## 2 Principle of the Proposed Algorithm and Analysis of Its Residual Errors  
### Principle and Process of the Proposed Algorithm  
The zonal method represents the recovered wavefront through a linear combination of gradient information near each sampling point, and the least-squares solution is obtained by solving a linear system of equations. Take the most widely used Southwell model as an example, shown in Figure 1.  
Using Newtonian interpolation integrals at two adjacent points, two sets of integration equations in the horizontal and vertical directions are obtained as follows:  
\[ \begin{split} W_{2}-W_{1}&=h\frac{\left(\frac{S_{1} ^{x}+S_{2}^{x}}{2}\right)}{2},\\ W_{3}-W_{1}&=h\frac{\left(\frac{S_{1}^{y}+S_{3}^{ y}}{2}\right)}{2},\end{split}  \tag{1}\]where \(W_{n}\) represents the wavefront value at position \(n\) of the sampling point, \(h\) is the sampling interval, and \(S_{n}{}^{x}\) and \(S_{n}{}^{y}\) represent the slope values of the wavefront in the \(x\) and \(y\) directions. To further introduce more reconstruction information and thus improve the reconstruction package accuracy, [PERSON] et al. [22] used Taylor expansion to introduce second-order curvature information based on the Southwell model, which only utilizes first-order gradient information in the horizontal and vertical directions, to improve the reconstruction accuracy. Meanwhile, sampling points in the diagonal and anti-diagonal directions are also introduced to improve the reconstruction noise immunity. As shown in Figure 2, the \(XY\) coordinate system is rotated by 45\({}^{\circ}\) to obtain the integration equations in the diagonal and anti-diagonal directions using a typical nine-sample point as an example as follows:  
\[ \begin{array}{l}W_{5}-W_{0}=h\times\frac{(S_{n}{}^{x}+S_{n}{}^{y})}{2}+h^{2} \times\frac{(C_{n}{}^{y}+C_{0}{}^{x})}{12},\\ W_{7}-W_{0}=h\times\frac{(S_{n}{}^{y}+S_{n}{}^{y})}{2}+h^{2}\times\frac{(C_{n}{ }^{y}+C_{0}{}^{y})}{12},\\ W_{3}-W_{0}=\sqrt{2}h\times\frac{(S_{n}{}^{y}+S_{0}{}^{y})}{2}+2h^{2}\times \frac{(C_{n}{}^{y}+C_{0}{}^{y})}{12},\\ W_{8}-W_{0}=\sqrt{2}h\times\frac{(S_{n}{}^{y}+S_{0}{}^{y})}{2}+2h^{2}\times \frac{(C_{n}{}^{y}+C_{0}{}^{y})}{12},\end{array}  \tag{2}\]  
where  
\[ \begin{array}{l}S_{n}{}^{+}=\frac{(S_{n}^{x}+S_{n}{}^{y})}{2},\\ S_{n}{}^{-}=\frac{(S_{n}^{y}-S_{n}{}^{y})}{2},\\ C_{n}{}^{+}=\frac{C_{n}{}^{x}+C_{n}^{y}}{2}+\frac{C_{n}^{y}}{2},\\ C_{n}{}^{-}=\frac{C_{n}^{x}}{2}-C_{n}^{x}+\frac{C_{n}^{y}}{2},\end{array}  \tag{3}\]  
\(C_{n}{}^{xx}\) represents the curvature in the \(x\) direction, \(C_{n}{}^{yy}\) represents the curvature in the \(y\) direction, and \(C_{n}{}^{xy}\) represents the twist curvature term at position \(n\) of the sampling point. The first two equations in Equation (2) represent the reconfiguration relationships in the horizontal and vertical directions, and the second two equations represent the reconfiguration relationships in the diagonal and anti-diagonal directions. By comparing Equations (1) and (2), we can see that the algorithm increases the number of integration equations compared to the traditional Southwell method, thus increasing the number of operations and computation time. The integration equations in the diagonal and anti-diagonal directions improve the noise immunity but increase the sampling interval, which affects the reconstruction accuracy. To solve the above problems, based on the above two models, this paper proposes a new way of constructing the difference operator. By dividing the sampling area into subregions in groups of four points, the first- and second-order derivative information in the diagonal and antagonistic directions is introduced in the integration equations by  
Figure 1: Geometric schematic of the Southwell model.