# Fragile Watermarking for Image Authentication Using the Characteristic of SVD
## 2 Singular Value Decomposition (SVD)  
As an effective method of algebraic feature extraction, SVD is widely applied in signal processing [15]. For a \(N\times N\) matrix \(\mathbf{A}\) with rank \(r\), SVD can be expressed as  
\[\mathbf{A} =\mathbf{US}\mathbf{V}^{\mathrm{T}}=[\mathbf{u}_{1},\mathbf{u}_{2},\cdots,\mathbf{u}_ {N}]\mathbf{S}[\mathbf{v}_{1},\mathbf{v}_{2},\cdots,\mathbf{v}_{N}]^{\mathrm{T}} \tag{1}\] \[=\left[ \begin{array}{cccc}u_{1,1}&\cdots&u_{1,N}\\ u_{2,1}&\cdots&u_{2,N}\\ \vdots&\ddots&\vdots\\ u_{N,1}&\cdots&u_{N,N}\end{array} \right]\left[ \begin{array}{cccc}\lambda_{1 }&0&\cdots&0\\ 0&\lambda_{2}&\cdots&0\\ \vdots&\vdots&\ddots&0\\ 0&0&\cdots&\lambda_{N}\end{array} \right]\left[ \begin{array}{cccc}v_{1,1}& \cdots&v_{1,N}\\ v_{2,1}&\cdots&v_{2,N}\\ \vdots&\ddots&\vdots\\ v_{N,1}&\cdots&v_{N,N}\end{array} \right]^{\mathrm{T}}\]  
where \(\mathbf{u}_{i}\) and \(\mathbf{v}_{i}\)\((i=1,2,\cdots,N)\) are column vectors of singular value matrices \(\mathbf{U}\) and \(\mathbf{V}\), and \(u_{i,j}\)\((i,j=1,2,\cdots,N)\) are their matrix elements, respectively. \(\mathbf{S}\) is a diagonal singular matrix made up of singular values \(\lambda_{i}\)\((i=1,2,\cdots,N)\), where \(\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{r}>\lambda_{r+1}=\cdots= \lambda_{N}=0\). The singular values have good stability that it can resist slight disturbances in signal processing. Moreover, the SVD can be implemented to arbitrary matrices without any restriction and the larger singular values contain the most information of an image. Due to these properties, SVD has been gradually applied in digital watermarking schemes [16; 17; 18]. Taking \(N=4\) for example, the SVD can be rewritten as  
\[ \begin{split} A&=\left[\begin{array}{cccc}a_{1,1}&a_{ 1,2}&a_{1,3}&a_{1,4}\\ a_{2,1}&a_{2,2}&a_{2,3}&a_{2,4}\\ a_{3,1}&a_{3,2}&a_{3,3}&a_{3,4}\\ a_{4,1}&a_{4,2}&a_{4,3}&a_{4,4}\end{array}\right]=\textbf{U}\textbf{S}\textbf{V} ^{\text{T}}\\ &=\left[\begin{array}{cccc}\boldsymbol{u}_{1},\boldsymbol{u}_{2}, \boldsymbol{u}_{3},\boldsymbol{u}_{4}\end{array}\right]\left[\begin{array}{ ccc}\lambda_{1}&&\\ &\lambda_{2}&&\\ &&\lambda_{3}&\\ &&\lambda_{4}\end{array}\right]\left[\begin{array}{cccc}\boldsymbol{v}_{1}, \boldsymbol{v}_{2},\boldsymbol{v}_{3},\boldsymbol{v}_{4}\end{array}\right]^{ \text{T}}\\ &=\lambda_{1}\boldsymbol{u}_{1}\boldsymbol{v}_{1}^{\text{T}}+\lambda_{2} \boldsymbol{u}_{2}\boldsymbol{v}_{2}^{\text{T}}+\lambda_{3}\boldsymbol{u}_{3} \boldsymbol{v}_{3}^{\text{T}}+\lambda_{4}\boldsymbol{u}_{4}\boldsymbol{v}_{4 }^{\text{T}}.\end{split}  \tag{2}\]  
Because \(\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\geq\lambda_{4}\), it is noted from Equation (2) that the first item \(\lambda_{1}\boldsymbol{u}_{1}\boldsymbol{v}_{1}^{\text{T}}\) plays an important role in matrix \(\boldsymbol{A}\). To reveal this relationship, let us consider an exceptional circumstance that all the elements in matrix \(\boldsymbol{A}\) are the same. Then, matrix \(\boldsymbol{A}\) can be derived as  
\[\boldsymbol{A}=\left[ \begin{array}{cccc}a&a&a&a\\ a&a&a&a\\ a&a&a&a\\ a&a&a&a\end{array} \right]=\lambda_{1}\left[ \begin{array}{c}u_{1,1}\\ u_{2,1}\\ u_{3,1}\\ u_{4,1}\end{array} \right]\left[ \begin{array}{cccc}v_{1,1}&v_{2,1}&v_{3,1}&v_ {4,1}\end{array} \right]=\lambda_{1}\boldsymbol{u}_{1}\boldsymbol{v}_{1}^{ \text{T}} \tag{3}\]  
where \(\lambda_{1}\) is the only singular value in \(\boldsymbol{S}\), i.e., \(\boldsymbol{S}=\text{diag}(\lambda_{1},0,0,0)\). It is generally known that for a matrix with size of \(N\times N\), \(\lambda_{1}\) equals \(a\times N\) in such a circumstance. According to Equation (3), the matrix \(\boldsymbol{u}_{1}\boldsymbol{v}_{1}^{\text{T}}\) can be calculated as shown in Equation (4):  
\[\boldsymbol{u}_{1}\boldsymbol{v}_{1}^{\text{T}}=\frac{1}{\lambda_{1}} \boldsymbol{A}=\frac{1}{a\times N}\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}{N}\left[ \begin{array}{cccc}1&1&1&1\\ 1&1&1&1\\ 1&1&1&1\\ 1&1&1&1\end{array} \right].