Improving wavefront reconstruction accuracy by using integration equations with higher-order truncation errors in the Southwell geometry

Least-squares (LS)-based integration computes the function values by solving a set of integration equations (IEs) in LS sense, and is widely used in wavefront reconstruction and other fields where the measured data forms a slope. It is considered that the applications of IEs with smaller truncation errors (TEs) will improve the reconstruction accuracy. This paper proposes a general method based on the Taylor theorem to derive all kinds of IEs, and finds that an IE with a smaller TE has a higher-order TE. Three specific IEs with higher-order TEs in the Southwell geometry are deduced using this method, and three LS-based integration algorithms corresponding to these three IEs are formulated. A series of simulations demonstrate the validity of applying IEs with higher-order TEs in improving reconstruction accuracy. In addition, the IEs with higher-order TEs in the Hudgin and Fried geometries are also deduced using the proposed method, and the performances of these IEs in wavefront reconstruction are presented.