## 2 Relevant Techniques
### Pseudo-Zernike Moment  
Pseudo-Zernike moments [31] are very effective orthogonal rotation invariant moments and pseudo-Zernike moments are robust image feature descriptors. The moments have several characteristics: (1) Redundancy of information expression is small. Since the basis of the Zernike moment is orthogonal polynomial, the extracted features can be guaranteed to have small correlation and redundancy. (2) Effectiveness of information expression. It has been proven that the set of pseudo-Zernike moments can provide a compact, fixed-length and computation effective representation of the image content, and only a small fixed number of compact pseudo-Zernike moments need to be stored to effectively characterize the image content. (3) Multilevel representation of information. Pseudo-Zernike Moments effectively represent the contour of an image. The low-order moments and middle-order moments of pseudo-Zernike moments describe the overall shape of an image, while the high-order moments describe the details of an image. The pseudo-Zernike moments [32] of order \(n\) with repetition \(m\) for a 2-d continuous function \(f(x,y)\) are expressed as follows:  
\[\begin{array}{ll}P_{mm}&=\frac{n+1}{\pi}\iint_{x^{2}+y^{2}\leq 1}f(x,y)V^{*}_{ nm}(x,y)dxdy\\ &=\frac{n+1}{\pi}\int_{0}^{2\pi}\int_{0}^{1}f(p,\mu)V^{*}_{nm}(x,y)dpd\mu.\end{array} \tag{11}\]  
where \(V^{*}_{nm}(x,y)\) is a complex conjugate of \(V_{nm}(x,y)\) and \(n\) is any positive, \(m\) is any positive and negative integer such that \(|m|<n\). The variables \(x\) and \(y\) are such that \(x^{2}+y^{2}\leq 1\), \(p=\sqrt{x^{2}+y^{2}}\), \(\mu=tan^{-1}(\frac{y}{x})\). Pseudo-Zernike polynomials [32]\(V_{nm}(x,y)\) of order \(n\) with repetitions \(m\) are expressed as follows:  
\[V_{mm}(x,y)=R_{nm}(x,y)e^{jm\mu}. \tag{12}\]  
where \(j=\sqrt{-1}\). The pseudo-Zernike radial polynomial [32]\(R_{nm}(x,y)\) is defined as follows:  
\[R_{nm}(x,y)=\sum_{s=0}^{(n-|m|)/2}(-1)^{s}\frac{(n-s)!(x^{2}+y^{2})^{(n-2s)/2} }{s!(\frac{n+|m|}{2}-s)!(\frac{n-|m|}{2}-s)!}. \tag{13}\]  
When \(f(x,y)\) is an image size of \(N\times N\), the pseudo-Zernike moments [33] are defined as follows:  
\[\begin{array}{ll}P_{mm}=\frac{n+1}{\lambda}\sum_{x=0}^{N-1}\sum_{y=0}^{N-1} f(x,y)R_{nm}(x,y)e^{-jm\mu}\\ \hskip 142.26378 pt=\frac{n+1}{\lambda}\sum_{x=0}^{N-1}\sum_{y=0}^{N-1}f(x,y)V ^{*}_{nm}(p,\mu).\end{array} \tag{14}\]where \(\lambda\) is the number of pixels in an image that are mapped into the unit circle.  
\[p=\frac{\sqrt{(2x-N+1)^{2}+(N-1-2y)^{2}}}{N}. \tag{15}\]  
\[\mu=tan^{-1}(\frac{N-1-2y}{2x-N+1}). \tag{16}\]  
Figure 2 shows the information expression of pseudo-Zernike moments for an image. It can be seen from the figure that the low-order moments of pseudo-Zernike moments can be used to construct the contour of the image.  
Considering global geometric distortions, we select six low-order pseudo-Zernike moments \(Z_{n,m}\), including \(Z_{0,0}\), \(Z_{2,2}\), \(Z_{4,4}\), \(Z_{8,8}\), \(Z_{9,9}\), and \(Z_{11,11}\) to reflect the global information of a digital image. The pseudo-Zernike moments are calculated as parameters to correct the geometric attack in the process of watermark extraction.