# A Robust and Secure Watermarking Approach Based on Hermite Transform and SVD-DCT
## 3 Materials and Methods
### SVD Analysis  
Singular-value decomposition, used in linear algebra, performs an expansion of a rectangular matrix \(A\in\mathbb{R}^{M\times N}\) in a coordinate system where \(M\) and \(N\) are the dimensions of \(A\) and the covariance matrix is diagonal. Equation (2) shows the SVD theorem:  
\[A=USV^{T}\] \[\begin{bmatrix}A_{\mathbf{M}\times\mathbf{N}}&\\ a_{11}&\dots&a_{1n}\\ &\ddots&\\ a_{m1}&\dots&a_{mn}\end{bmatrix}=\] \[\begin{bmatrix}\mathbf{U}_{\mathbf{M}\times\mathbf{M}}&\mathbf{S}_{\mathbf{M}\times \mathbf{N}}&\mathbf{V}_{\mathbf{N}\times\mathbf{N}}\\ \ddots&\ddots&\\ u_{1m}&\dots&u_{mm}\end{bmatrix}\begin{bmatrix}\sigma_{1}&\dots&0\\ &\ddots&\\ 0&\dots&0\end{bmatrix}\begin{bmatrix}v_{11}&\dots&v_{1n}\\ &\ddots&\\ v_{n1}&\dots&v_{nn}\end{bmatrix},\]  
where \(U\in\mathbb{R}^{M\times M}\) and \(V\in\mathbb{R}^{N\times N}\) are orthogonal matrices defined by:  
\[U^{T}U =I_{M\times M}\] \[V^{T}V =I_{N\times N}.\]  
\(S\in\mathbb{R}^{M\times N}\) is a diagonal matrix, and \(\sigma_{i}\) with \(i=1,\dots,M\) are the singular values that satisfy \(\sigma_{1}\geq\sigma_{2}\geq\dots\geq\sigma_{r}=\sigma_{r+1}=\dots\sigma_{M \times N}=0\) with \(r\leq M\times N\) representing the rank of the matrix \(A\).  
Thus, SVD calculates the eigenvalues of \(AA^{T}\), forming the columns of \(V\)and the eigenvectors of \(A^{T}A\) and generating the columns of \(U\), and the singular values in \(S\) are obtained by the square roots of the eigenvalues from \(AA^{T}\) and \(A^{T}A\).  
SVD has been successfully used for a variety of applications. In particular, in the signal- and image-processing areas, SVD has been applied to image compression and image completion [45], dimensionality reduction, facial recognition, background video removal, image noise reduction, cryptography, and digital watermarking [46].  
In the following, we describe some properties of SVD:  
* A few singular values contain the majority of the signal's energy, which has been exploited in compression applications.
* The decomposition/reconstruction could be applied to both square and non-square images.
* When a slight interference, e.g., noise, alters the values of the image, the singular values remain relatively stable.
* Singular values of an image represent its intrinsic algebra.  
SVD generates the sorted matrices \(U\), \(S\), and \(V\), following how they contribute to the matrix \(A\). Thus, we obtained an approximation of the input image when only a number \(k\) of singular values was used. In addition, if \(k\) is very near \(M\), the quality of the reconstructed image increases. From Equation (2), an image approximation is obtained taking \(r\) columns of \(U\) and \(V\) and the upper left \(r\times r\) square of \(S\).  
Figure 6 shows the image approximation using SVD over a grayscale image of \(256\times 256\) for \(r=8,32,64,128,256\), and Table 1 reports the correlation coefficient (\(R\)), which is calculated using the original image and the reconstructed image, where using only \(25\%\) of the singular values (\(r=64\) of \(256\)), a correlation value of \(0.994\) is achieved. In addition, Figure 7 shows the zooming of a region of the original image, where both homogeneous and texture regions are presented.  
Figure 6: Image approximation using a different number of singular values (\(r\)) of SVD.