## 3 Proposed Method
### Message Embedding
#### 3.1.1 Phase-1: Optimal Weighted Color-Transfer Embedding (OWCTE)  
In this phase, we input the source image, \(I_{\text{s}ou}\), and the target image, \(I_{\text{t}ar}\), and conceal two secret messages in two steps when conducting the optimal weighted color transfer. We detail the message embedding in this phase from the viewpoint of a single pixel. Algorithm 1 shows the OWCTE, where \(\mathbf{FP}\) = (r, g, b) denotes a single floating-point pixel, represented by three floating-point values, where all of them are within the range of [0.0, 255.0]. Readers are recommended to refer to [33] for a detailed description about the optimal color-transfer process. Here, we describe the process of producing a single floating-point pixel in brief, and adopt the Red channel as an example. We assume that an optimal weight, \(W^{*}\), has been determined. Given a pixel, \(P_{\text{R}}(x,\;y)\), in the source image at the position \((x,\;y)\), we can produce a color-transferred pixel with the floating-point value, \(FP_{\text{R}}(x,\;y)\), in the Red channel by Equation (5), where \(\sigma_{s}\) and \(\sigma_{t}\) indicate the respective standard deviations of the source and target image in the Red channel, and \(\mu_{s}\) and \(\mu_{t}\) denote the means of the source and target image in the Red channel, respectively. \[FP_{\text{R}}(x,\;y)=\frac{W^{*}\sigma_{t}+(1-W^{*})\sigma_{s}}{\sigma_{s}}(P_ {\text{R}}(x,\;y)-\mu_{s})+W^{*}\mu_{t}+(1-W^{*})\mu_{s} \tag{5}\]  
As an example, we consider a pixel in the position \((x,\;y)=(18,\;33)\), where the pixel in the Red channel has the value \(P_{\text{R}}(18,\;33)=14\). Let \(W^{*}=0.493\), \(\sigma_{s}=24.56\), \(\sigma_{t}=23.89\), \(\mu_{s}=120.7\), and \(\mu_{t}=124.5\). Given these parameters, we can adopt Equation (5) to derive \(FP_{\text{R}}(18,\;33)=17.3\). We can calculate the floating-point pixel values in the Green and Blue channels accordingly, thereby producing the available \(\mathbf{FP}\) = (r, g, b) for Algorithm 1. Note that \(\mathbf{FP}\) = (r, g, b) is produced in the mid-process of the optimal color transfer when pixels have not been rounded up or down to produce the integer values. ```
Input: \(\mathbf{FP}\) = (r, g, b), \(\mathbf{W}_{a}\) = [\(w_{1}\), \(w_{2}\), \(w_{3}\)], \(\mathbf{d}_{\mathbf{sb}}\) = \(d_{\mathbf{u}}^{1}\), \(d_{\mathbf{n}}^{2}\), \(d_{\mathbf{n}}^{3}\)), and \(d_{\mathbf{w}m}\) Output: \(\mathbf{P}_{\text{oucet}}^{*}\)
1. Determine the minimal distortion vector, \(\mathbf{V}_{\mathbf{sb}}\) = (\(v_{r}\), \(v_{g}\), \(v_{b}\)), to satisfy Equation (6). 2. Determine the coefficient, \(\mathbf{C}_{\mathbf{w}m}\) = (\(c_{r}\), \(c_{g}\), \(c_{b}\)), to satisfy Equations (7) and (8). 3. Produce \(\mathbf{P}_{\text{oucet}}^{*}\) = \(\mathbf{FP}+\mathbf{V}_{\mathbf{sb}}\) + n\(\mathbf{C}_{\mathbf{w}m}\). ```  
**Algorithm 1** OWCTE algorithm  
Let \(\mathbf{P}_{\text{oucet}}^{\prime}\) = (\(r^{\prime}\), \(g^{\prime}\), \(b^{\prime}\)) be the stego pixel which has a concealed \(n\)-ary secret message vector, \(\mathbf{d}_{\mathbf{sb}}\). Let \(\mathbf{P}_{\text{oucet}}^{\prime}\) represent the stego pixel produced in Phase-1, which has both \(\mathbf{d}_{\mathbf{sb}}\) and \(d_{\mathbf{w}m}\) concealed. The OWCTE algorithm describes the message embedding in this phase, which consists of three steps. **Step 1**: Producing the stego pixel \(\mathbf{P}_{\text{oucet}}^{\prime}\) = (\(r^{\prime}\), \(g^{\prime}\), \(b^{\prime}\)). Given \(\mathbf{FP}\) = (r, g, b), we determine the floating-point distortion vector, \(\mathbf{V}_{\mathbf{sb}}\) = (\(v_{r}\), \(v_{g}\), \(v_{b}\)), such that \(\mathbf{P}_{\text{oucet}}^{\prime}\) mod = (\(d_{\mathbf{n}}^{1}\), \(d_{\mathbf{n}}^{2}\), \(d_{\mathbf{n}}^{3}\)) and \(\mathbf{V}_{\mathbf{sb}}\) have the minimal length, \(L_{min}\) = [\((v_{r})^{2}+(v_{g})^{2}+(v_{b})^{2}\)]\({}^{0.5}\), as shown in Equation (6), where \(\mathbf{P}_{\text{oucet}}^{\prime}\) = \(\mathbf{FP}+\mathbf{V}_{\mathbf{sb}}\). Note that there are a number of distortion vectors which satisfy the above modulus operation.