## 2 Principle of the Proposed Algorithm and Analysis of Its Residual Errors
### Principle and Process of the Proposed Algorithm
canceling the same parameters, thus shortening the sampling interval in the reconstruction process in the diagonal and antagonistic directions, as shown in Figure 3.  
Firstly, a Taylor expansion is used in the horizontal and vertical directions in the divided subregions to obtain a relationship between the wavefront with minimum truncation error and the first- and second-order derivatives of the sampling points, as derived below. \[AW_{1}+BW_{2}=h\times(ES_{1}+FS_{2})+h^{2}\times(GC_{1}+HC_{2}), \tag{4}\]  
where \(W_{n}\) represents the estimated wavefront at the sample location, \(S_{n}\) represents the first-order derivative at location \(n\) along the adjacent sampling direction, \(C_{n}\) represents the second-order derivative at location \(n\) along the adjacent sampling direction, expands both sides of Equation (4) at the same time at the sample midpoint Taylor location, and lets the coefficients of the same terms corresponding to both sides of the equation be equal to solve for the following:  
\[W_{2}-W_{1}=\frac{h}{2}\times(S_{1}+S_{2})+\frac{h^{2}}{12}\times(C_{1}-C_{2}), \tag{5}\]  
Secondly, to introduce information on sampled points in the diagonal and anti-diagonal directions, four sets of reconstruction relations are obtained in turn using  
Figure 3: Geometric schematic of proposed algorithm. Figure 2: Geometric schematic of the hybrid wavefront reconstruction algorithm using multi-directional slope and curvature information. Equation (5) within the subregions consisting of the four points divided as described in Figure 3.  
\[ \begin{array}{l}W_{i,j+1}-W_{i,j}=\frac{h}{2}\Big{(}S_{i,j+1}^{x}+S_{i,j}^{x} \Big{)}+\frac{h^{2}}{24}\Big{(}C_{ij}^{xx}-C_{i,j+1}^{xx}\Big{)},\\ W_{i+1,j+1}-W_{i+1,j}=\frac{h}{2}\Big{(}S_{i+1,j+1}^{x}+S_{i+1,j}^{x}\Big{)}+ \frac{h^{2}}{2}\Big{(}C_{i+1,j}^{xx}-C_{i+1,j+1}^{xx}\Big{)},\\ W_{i+1,j}-W_{i,j}=\frac{h}{2}\Big{(}S_{i+1,j}^{y}+S_{i,j}^{y}\Big{)}+\frac{h^{2} }{24}\Big{(}C_{ij}^{yy}-C_{i+1,j}^{yy}\Big{)},\\ W_{i+1,j+1}-W_{i,j+1}=\frac{h}{2}\Big{(}S_{i+1,j+1}^{y}+S_{i,j+1}^{y}\Big{)}+ \frac{h^{2}}{12}\Big{(}C_{ij,j+1}^{yy}-C_{i+1,j+1}^{yy}\Big{)},\end{array}  \tag{6}\]  
where \(i\) represents the row position of the sampled point, and \(j\) represents the column position of the sampled point. Add the four equations in Equation (6) and divide by two to obtain the first equation in Equation (7). Add the first and third equations in Equation (6) and subtract the second and fourth equations and divide by 2 to obtain the second equation in Equation (7). \[ \begin{array}{l}W_{i+1,j+1}-W_{i,j}=\frac{h}{4}\left(\begin{array}{c}S_{i,j +1}^{x}+S_{i,j}^{x}\\ +S_{i+1,j+1}^{x}+S_{i+1,j}^{x}\\ +S_{i+1,j}^{y}+S_{i,j}^{y}\\ +S_{i+1,j+1}^{y}+S_{i,j+1}^{y}\end{array}\right)+\frac{h^{2}}{24}\left(\begin{array} []{c}C_{i,j}^{x}-C_{i,j+1}^{x}\\ +C_{i+1,j}^{x}-C_{i+1,j+1}^{xx}\\ +C_{i,j+1}^{y}-C_{i+1,j+1}^{y}\\ +C_{i,j}^{y}-C_{i+1,j}^{y}\end{array}\right),\\ W_{i,j+1}-W_{i+1,j}=\frac{h}{4}\left(\begin{array}{c}S_{i,j+1}^{x}+S_{i,j}^{ x}\\ +S_{i+1,j+1}^{x}+S_{i+1,j}^{x}\\ -S_{i+1,j}^{y}-S_{i,j}^{y}\\ -S_{i+1,j+1}^{y}-S_{i,j+1}^{y}\end{array}\right)+\frac{h^{2}}{24}\left(\begin{array} []{c}C_{i,j}^{x}-C_{i,j+1}^{x}\\ +C_{i+1,j}^{x}-C_{i+1,j+1}^{x}\\ -C_{i,j+1}^{y}+C_{i+1,j+1}^{y}\\ -C_{i,j}^{y}+C_{i+1,j}^{y}\end{array}\right),\end{array}  \tag{7}\]  
Equation (7) is the newly constructed difference operator in each subregion. By comparing with Equation (2), it can be seen that the algorithm proposed in this paper introduces sampling information in four directions using only two sets of equations, reducing the computational effort while ensuring the introduction of the same amount of information. After obtaining the newly reconstructed difference operator, Equation (7) is extended to all sampling points and, written in the form of a matrix, yields  
\[UW=V, \tag{8}\]  
where \(U\) is the coefficient matrix, W is the wavefront matrix, and \(V\) is the slope and curvature matrix in the form of  
\[U=\left[