# A Robust Image Watermarking Scheme Based on SVD in the Spatial Domain
## 3 SVD-Based Watermarking and Its Impact on the Spatial Domain
### The Impact of SVD-Based Watermarking on the Pixel Values in the Spatial Domain  
From the above analysis, the watermark embedding process of the SVD-based watermarking scheme can be further expressed as:  
\[\mathbf{A}_{\mathrm{w}}=\mathbf{U}(\mathbf{S}+\Delta\mathbf{S})\mathbf{V}^{\mathrm{T}}=\mathbf{U}\mathbf{S} \mathbf{V}^{\mathrm{T}}+\mathbf{U}\Delta\mathbf{S}\mathbf{V}^{\mathrm{T}}=\mathbf{A}+\Delta\mathbf{A}, \tag{7}\]  
where \(\mathbf{A}\) and \(\mathbf{A}_{\mathrm{w}}\) denote the image blocks before and after watermark embedding, respectively. \(\Delta\mathbf{S}\) denotes the watermark information, and \(\Delta\mathbf{A}\) is the difference matrix between \(\mathbf{A}\) and \(\mathbf{A}_{\mathrm{w}}\). From Equation (7), it is noted that the essence of SVD-based watermarking is to distribute the energy of embedded information over all pixels. If we obtain the difference matrix \(\Delta\mathbf{A}\), SVD-based watermarking could be achieved by directly modifying the pixel values in the spatial domain. Taking a \(4\,\times\,4\) image block for example, \(\Delta\mathbf{A}\) can be calculated by:  
\[\Delta\mathbf{A} =\mathbf{U}\Delta\mathbf{S}\mathbf{V}^{\mathrm{T}}=\left[ \begin{array}{ccc} \mathbf{u}_{1}&\mathbf{u}_{2}&\mathbf{u}_{3}&\mathbf{u}_{4}\\ \mathbf{u}_{5}&\mathbf{u}_{6}&\mathbf{u}_{7}&\mathbf{u}_{8}\\ \mathbf{u}_{9}&\mathbf{u}_{10}&\mathbf{u}_{11}&\mathbf{u}_{12}\\ \mathbf{u}_{13}&\mathbf{u}_{14}&\mathbf{u}_{15}&\mathbf{u}_{16}\end{array} \right]\left[ \begin{array}{ccc}w&&\\ &0&&\\ &&0&\\ &&0\end{array} \right]\left[ \begin{array}{ccc}v_{1}&v_{2}&v_{3}&v_{4}\\ v_{5}&v_{6}&v_{7}&\mathbf{v}_{8}\\ v_{9}&\mathbf{v}_{10}&\mathbf{v}_{11}&\mathbf{v}_{12}\\ v_{13}&\mathbf{v}_{14}&\mathbf{v}_{15}&\mathbf{v}_{16}\end{array} \right]^{\mathrm{T}}, \tag{8}\] \[=w\left[ \begin{array}{c}\mathbf{u}_{1}\\ \mathbf{u}_{5}\\ \mathbf{u}_{9}\\ \mathbf{u}_{13}\end{array} \right]\left[ \begin{array}{ccc}v_{1}&\mathbf{v}_{5}&\mathbf{v} _{9}&\mathbf{v}_{13}\end{array} \right]=w\mathbf{u}_{1}\mathbf{v}_{1}^{\mathrm{T}}\]  
where \(\mathbf{u}_{i}\) and \(\mathbf{v}_{i}\)\((i=1,2,3,4)\) are column vectors in matrices \(\mathbf{U}\) and \(\mathbf{V}\); and \(\mathbf{u}_{i}\) and \(\mathbf{v}_{i}\)\((i=1,2,\cdots,16)\) are their matrix elements, respectively. \(w\) denotes the embedded watermark information. It can be seen from Equation (8) that the calculation of \(\Delta\mathbf{A}\) is equivalent to the solving process of \(\mathbf{u}_{1}\mathbf{v}_{1}^{\mathrm{T}}\). In [24], [PERSON] et al. found that the values in \(\mathbf{u}_{1}\mathbf{v}_{1}^{\mathrm{T}}\) are closely associated with its matrix size, especially when a matrix has similar matrix elements. In this paper, we use this property to develop our proposed scheme. Considering a special case that the matrix \(\mathbf{A}\) has the same elements, the matrix \(\mathbf{A}\) can be derived as:  
\[\mathbf{A}=\left[ \begin{array}{cccc}a&a&a&a\\ a&a&a&a\\ a&a&a&a\\ a&a&a&a\end{array} \right]=\left[ \begin{array}{cccc}\mathbf{u}_{1}&\mathbf{u}_{2}& \mathbf{u}_{3}&\mathbf{u}_{4}\end{array} \right]\mathbf{S}\left[ \begin{array}{cccc}\mathbf{v }_{1}&\mathbf{v}_{2}&\mathbf{v}_{3}&\mathbf{v}_{4}\end{array} \right]^{\mathrm{T}}=\lambda_{ 1}\mathbf{u}_{1}\mathbf{v}_{1}^{\mathrm{T}}, \tag{9}\]  
where \(\lambda_{1}\) is the unique singular value in the diagonal matrix \(\mathbf{S}\). For an image block with a size of \(m\times m\), the value of \(\lambda_{1}\) in this particular case is equal to the product between matrix element \(a\) and block size \(m\). By Equation (10), the matrix \(\mathbf{u}_{1}\mathbf{v}_{1}^{\mathrm{T}}\) can be obtained. \[\mathbf{u}_{1}\mathbf{v}_{1}^{\mathrm{T}}=\frac{1}{\lambda_{1}}\mathbf{A}=\frac{1}{a\times m }\left[ \begin{array}{cccc}a&a&a&a\\ a&a&a&a\\ a&a&a&a\\ a&a&a&a\end{array} \right]=\frac{1}{m}\left[ \begin{array}{cccc}1&1&1&1\\ 1&1&1&1\\ 1&1&1&1\\ 1&1&1&1\end{array} \right]. \tag{10}\]  
It is generally known that the local pixel values in each image block have similar pixel values. Based on this characteristic, we promote [PERSON] et al.