# A Hybrid Domain Color Image Watermarking Scheme Based on Hyperchaotic Mapping
## 3 The Proposed Watermarking Scheme
### Watermark Embedding  
In this paper, the cover image is a color image, and the watermark image is a gray image. In order to improve the security of the watermarking algorithm, we use Lorenzhyperchaotic mapping to encrypt the watermark image. Then, the singular value decomposition operation is performed on the encrypted image. In order to enhance the imperceptibility of the watermarking algorithm, we convert the cover image to the \(YCbCr\) color space, and select the \(Y\) component for discrete wavelet decomposition. The selection of the \(YCbCr\) color space is designed to improve the visual transparency of the watermarking algorithm, and the \(Y\) component is selected because the \(Y\) component has stronger robustness when embedding watermarks than the \(Cb\) and \(Cr\) components. Then, we perform discrete wavelet decomposition on the \(Y\) component and perform discrete cosine transform on the obtained LL sub-band. Finally, we perform singular value decomposition on the image after discrete cosine transform. The embedding factor is adaptively calculated by the Bhattacharyya distance of the cover image and the watermark image to perform Alpha hybrid technology to embed the watermark. The steps of watermark embedding are shown in Figure 4.  
The specific steps of watermark embedding are as follows:  
Step 1: Read the gray watermark image and encrypt it with Lorenz hyperchaotic map. The control parameters of the chaotic system are \(a~{}=~{}10\), \(b~{}=~{}8/3\), \(c~{}=~{}28\) and \(r~{}=~{}1\). The initial values of the chaotic system are \(x_{0}~{}=~{}3.5739\), \(y_{0}~{}=~{}3.7979\), \(z_{0}~{}=~{}4.2980\) and \(w_{0}~{}=~{}3.9282\).  
\[W_{E}=Logrenz(W_{1}) \tag{11}\]  
Step 2: Using singular value decomposition, the image \(W_{E}\) is decomposed into three matrices \(U_{w}\), \(S_{w}\) and \(V_{w}\):  
\[[U_{w},S_{w},V_{w}]=SVD(W_{E}) \tag{12}\]  
Step 3: Read the color cover image and convert it to the \(YCbCr\) color space, and then select the \(Y\) component:  
\[[H_{Y},H_{Cb},H_{Cr}]=RGB2 YCbCr(H_{RGB}) \tag{13}\]  
Step 4: Use Equation (10) to adaptively calculate the embedding factor a.  
Step 5: The Y component \((H_{Y})\) of the cover image is decomposed by the first-order Haar wavelet, and it is decomposed into four sub-bands: \(LL\), \(HL\), \(LH\) and \(HH\):  
\[[LL,HL,LH,HH]=DWT(H_{Y}) \tag{14}\]  
Figure 4: Watermark embedding process.  
Step 6: Perform discrete cosine transform on \(LL\) to obtain  
\[LL_{D}=DCT(LL) \tag{15}\]  
Step 7: The singular value decomposition of the image after discrete cosine transform is performed to obtain three matrices \(U_{H}\), \(S_{H}\) and \(V_{H}\):  
\[[U_{H},S_{H},V_{H}]=SVD(LL_{D}) \tag{16}\]  
Step 8: Watermark embedding is performed by Alpha mixing the result \(S_{w}\) of the singular value decomposition in step 2 with the result \(S_{H}\) of singular value decomposition in step 7:  
\[S_{H}^{\prime}=\alpha\times S_{w}+(1-\alpha)\times S_{H} \tag{17}\]  
Step 9: According to the matrix \(U_{H}\), \(S_{H}^{\prime}\) and \(V_{H}\), the \(LL_{D}\) part is reconstructed by inverse singular value decomposition:  
\[LL_{D}^{\prime}=U_{H}\times S_{H}^{\prime}\times V_{H}^{T} \tag{18}\]  
Step 10: The \(LL^{\prime}\) sub-band is reconstructed by inverse discrete cosine transform:  
\[LL^{\prime}=IDCT(LL_{D}^{\prime}) \tag{19}\]  
Step 11: According to the four sub-bands of \(LL^{\prime}\), \(HL\), \(LH\) and \(HH\), the inverse discrete wavelet transform is performed to obtain the Y channel image \(Y^{\prime}\) embedded with watermark:  
\[Y^{\prime}=IDWT[LL^{\prime},HL,LH,HH] \tag{20}\]  
Step 12: Use \(Cb\), \(Cr\) and \(Y^{\prime}\) to convert the image into \(RGB\) color space to obtain the image after embedding the watermark:  
\[H_{RGB}^{\prime}=YCbCr2 RGB[H_{Y},H_{Cb},H_{Cr}] \tag{21}\]