# Robust Image Watermarking in Spatial Domain Utilizing Features Equivalent to SVD Transform
## 3 Proposed Watermarking Scheme
### Block-Wise Invariant Maximum Singular Value in the Spatial Domain  
Several image-watermarking schemes by modifying the singular values to embed the watermark are proposed in the literature [10; 15]. While these approaches can withstand some typical image distortion attacks, there is still room for improvement in a number of areas, such as image flipping and rotation by multiples of ninety degrees, where these schemes do not produce satisfactory results. To destroy the embedded watermark information, these basic techniques modify the spatial positions of the pixels in a watermarked image without affecting their intensities. The singular value of a matrix only considers the matrix elements irrespective of their locations. Keeping this in mind, the image pixels are redistributed in such a way that in each block, the pixel values remain the same irrespective of their locations to obtain the same singular value after the attacks.  
This goal is achieved by utilizing the approach developed by [PERSON] et al. [27] and adopted by others [53]. A square image 'A' of size \(m\)\(\times\)\(m\) is divided into four sub-images of equal size, and the mean intensities of each are computed and collected in a matrix as \(\begin{bmatrix}\mu_{1}&\mu_{2}\\ \mu_{3}&\mu_{4}\end{bmatrix}\),\(\mu_{i}\geq 0\). A normalization matrix (\(N\)) is generated as follows using these means:  
Figure 2: Illustration of SVD-based watermarking and spatial domain watermarking.  
\[N=\begin{bmatrix}N_{1}&N_{2}\\ N_{3}&N_{4}\end{bmatrix}=\begin{bmatrix}\mu_{1}+\mu_{2}+\mu_{3}+\mu_{4}&\mu_{1}- \mu_{2}+\mu_{3}-\mu_{4}\\ \mu_{1}-\mu_{2}+\mu_{3}-\mu_{4}&\mu_{1}-\mu_{2}+\mu_{3}-\mu_{4}\end{bmatrix} \tag{15}\]  
Using the distribution rule described in Equation (16), the pixels of the given image (\(A\)) are relocated to new locations to obtain a new image (\(B\)).  
\[\left\{\begin{array}{ll}B(2i-1,2j-1)=A(i,j),&if&1\leq i\leq m/2,1\leq j\leq m /2\\ B(2i-1,2j-m)=A(i,3m/2-j+1),&if&1\leq i\leq m/2,m/2\leq j\leq m\\ B(2i-m,2j-1)=A(3m/2-i+1,j),&if&m/2\leq i\leq m,1\leq j\leq m/2\\ B(2i-m,2j-m)=A(3m/2-i+1,3m/2-j+1),&if&m/2\leq i\leq m,m/2\leq j\leq m\end{array}\right. \tag{16}\]  
If the absolute value of \(N_{2}\) is more than \(N_{3}\) in Equation (15), matrix \(B\) is left unchanged; otherwise, the transpose of \(B\) is taken to obtain the normalized image. Image \(B\) is divided into square blocks of size \(2^{n}\) (\(n=1\), \(2\), \(3\)... \(|2^{n}<m/2\)) (in this study, \(n=2\)), which are represented by \(B_{ij}\) (\(i=1\), \(2\),... \(m/4\); \(j=1\), \(2\),... \(m/4\)). The largest singular value \(\sigma_{max}\) of each image block is calculated using the matrix 2-norm given in Equation (4) in the spatial domain instead of utilizing the SVD transform. Figure 3 provides an illustration of the concept described above using the example of a square image with dimensions of 32 by 32 pixels, which has been split into blocks with dimensions of 4 by 4 pixels. It is evident from the figure that the largest singular value of the image blocks remains the same under different attacks.  
Figure 3: Illustration of block-wise invariant maximum singular values under various attacks.