## 1 Introduction  
The Shack-Hartmann wavefront sensor is one of the most widely used wavefront sensors because of its great energy utilization, fast real-time sensing, and strong anti-noise ability [1]. It is widely employed in the detection of human eye aberrations [2, 3], astronomical observation [4, 5], laser beam purification [6, 7], optical element identification [8, 9], and other fields. To enhance detection performance, researchers have modified the instrument structure to introduce more information and proposed slope and curvature hybrid sensors, which have significant research value. The selection of the reconstruction algorithm is of great importance to the performance of wavefront sensors in various application areas. To meet the practical engineering requirements, the reconstruction time should be shortened as much as possible while satisfying the reconstruction accuracy requirements and ensuring good noise resistance to adapt to various measurement environments [10, 11, 12, 13, 14, 15, 16, 17]. Researchers have modified the classic zonal reconstruction method to improve the performance of the reconstruction method even further. On the one hand, a breakthrough point may be the discovery of a novel difference operator creation approach using mathematical methods. For example, [PERSON] et al. [18] employed the one-dimensional Taylor's law to integrate the relationship among wavefront value, slope, and curvature, with smaller truncation errors. However, the arrangement of sample points affects reconstruction accuracy and noise immunity. [PERSON] et al. [19] enhanced the approximation accuracy of the difference operator in the interpolation process by inserting a spline function. Nonetheless, this approach is solved using an iterative method that requires a long computing time and shows relatively poor noise immunity. On the other hand, increasing the amount of measurement information in the reconstruction algorithm can improve reconstruction accuracy and noise immunity. For example, [PERSON] et al. [20] proposed the least-squares reconstruction method, which utilizes slope, Laplacian curvature, and torsional curvature information comprehensively. This method effectively improves the noise resistance, and the calculation speed is relatively fast, but the reconstruction accuracy could be improved further. To obtain additional data for reconstruction, [PERSON] et al. [21] incorporated the diagonal phase and slope into the zonal reconstruction. Based on the work of [PERSON] et al. [18], [PERSON] et al. [22] modified the method using the difference operator by introducing more information into the reconstruction process, which could not only increase the accuracy of the reconstruction algorithm but also improved its noise immunity. However, the algorithm increased the reconstruction time due to the addition of a new set of integral equations, and the error of the diagonal integration equation increased due to the increase in sampling point interval, both of which have the potential for further improvement. As a result, to better meet the three criteria of the reconstruction algorithm indicated above, this paper proposes a fast slope-curvature hybrid reconstruction algorithm. By dividing the surroundings of the reconstruction point into multiple subregions, an integration equation with the minimum truncation error can be constructed for each subregion. This can be achieved using the Taylor expansion method. The integration equations of adjacent subregions are pairwise canceled to introduce first-order and second-order derivative information in the horizontal, vertical, diagonal, and anti-diagonal directions. Through the cancelation, more wavefront reconstruction information can be introduced while ensuring that the computational complexity remains unchanged, thereby improving the accuracy of wavefront estimation.