## 3 Proposed Method
### Message Embedding
#### 3.1.1 Phase-1: Optimal Weighted Color-Transfer Embedding (OWCTE)
We determine the one that has the minimal length in aiming to reduce the pixel modification to as small as possible. **Example-1**: If \(\mathbf{FP}\) = (17.3, 3.7, 20.6) and \(\mathbf{d}_{\mathbf{sb}}\) = (13, 2, 0.3) represents a 3-ary message vector (\(n\) = 3) to be concealed. We determine \(\mathbf{V}_{\mathbf{sb}}\) = (\(-\)1.3, 1.3, 0.4), leading to \(\mathbf{P}_{\text{oucet}}^{\prime}\) = \(\mathbf{FP}+\mathbf{V}_{\mathbf{sb}}\) = (\(r^{\prime}\), \(g^{\prime}\), \(b^{\prime}\))= (16, 5, 21). Clearly, \(\mathbf{P}_{\text{oucet}}^{\prime}\) mod \(n\) = (16 mod 3, 5 mod 3, 21 mod 3) = (1, 2, 0) satisfies Equation (6). In addition, \(\mathbf{V}_{\mathbf{sb}}\) has the minimal length, where \(L_{min}\) = [\((-1.3)^{2}+(1.3)^{2}+(0.4)^{2}\)]\({}^{0.5}\). If we instead determine \(\mathbf{V}_{\mathbf{sb}}\) = (1.7, 1.3, 0.4), this leads to \(\mathbf{P}_{\text{oucet}}^{\prime}\) = (19, 5,21). Although \(\mathbf{P}^{\prime}_{\textit{oucet}}\) satisfies the modulus operation, the length of \(\mathbf{V_{sb}}\), denoted as \(L\), is not minimal because \(L=[(1.7)^{2}+(1.3)^{2}+(0.4)^{2}]^{0.5}>L_{min}\). **Step 2:** Producing the final stego pixel \(\mathbf{P}^{\prime}_{\textit{oucet}}\) = \((r^{\prime\prime},g^{\prime\prime},b^{\prime\prime})\). We determine the distortion vector \(\mathbf{C_{\textit{cum}}}\) = \((c_{r},c_{g},c_{b})\) such that \(\mathbf{P}^{\prime}_{\textit{oucet}}\) satisfies the following two conditions. First, it complies with the modulus operation, \((\mathbf{P}^{\prime}_{\textit{oucet}}\) dot \(\mathbf{W_{a}})\) mod \(m=d_{\textit{wm}}\) and \(n\times\mathbf{C_{\textit{cum}}}\) has minimal length, \(L_{min}=[n^{2}(c_{r}^{2}+c_{g}^{2}+c_{b}^{2})]^{0.5}\) as shown in Equation (7), where \(\mathbf{P}^{\prime}_{\textit{oucet}}\) = \(\mathbf{P}^{\prime}_{\textit{oucet}}\)+ \(n\times\mathbf{C_{\textit{cum}}}\). Note \(\mathbf{W_{a}}\) = \([w_{a}^{1},w_{a}^{2},w_{a}^{3}]\) represents the weight vector given by the end user for modulus operation. Any distortion vectors satisfying Equation (7) indicate that the stego pixel \(\mathbf{P}^{\prime}_{\textit{oucet}}\) has a concealed secret message, \(d_{\textit{wm}}\), which can later be extracted on the receiver side. The second condition, shown in Equation (8), states that \(\mathbf{P}^{\prime}_{\textit{oucet}}\) must be within the eligible range, [LB(\(k\)), UB(\(k\))], where LB(\(k\)) and UB(\(k\)) represent the lower bound and upper bound of the range, respectively, The eligible range depends on parameter \(k\), which represents the notational system based on which a secret message will be concealed in Phase-2. This condition ensures that the current stego pixel, \(\mathbf{P}^{\prime}_{\textit{oucet}}\), is eligible to conceal a \(k\)-ary secret message in Phase-2 without affecting previous hidden messages and encountering an overflow or underflow problem. Specifically, for example, if we intend to conceal a 3-ary secret message in Phase-2 (\(k=3\)), the pixel range is restricted to [0, 254], because embedding such a secret message will cause a pixel change in the range of [0, +1]; thereby, the lower bound is 0 and the upper bound is 254. If we instead convey a 5-ary secret message (\(k=5\)), the pixel change is in the range [1, 254] because the maximal magnitude of change due to message concealment is plus or minus 1 (\(\pm 1\)). Following up on Example-1, a stego pixel, \(\mathbf{P}^{\prime}_{\textit{oucet}}\) = (16, 5, 21), has been produced after concealing the 3-ary message vector, \(\mathbf{d_{sb}}\) = (1\({}_{3}\), 2\({}_{3}\), 0\({}_{3}\)). Let \(d_{\textit{wm}}\) = 2\({}_{7}\) represent a 7-ary secret message to be concealed in this step (\(m=7\)). Let \(\mathbf{W_{a}}\) = [1, 2, 3] represent the weight vector for message concealment. Without loss of generality, we assume the secret message to be conveyed in Phase-2 is in the 5-ary notational system (\(k=5\)); thereby, the eligible range is [1, 254].