## 2 Related Work
### Brief Introduction to [PERSON] et al.'s Method  
The process of watermark embedding is marked with a red dotted box in Figure 1. To generate watermark information, the first step is to divide the original image into \(2\times 2\) nonoverlapping blocks and classify them. This is achieved by SVD, as shown in Equation (1):  
\[\mathbf{A}=\left(a_{ij}\right)_{2\times 2}=\mathbf{U}\mathbf{\lambda}\mathbf{V}^{\mathrm{T}}= \left[\begin{array}{cc}\mathbf{u}_{1}&\mathbf{u}_{2}\end{array}\right]\left[ \begin{array}{cc}\lambda_{1}&\mathbf{v}_{1}\\ \lambda_{2}&\mathbf{v}_{2}\end{array}\right]=\lambda_{1}\mathbf{u}_{1}\mathbf{v}_{1}^{ \mathrm{T}}+\lambda_{2}\mathbf{u}_{2}\mathbf{v}_{2}^{\mathrm{T}}, \tag{1}\]  
where \(\mathbf{A}\) is a \(2\times 2\) matrix of pixels in a block. The block is divided into three matrices \(\mathbf{U}\), \(\mathbf{A}\), \(\mathbf{V}\), where \(\mathbf{u}_{i}\) and \(\mathbf{v}_{i}\) are the column vectors of \(\mathbf{U}\) and \(\mathbf{V}\), respectively. Thus \(\mathbf{u}_{i}\mathbf{v}_{i}^{\mathrm{T}}\) (\(i=1,2\)) are sub-images of the input image. The first sub-image, \(\mathbf{u}_{1}\mathbf{v}_{1}^{\mathrm{T}}\), contains most of the information and for smooth blocks, because the intensities are similar, all the element values in the sub-image matrix are concentrated around 0.5. The authentication watermark is generated by the weight matrix \(\mathbf{\lambda}\) and the information watermark is either the average pixel value in the smooth blocks or the coefficient after DCT [20] in the texture blocks. The final watermark consists of both authentication and information watermarks.  
The second step is to insert the watermark into the 3 LSBs of each pixel. Different embedding schemes are implemented according to the classification results. Finally, a map is used to scramble the embedding position of the watermark to enhance security, as presented in Equation (2):  
\[P^{\prime}=(K\times P)\mathrm{mod}\;N+1, \tag{2}\]  
where \(P\) and \(P^{\prime}\) indicate the position of a block before and after mapping, respectively. \(N\) is the number of blocks and \(K\) is a custom prime ranging from 1 to \(N-1\).  
Figure 1: A brief overview of the method in Reference [7].  
During the tamper detection process, [PERSON] et al. [7] presented a three-level detection mechanism to achieve higher location accuracy, marked with a green dotted box in Figure 1. This mechanism repeats the same operations used for the watermark generation process on the detected image and then compares the newly generated watermark with the extracted original watermark from two aspects: the authentication watermark and the information watermark. The last step is to adjust the marking status according to the neighborhood conditions, effectively reducing the false alarm rate.  
Finally, the tampered region is recovered using the methods shown in the blue dotted box in Figure 1. During the inverse watermark embedding process, the information watermark is fully used to reconstruct the pixel values in the tampered areas. Later, any blocks that were not successfully recovered can be determined through neighborhood adjustment. This method produces final results that perform better than other methods under most tampering attack conditions.