# Robust Image Watermarking in Spatial Domain Utilizing Features Equivalent to SVD Transform
## 3 Proposed Watermarking Scheme
### Watermark Embedding Process  
The process of embedding a watermark may be broken down into the following steps for easier understanding:  
* Step 1: Encrypt the watermark image W using the piecewise linear chaotic map (PWLCM) [54] prior to embedding it into the cover image using a secret key (k) to increase the security of the watermarking approach by adding an additional layer. This is one of the chaotic maps that has recently gained popularity due to its dynamic nature, simple form, and effective implementation. Without the correct security key, an impostor or unauthorized user cannot detect the watermark in the watermarked image.
* Step 2: Image B is created by rearranging the pixels of cover image 'A' in order to obtain the invariant features described in Section 3.1, and this image is split into 4 \(\times\) 4 non-overlapping blocks \(B_{i,j}\) (\(i=1,2,\ldots,m/4;j=1,2,\ldots,m/4\)). Because a watermark is introduced one bit at a time into each block, the number of non-overlapping blocks must be greater than or equal to the number of watermark bits. Using the matrix 2-norm (Equation (4)), the invariant largest singular value \(\sigma_{max}\) of each image block of redistributed image \(B\) is calculated in the spatial domain instead of applying the SVD transform. In the proposed scheme, the number of blocks is more than the watermark's size, so these singular values are arranged in descending order, and the largest of these are selected according to the watermark's capacity. The selection of these singular values is motivated by their good imperceptibility relative to other singular values.
* Step 3: Modification magnitudes \(T_{1}\) and \(T_{2}\), which are decided based on the watermark information, are given by Equation (17): \[\begin{array}{l}T_{1}=0.5Q,\quad T_{2}=-1.5Q\quad if\ w=1\\ T_{1}=-0.5Q,\quad T_{2}=1.5Q\quad otherwise\end{array}\] (17) where '\(Q\)' is the quantization factor, which controls imperceptibility and robustness.
* Step 4: The potential quantization results \(Q_{1}\) and \(Q_{2}\), utilizing these magnitudes \(T_{1}\) and \(T_{2}\), are now calculated as follows: \[Q_{1}=2 kQ+T_{1},\quad Q_{2}=2 kQ+T_{2}\] (18) where \(k\) is an integer such that \(k=\textit{floor}(ceil(\sigma_{\max}/Q)/2)\), \(ceil(\text{*})\), and \(\textit{floor}(\text{*})\) represents the smallest integer that is greater than or equal to the given value and the greatest integer that is less than or equal to the given value, respectively.
* Step 5: Based on \(Q_{1}\) and \(Q_{2}\), the modified singular value \(\sigma_{\max}^{\prime}\) corresponding to the singular value \(\sigma_{\max}\) is obtained as follows: \[\sigma_{\max}^{\prime}=\begin{cases}Q2\quad if(|\sigma_{\max}-Q2|<|\sigma_{ \max}-Q1|)\\ Q1\quad else\end{cases}\] (19)
* Step 6: Singular value difference \(\Delta\sigma\) between modified singular value \(\sigma_{\max}^{\prime}\) and its corresponding singular value, \(\sigma_{\max}\), is calculated using Equation (20).  
\[\Delta\sigma=\sigma_{\max}^{\prime}-\sigma_{\max} \tag{20}\]
* Step 7: Difference matrix \(\Delta\Delta\) can be achieved with the help of the difference in singular value \(\Delta\sigma\) using Equation (12) as \(\Delta A=\Delta\sigma u_{1}v_{1}^{\intercal}\). With the help of this difference matrix and using Equation (9), watermark bits are directly inserted into image block \(B_{i,j}\) by altering the image pixels. The procedure is repeated until all watermark bits are embedded, and then the reverse process is applied to put back the pixels in their original places to create watermarked image \(Aw\).