# Robust Image Watermarking in Spatial Domain Utilizing Features Equivalent to SVD Transform
## 2 Review of the Algorithmic Concepts
### SVD-Based Watermarking Scheme  
In SVD-based image watermarking [49], watermark \(W\) is divided into three components by SVD to embed the principal component into the host image by modifying its singular values. \[W\Rightarrow U_{w}S_{w}V_{w}^{T} \tag{5}\]  
The host image is transformed into the frequency domain by DWT, and the low-frequency sub-band (\(LL\)) is selected to divide it into several non-overlapping blocks. SVD is applied to each block to obtain the largest singular value to embed the principal component of the watermark image. It can be given by the following expression:  
\[\sigma_{\max}^{\prime}=\sigma_{\max}+\alpha U_{w}S_{w} \tag{6}\]where \(\sigma_{\max}\) and \(\sigma_{\max}^{\prime}\) are the original and modified largest singular values of the host image block, and \(\alpha\) is the embedding intensity control factor. After the modification of the largest singular value of each block of the host image, the watermarked image is achieved by the reverse process. The principal component of the embedded watermark can be extracted from the watermarked image by the following expression:  
\[U_{w}S_{w}=\frac{1}{\alpha}(\sigma_{\max}^{\prime}-\sigma_{\max}) \tag{7}\]  
With the help of side information \(V_{w}\) from the watermark insertion phase and by the principal component obtained in Equation (7), the watermark can be extracted by the following equation:  
\[U_{w}S_{w}V_{w}^{T}\Rightarrow W \tag{8}\]  
According to [PERSON] et al. [49], changes equivalent to the watermarking in the SVD domain explained above can be achieved in the spatial domain as follows:  
\[A_{w}=U(S+\Delta S)V^{T}=USV^{T}+U\Delta S)V^{T}=A+\Delta A \tag{9}\]  
where \(\Delta A\) is the difference between host image \(A\) and watermarked image \(A_{w}\), and \(\Delta S\) represents the watermark information. According to Equation (9), the goal of watermarking schemes in the SVD domain is to distribute the watermark information across all pixels of the cover image. Therefore, SVD-based watermarking is possible by directly altering pixel values in the spatial domain if it is possible to obtain the difference matrix \(\Delta A\). For example, \(\Delta A\) can be calculated as follows by taking a 4 \(\times\) 4 image block:  
\[\Delta A=U\Delta S\ V^{T}= \begin{bmatrix}u_{1,1}&u_{1,2}&u_{1,3}&u_{1,4}\\ u_{2,1}&u_{2,2}&u_{2,3}&u_{2,4}\\ u_{3,1}&u_{3,2}&u_{3,3}&u_{3,4}\\ u_{4,1}&u_{4,2}&u_{4,3}&u_{4,4}\end{bmatrix} \times\Delta S\times \begin{bmatrix} v_{1,1}&v_{1,2}&v_{1,3}&v_{1,4}\\ v_{2,1}&v_{2,2}&v_{2,3}&v_{2,4}\\ v_{3,1}&v_{3,2}&v_{3,3}&v_{3,4}\\ v_{4,1}&v_{4,2}&v_{4,3}&v_{4,4}\end{bmatrix} ^{T} \tag{10}\]  
\[\Delta A=U\Delta S\ V^{T}= \begin{bmatrix}u_{1,1}&u_{1,2}&u_{1,3}&u_{1,4}\\ u_{2,1}&u_{2,2}&u_{2,3}&u_{2,4}\\ u_{3,1}&u_{3,2}&u_{3,3}&u_{3,4}\\ u_{4,1}&u_{4,2}&u_{4,3}&u_{4,4}\end{bmatrix} \times \begin{bmatrix}w&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix} \times \begin{bmatrix}v_{1,1}&v_{1,2}&v_{1,3}&v_{1,4}\\ v_{2,1}&v_{2,2}&v_{2,3}&v_{2,4}\\ v_{3,1}&v_{3,2}&v_{3,3}&v_{3,4}\\ v_{4,1}&v_{4,2}&v_{4,3}&v_{4,4}\end{bmatrix} ^{T} \tag{11}\]  
\[\Delta A=U\Delta S\ V^{T}=w \begin{bmatrix}u_{1,1}\\ u_{2,1}\\ u_{3,1}\\ u_{4,1}\end{bmatrix} \times \begin{bmatrix}v_{1,1}&v_{2,1}&v_{3,1}&v_{4,1} \end{bmatrix} =wu_{1}v_{1}^{T} \tag{12}\]  
where \(w\) indicates the watermark information, and \(u_{1}\) and \(v_{1}\) are column vectors in the matrices \(U\) and \(V\). It is clear in Equation (12) that difference matrix \(\Delta A\) can be achieved by the multiplication of the watermark's information and eigenvectors \(u_{1}v_{1}^{T}\). [PERSON] et al. [49] claimed that when the elements of a matrix are similar, \(u_{1}v_{1}^{T}\) is closely associated with its matrix size. Using this property and taking a special case where all elements of a square matrix \(A\) are the same, it is expressed as follows:  
\[A= \begin{bmatrix}a&a&a&a\\ a&a&a&a\\ a&a&a&a\\ a&a&a&a\end{bmatrix} =