## 3 The Proposed Approach
### The Extraction and Recovery Phase  
Through the communication channel, the stego-image and the trained MLP model are received. Next, we consider the secret data extraction from the stego-image and the stego-image recovery. The scheme then enters the extraction and recovery phase. In the extraction and recovery stage, the procedure of the secret data extraction and the stego-image recovery is similar to the procedure of embedding and shifting. The secret data extraction process is briefly described as follows. First, rhombus prediction and double-layered embedding is adopted to divide the stego-image into two sets denoted as \"star\" and \"dot\" (as shown in Figure 4a), and half of the secret data will be extracted from the star and dot sets, respectively. Except for the pixels located in borders, the pixels of the star or dot set are scanned from top-left to bottom-right to derive the stego sequence \((p_{1}^{\prime},\dots,p_{n}^{\prime})\). Then, the 4-neighbor dots of each \(p_{i}^{\prime}\) are introduced to the trained MLP to obtain its predicted value \(\hat{p}_{i}^{\prime}\). The predicted value is used to determine the prediction-error sequence \((e_{1}^{\prime},\dots,e_{n}^{\prime})\), and the sequence is divided into the prediction-error triples \((e_{x}^{\prime},e_{y}^{\prime},e_{z}^{\prime})\). The prediction-error \(e_{i}^{\prime}\) can be obtained as  
\[e_{i}^{\prime}=p_{i}^{\prime}-\hat{p}_{i}^{\prime}.\]  
Finally, each recovered triple \((p_{x}^{\prime},p_{y}^{\prime},p_{z}^{\prime})\) is extracted based on the 3D-PEH as the method shown in Table 6-8. The 3D-PEH recovery method is divided into seven types: Type \(A^{\prime}\) to Type \(G^{\prime}\). Besides, \((e_{x}^{\prime},e_{y}^{\prime},e_{z}^{\prime})\) should be the prediction-errors between the \"marked pixels\" (in the stego-image) and the prediction of the \"marked pixels\". When the prediction-error \(e_{i}^{\prime}\) is 1, the recovered value is \(p_{i}=p_{i}^{\prime}-1\), and when the prediction-error \(e_{i}^{\prime}\) is \(-1\), the recovered value is \(p_{i}=p_{i}^{\prime}+1\). The secret data bits are extracted by type \(A^{\prime}\) to type \(G^{\prime}\) according to the condition of the prediction-error group and the stego pixel group \((p_{x}^{\prime},p_{y}^{\prime},p_{z}^{\prime})\) is recovered to the recover pixel groups that have the same pixel values as the cover pixel groups \((p_{x},p_{y},p_{z})\). In addition, the type \(G^{\prime}\) has no embedded data bits, so only recover the stego pixel groups to the recover pixel groups without secret data extraction. Through the extraction and recovery phase, the secret data and the recovered image are obtained. **Example 2**. : _Let \(P^{\prime}=\{210,99,131,64,72,162,17,19,25,162,24,71,87,96,47\}\), and \(E^{\prime}=\{0,0,0,\\ -1,0,0,1,1,-1,2,-1,0,1,2,0\}\)._  
* _Step 1:_ 1. _Get the three bits from_ \(E^{\prime}=\{\mathbf{0,0,0,-1,0,0,1,1,-1,2,-1,0,1,2,0\}\)_:_ \((e^{\prime}_{x},e^{\prime}_{y},e^{\prime}_{z})=(0,0,0)\)_. This is a Type-_\(A^{\prime}\) _case. The extracted secret bits are_ \((0,0,0)\)_._
* _Get three bits from_ \(P^{\prime}=\{\mathbf{210,99,131,64,72,162,17,19,25,162,24,71,87,96,47\}\)_. Then, we can derive_ \((p_{x},p_{y},p_{z})=(210,99,131)\)_._ _The results of this step are_ \(S=\{0,0,0,\ldots\}\) _and_ \(P=\{210,99,131,\ldots\}\)_._
* _Step 2:_ 1. _Get the three bits from_ \(E^{\prime}=\{0,0,0,-1,\mathbf{0,0,1,1,-1,2,-1,0,1,2,0\}\)_:_ \((e^{\prime}_{x},e^{\prime}_{y},e^{\prime}_{z})=(-1,0,0)\)_. This is a Type-_\(A^{\prime}\) _case. The extracted secret bits are_ \([1],[1]\)_._
* _Get three bits from_ \(P^{\prime}=\{210,99,131,64,72,162,17,19,25,162,24,71,87,96,47\}\)_. Then, we can derive_ \((p_{x},p_{y},p_{z})=(64+1,72,162)=(65,72,162)\)_._ _The results of this step are_ \(S=\{0,0,0,1,1,\ldots\}\) _and_ \(P=\{210,99,131,65,72,162\ldots\}\)_._
* _Step 3:_ 1. _Get the three bits from_ \(E^{\prime}=\{0,0,0,-1,0,0,1,1,-1,2,-1,0,1,2,0\}\)_:_ \((e^{\prime}_{x},e^{\prime}_{y},e^{\prime}_{z})=(1,1,-1)\)_. This is a Type-_\(B^{\prime}\) _case.