# A Self-Recovery Fragile Image Watermarking with Variable Watermark Capacity
## 2 SVD Transform and Its Characteristics  
In this section, the SVD transform and its characteristics are analyzed in detail. Based on the characteristics, a new image block classification method is introduced, which provides a basis for the design of the variable capacity self-recovery watermarking. In recent years, the SVD transform has been broadly used in digital watermarking scheme [30; 31] due to its good properties in image processing. By SVD transform, a matrix could be decomposed into three parts: two orthogonal matrices \(\mathbf{U}\) and \(\mathbf{V}\) (also known as left and right singular matrices), and a diagonal matrix \(\mathbf{S}\) which is also called the singular value matrix. Taking a \(4\times 4\) matrix \(\mathbf{A}=\left(a_{ij}\right)_{4\times 4}\) for example, the formula of SVD transform can be expressed as:\[\mathbf{A} = \left[ \begin{array}{cccc}a_{11}&a_{12}&a_{13}&a_{14}\\ a_{21}&a_{22}&a_{23}&a_{24}\\ a_{31}&a_{32}&a_{33}&a_{34}\\ a_{41}&a_{42}&a_{43}&a_{44}\end{array} \right]=\mathbf{U}\mathbf{S}\mathbf{V}^{\mathrm{T}}= \left[ \begin{array}{cccc}\mathbf{u}_{1}&\mathbf{u}_{2}&\mathbf{u}_{3}&\mathbf{u}_{4}\end{array}  \right]\mathbf{S}\left[ \begin{array}{cccc}\mathbf{v}_{1}&\mathbf{v}_{2}&\mathbf{v}_{3}&\mathbf{v }_{4}\end{array} \right]^{\mathrm{T}}\] \[= \left[ \begin{array}{cccc}\mathbf{u}_{1}&\mathbf{u}_{2}&\mathbf{u}_{3}&\mathbf{u}_ {4}\end{array} \right]\left[ \begin{array}{cccc}\lambda_{1}\\ &\lambda_{2}\\ &&\lambda_{3}\\ &&\lambda_{4}\end{array} \right]\left[ \begin{array}{cccc}\mathbf{v}_{1}&\mathbf{v}_ {2}&\mathbf{v}_{3}&\mathbf{v}_{4}\end{array} \right]^{\mathrm{T}}\] \[= \lambda_{1}\mathbf{u}_{1}\mathbf{v}_{1}^{\mathrm{T}}+\lambda_{2}\mathbf{u}_{2 }\mathbf{v}_{2}^{\mathrm{T}}+\lambda_{3}\mathbf{u}_{3}\mathbf{v}_{3}^{\mathrm{T}}+ \lambda_{4}\mathbf{u}_{4}\mathbf{v}_{4}^{\mathrm{T}},\]  
where \(\mathbf{u}_{i}\) and \(\mathbf{v}_{i}\)\((i=1,2,3,4)\) are the column vectors of \(\mathbf{U}\) and \(\mathbf{V}\), respectively; \(\lambda_{i}\)\((i=1,2,3,4)\) is the singular values in matrix \(\mathbf{S}\), and \(\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\geq\lambda_{4}\geq 0\). From Equation (1), we can see that the matrix \(\mathbf{A}\) can be expressed as the sum of four sub-images \(\mathbf{u}_{i}\mathbf{v}_{i}^{\mathrm{T}}\)\((i=1,2,3,4)\) and the singular values \(\lambda_{i}\)\((i=1,2,3,4)\) are their weight coefficients. Besides, the larger the singular values are, the greater the proportion of the sub-image on the matrix decomposition will be. To further analyze the roles of singular values and sub-images in matrix decomposition, we calculate the energy of the matrix by using the square of F-norm, as shown in Equation (2):  
\[\|\mathbf{A}\|_{\mathrm{F}}^{2}=\left[(\sum_{i=1}^{4}\sum_{j=1}^{4}\lvert a_{ij} \rvert^{2})^{\frac{1}{2}}\right]^{2}=\mathrm{tr}(\mathbf{A}^{\mathrm{H}}\mathbf{A})= \mathrm{tr}(\mathbf{V}\mathbf{S}\mathbf{U}^{\mathrm{H}}\mathbf{U}\mathbf{S}\mathbf{V}^{\mathrm{H}})= \mathrm{tr}(\mathbf{V}\mathbf{S}\mathbf{S}\mathbf{V}^{\mathrm{H}})=\sum_{i=1}^{4}\lambda_{i}^{2}, \tag{2}\]  
where \(\mathrm{tr}(\cdot)\) is used to calculate the trace of a matrix and \(\mathbf{A}^{\mathrm{H}}\) is the conjugate transpose matrix of \(\mathbf{A}\). From Equation (2), we can draw a conclusion that the energy information of a matrix is concentrated on the singular values \(\lambda_{i}\)\((i=1,2,3,4)\), while the geometry and texture information are concentrated on its sub-images \(\mathbf{u}_{i}\mathbf{v}_{i}^{\mathrm{T}}\)\((i=1,2,3,4)\). To show the contributions of these sub-images for image texture information, Figure 1 gives the residual images formed by these sub-images, which takes image Lena as an example. From Figure 1, it can be observed that the residual image formed by the first sub-image \(\mathbf{u}_{1}\mathbf{v}_{1}^{\mathrm{T}}\) contains the most texture information of image Lena, and only a little information is remained in other residual images. For other test images, the same conclusion can be obtained.