## 3 Proposed Method
### Message Embedding
#### 3.1.2 Phase-2: Weight Modulus Reversible Data-Hiding Embedding (WMRDHE)
We use an individual secret message in \(\mathbf{d}_{du}\) as the row index and the weight vector, \(\mathbf{W}_{b}\) = [\(w_{1}\), \(w_{2}\)], as the color index to derive the corresponding 2-tuple distortion vector, \(\mathbf{V}_{d_{k}}\), stored in the distortion vector table (DVT). For example, let \(\mathbf{d}_{du}\) = \((d_{k}^{1},d_{k}^{2},d_{k}^{3})\)= (\(3_{5}\), \(2_{5}\), \(4_{5}\)) be the 5-ary secret message vector (\(k\) = 5) and \(\mathbf{W}_{b}\) = [1,4] be the weight vector. The distortion vector table, shown in Table 3, indicates that \(\mathbf{V}_{3}\) = [\(-\)1, 1] is the distortion vector corresponding to 3\({}_{5}\). Similarly, the distortion vectors corresponding to 2\({}_{5}\) and 4\({}_{5}\) are \(\mathbf{V}_{2}\) = [1, \(-\)1] and \(\mathbf{V}_{4}\) = [0, 1], respectively. We simplify the process in Step 2 as the DVT function, where its entry is a parameter pair, \((d_{k}\), \(\mathbf{W}_{b})\), and the output is the distortion vector, \(\mathbf{V}_{d_{k}}\), as show in Equation (9). **Step 3:** Producing the stego component pairs in three channels, \(\mathbf{P}_{r1}^{\prime\prime}\), \(\mathbf{P}_{g2}^{\prime\prime}\), and \(\mathbf{P}_{b3}^{\prime\prime}\). Using the vector addition, shown in Equation (10), we can embed a secret message vector \(\mathbf{d}_{du}\) into the dual-pixel \(\mathbf{P}_{d\mu}^{\prime\prime}\) and produce the stego dual-pixel, \(\mathbf{P}_{d\mu}^{\prime\prime}\), which contains the stego component pairs, \(\mathbf{P}_{r1}^{\prime\prime}\), \(\mathbf{P}_{g2}^{\prime\prime}\), and \(\mathbf{P}_{b3}^{\prime\prime}\). Following up on Example-1, we have constructed the dual-pixel, \(\mathbf{P}_{du}^{\prime\prime}\) = ((13, 13), (5, 5), (21, 21)). Let \(\mathbf{d}_{du}\) = (\(3_{5}\), \(2_{5}\), \(4_{5}\)) be the secret message vector, and \(\mathbf{V}_{3}\) = [\(-\)1, 1], \(\mathbf{V}_{2}\) = [1, \(-\)1], and \(\mathbf{V}_{4}\) = [0, 1] represent the corresponding distortion vectors. We can produce the stego component pair, \(\mathbf{P}_{A}^{\prime\prime}\) = (13, 13) + (\(-\)1, 1) = (12, 14). Similarly, we can produce the other two stego component pairs, \(\mathbf{P}_{g2}^{\prime\prime}\) = (5, 5) + (1, \(-\)1)= (6, 4) and \(\mathbf{P}_{b3}^{\prime\prime}\) = (21, 21) + (0, 1) = (21, 22). Therefore, the stego dual-pixel produced becomes \(\mathbf{P}_{d\mu}^{\prime\prime}\) = ((12, 14), (6, 4), (21, 22)). Note that the corresponding stego pixels in \(I_{s1}^{\prime\prime}\) and \(I_{s2}^{\prime\prime}\) are \(\mathbf{P}_{s1}^{\prime\prime}\) = (12, 6, 21) and \(\mathbf{P}_{s2}^{\prime\prime}\) = (14, 4, 22), respectively. We remark that in Phase-1, these two stego pixels have a 3-ary secret message vector, \(\mathbf{d}_{\mathbf{sb}}\) = (\(1_{3}\), \(2_{3}\), \(0_{3}\)), concealed, and a 7-ary secret message, \(2_{7}\). In addition, in Phase-2, they have also conveyed a 5-ary secrete message vector, \(\mathbf{d}_{du}\) = (\(3_{5}\), \(2_{5}\), \(4_{5}\)). Clearly, we can produce the stego component pairs for every pixel in three channels: \(\mathbf{P}_{r1}^{\prime\prime}\), \(\mathbf{P}_{g2}^{\prime\prime}\), and \(\mathbf{P}_{b3}^{\prime\prime}\) by a vector addition shown in Equation (10). Consequently, dual stego images, \(I_{s1}^{\prime\prime\prime}\) and \(I_{s2}^{\prime\prime}\), will be produced in Phase-2. \[\mathbf{V}_{d_{k}} =\text{DVT}(d_{k}\,\ \mathbf{W}_{b}) \tag{9}\] \[\left\{ \begin{array}{l}\mathbf{P}_{s1}^{\prime\prime}=\mathbf{P}_{s1}^{ \prime\prime}+\text{DVT}(d_{k}^{1}\,\ \mathbf{W}_{b})\\ \mathbf{P}_{g2}^{\prime\prime}=\mathbf{P}_{g2}^{\prime\prime}+\text{DVT}(d_{k}^{2}\,\ \mathbf{W}_{b})\\ \mathbf{P}_{b3}^{\prime\prime}=\mathbf{P}_{b3}^{\prime\prime}+\text{DVT}(d_{k}^{3}\,\ \mathbf{W}_{b})\end{array} \right.