## 3 Watermarking in Tensor Domain
### Correlation Analysis among Components of QDFT  
A color-image is decomposed by QDFT to obtain four-dimensional frequency components, including a real component \(A\) and three imaginary components \(C\), \(D\), and \(E\). The imaginary three-dimensional frequency components have a strong correlation. The part proves the correlation among the imaginary three-dimensional frequency components of QDFT. Based on the analysis of its theory, the relationship of the three-dimensional frequency components is proved. Most images have close correlation among the three channels in the RGB color space. The color channels are derived from the same physical model, which determines that images not only have similarity among adjacent pixels, but also have close correlation among the color channels of each pixel [35; 36]. Then, any channels of the color image, red, green, and blue, replaces another channel, such as red, green, and green. We find the reconstructed image is still clear, and no blur distortion occurs. Thus, the research fully shows that color image similarities among adjacent pixels, and the three channels of each pixel have a close correlation. Furthermore, the difference between the two color-channels are almost the same or very close, the results are shown in Figure 5.  
Figure 4: [PERSON]’s slices map. \(C\), \(D\), and \(E\) all have red, green, and blue channels, these are combined by different coefficients. We substitute Equation (8) into Equation (6) as follows:  
\[ \begin{array}{ll}F_{L}(\lambda,v)=i\frac{1}{\sqrt{\chi,Y}}\sum_{\chi=0}^{X-1} \sum_{y=0}^{Y-1}e^{-\theta 2\pi(\frac{\lambda}{X}+\frac{\psi w}{Y})}R(x,y)&+j\frac{1}{ \sqrt{\chi,Y}}\sum_{\chi=0}^{X-1}\sum_{y=0}^{Y-1}e^{-\theta 2\pi(\frac{ \lambda}{X}+\frac{\psi w}{Y})}G(x,y)\\ &+k\frac{1}{\sqrt{\chi,Y}}\sum_{\chi=0}^{X-1}\sum_{y=0}^{Y-1}e^{-\theta 2\pi(\frac{ \lambda}{X}+\frac{\psi}{Y})}B(x,y).\end{array}  \tag{18}\]  
\[ \begin{array}{ll}F_{L}(\lambda,v)=i[\Theta(F_{r}(\lambda,v))+\theta\bullet I (F_{r}(\lambda,v))]+j[\Theta(F_{g}(\lambda,v))+\theta\bullet I(F_{g}(\lambda, v))]\\ &+k[\Theta(F_{b}(\lambda,v))+\theta\bullet I(F_{b}(\lambda,v))]\\ =A(\lambda,v)+C(\lambda,v)i+D(\lambda,v)j+E(\lambda,v)k.\end{array}  \tag{19}\]  
\[A(\lambda,v)=\frac{1}{\sqrt{3}}sin(2\pi(\frac{x\lambda}{X}+\frac{yv}{Y})(R_{( x,y)}+G_{(x,y)}+B_{(x,y)}). \tag{20}\]  
\[ \begin{array}{ll}C(\lambda,v)=i(R(x,y)cos(2\pi(\frac{x\lambda}{X}+\frac{yv}{ Y})+\frac{1}{\sqrt{3}}G(x,y)sin(2\pi(\frac{x\lambda}{X}+\frac{yv}{Y})\\ +\frac{1}{\sqrt{3}}B(x,y)sin(2\pi(\frac{x\lambda}{X}+\frac{yv}{Y})).\end{array}  \tag{21}\]  
\[ \begin{array}{ll}D(\lambda,v)=j(\frac{1}{\sqrt{3}}R(x,y)sin(2\pi(\frac{x \lambda}{X}+\frac{yv}{Y})+G(\lambda,v)cos(2\pi(\frac{x\lambda}{X}+\frac{yv}{ Y})\\ -\frac{1}{\sqrt{3}}B(x,y)sin(2\pi(\frac{x\lambda}{X}+\frac{yv}{Y})).\end{array}  \tag{22}\]  
\[ \begin{array}{ll}E(x,y)=k(\frac{1}{\sqrt{3}}R(x,y)sin(2\pi(\frac{x\lambda}{ X}+\frac{yv}{Y})+\frac{1}{\sqrt{3}}G(x,y)sin(2\pi(\frac{x\lambda}{X}+\frac{yv}{Y})\\ +B(x,y)cos(2\pi(\frac{x\lambda}{X}+\frac{yv}{Y})).\end{array}  \tag{23}\]  
where \(\Theta(a+bi)=\) a, \(I(a+bi)=b\), and \(i\), \(j\), and \(k\) are all orthogonal to each other. On the other hand, the correlation of the three-dimensional imaginary components is proved by data distribution characteristics. We randomly select image block \(I_{r}\) of size \(16\times 16\) in Lena, Table 1 shows the statistical characteristics of the RGB color space and QDFT frequency space for \(I_{r}\). Then, QDFT transformation operates on the image block \(I_{r}\). The distribution of \(C\), \(D\), and \(E\) are similar, the results are shown in Figure 6. The \(C(:,r)\) is  
Figure 5: RGB channels combination and the difference between channels. the \(r\) column value of C component, \(D(:,r)\) is \(r\) column value of D component, and \(E(:,r)\) is \(r\) column value of E component, where \(1\leq r\leq 16\). It can be found from Table 1 that the max value of C is 57,738, as shown in Figure 6a, the max value of first column is also 57,738.