Error analysis for the Fourier domain offset estimation algorithm

The offset estimation algorithm is crucial for the accuracy of the Shack-Hartmann wave-front sensor. Recently, the Fourier Domain Offset (FDO) algorithm has been proposed for offset estimation. Similar to other algorithms, the accuracy of FDO is affected by noise such as background noise, photon noise, and 'fake' spots. However, no adequate quantitative error analysis has been performed for FDO in previous studies, which is of great importance for practical applications of the FDO. In this study, we quantitatively analysed how the estimation error of FDO is affected by noise based on theoretical deduction, numerical simulation, and experiments. The results demonstrate that the standard deviation of the wobbling error is: (1) inversely proportional to the raw signal to noise ratio, and proportional to the square of the sub-aperture size in the presence of background noise; and (2) proportional to the square root of the intensity in the presence of photonic noise. Furthermore, the upper bound of the estimation error is proportional to the intensity of 'fake' spots and the sub-aperture size. The results of the simulation and experiments agreed with the theoretical analysis.