## 2 SVD Transform  
In linear algebra, the Singular Value Decomposition (SVD) is a well-known technique for factorizing a rectangular matrix, real or complex, which has been widely employed in signal processing, like image compression [17, 18], noise reduction or image watermarking. Recently, the SVD transform was used to measure the image quality under different types of distortions [19].  
Suppose to have an image represented as a matrix of size \(m\) rows by \(n\) columns, \(A_{mxn}\); applying the SVD on matrix \(A\) will result in the three decomposition matrices \(U_{mxm}\), \(S_{mxn}\) and \(V_{nxn}\), as shown in (2):  
\[SVD(A_{mxn})=[U_{mxm}S_{mxn}V_{nxn}] \tag{2}\]  
By multiplying \(U\), \(S\) and \(V^{T}\) (where \(V^{T}\) means the transpose of \(V\)) we obtain the matrix \(A\):  
\[A_{mxn}=U_{mxm}\cdot S_{mxn}\cdot V^{T}_{nxn}=\sum_{i=1}^{\min(m,n)}\sigma_{i} \cdot u_{i}\cdot v^{T}_{i}\, \tag{3}\]  
where \(\sigma_{i}\in\Re_{+}\), \(i=1\ldots\min(m,n)\) are the _singular values_, i.e., the available diagonal elements of the matrix \(S\) sorted in descending order, \(u_{i}\) are the _left singular vectors_, i.e., the columns of \(U\), and \(v_{i}\) are the _right singular vectors_, i.e., the rows of \(V^{T}\) (or columns of \(V\)). \(U\) and \(V\) are unitary matrices, that means \(U\cdot U^{T}=I_{mxm}\), \(V\cdot V^{T}=I_{nxn}\), where \(I_{mxm}\) and \(I_{nxn}\) are the unit matrices.  
To calculate the SVD we need to compute the eigenvalues and eigenvectors of \(A\cdot A^{T}\)and \(A^{T}\cdot A\). The eigenvectors of \(A\cdot A^{T}\) form the columns of \(U\), whilst the eigenvectors of \(A^{T}\cdot A\) form the columns of \(V\). Moreover, the singular values (SVs) in \(S\) are the square roots of the eigenvalues of \(A^{T}\cdot A\) or \(A\cdot A^{T}\).  
Note that increasing the magnitude of the singular values of matrix \(S\) will increase the image luminance, while lowering the magnitude will decrease the image luminance. Therefore, it is correct to state that \(S\) is in close relation with the image luminance, while the intrinsic \"geometry\" of the image depends upon orthogonal matrices \(U\) and \(V\) which represent, respectively, the _horizontal_ and _vertical details_ (edges) of the image [20].  
From (3) we can observe that each SV is multiplied by the corresponding left and right singular vectors. Hence, this creates different image layers, i.e., a sum of rank-one matrices, where the first image layer (generated multiplying the first SV by the left and right singular vectors) represents the image profile, which concentrates a large amount of the energy contained into the final image [15]. Left and right singular vectors which correspond to the largest SV represent the shape (i.e., strong edges) of the image, while the rest of singular vectors expresses edges and texture regions.  
Another interesting feature of the SVD is the invariance of SVs to common image processing operations (except for noise addition) and geometric transforms, like rotation, translation and scaling [21]. Due to these properties, the SVD has been used (also combined with other techniques) for devising watermarking algorithms particularly resistant to geometric attacks.