are received as values of the minimum and maximum pixel pair. Thus, the hidden secret data bit '001' and original image are both obtained without loss.