# Piston Error Measurement for Segmented Telescopes with an Artificial Neural Network
## 1 Introduction
The classic focal-plane detection methods, such as phase diversity [16; 17; 18] and phase retrieval [19; 20], have high measurement accuracy but narrow detection range and are very time-consuming. In 2015, [PERSON] [21] found out that the modulation transfer function (MTF) amplitude part of the optical transfer function (OTF) has a relationship with the piston error based on the coherence measurement of a star image when using broadband input light, using Gaussian function to fit the relationship between the MTF and the piston error. The capture range of this method is close to the coherence length of the input light, but its accuracy decreases sharply when approaching the extremes of capture range. [PERSON] et al. [22] utilized piecewise quartic polynomial function to fit the relationship; the accuracy of this method is higher than that of Gaussian fitting but the complexity of curve fitting is increased since different mathematical expressions are needed in different intervals of the capture range. This paper proposes a different method to detect the piston error based on analyzing the broadband intensity distribution on the image plane of a star image with a back-propagation (BP) artificial neural network. By attaching a mask with a sparse multi-subaperture configuration in the exit pupil plane of the segmented telescope, we can get the non-redundant MTF sidelobes distribution. Then, based on Fourier optics principle [23], the theoretical relation between peak heights of MTF sidelobes and piston error is rigorously derived, and we have obtained a more universally applicable and accurate theoretical conclusion than the work presented in paper [22], which clearly indicates that the peak height of the MTF sidelobe is only related to the number of sub-pupils, input wavelength and the piston error between segments. Instead of fitting the relation by Gaussian function or two piecewise quartic polynomials, we utilize the BP artificial neural network to establish the nonlinear mapping relationship between the piston error and the amplitudes of the MTF's surrounding peaks. The BP artificial neural network [24; 25] is simple in structure and very convenient to train. Here, the peak heights of MTF sidelobes directly calculated from the obtained theoretical formula served as the inputs and the introduced piston errors served as outputs, respectively, to train the artificial neural network. Once well trained, it can estimate the piston error to a good precision with high efficiency and robustness usingthe PSF image collected from the optics imaging system as input. The detection accuracy is both higher than Gaussian function fitting and piecewise quartic polynomial function fitting, and the implementation complexity is lower than piecewise quartic polynomial function fitting since there is no need to build different computation models in different intervals of the capture range. The capture range is close to the coherence length of the input light; thus the piston error detection no longer needs to be divided into coarse and fine regimes which involve separate dedicated hardware solutions. The influence of the CCD camera noise and the existence of tip-tilt aberrations on the accuracy of piston error detection are discussed. The results indicate that the proposed piston error detection algorithm has very good noise immunity, but the existence of tip-tilt aberrations has a great influence on the detection accuracy. Besides, the generalization ability of the network is also discussed by changing the F-number (F\({}^{\sharp}\)) of the imaging system to generate different intensity images as testing sets to examine the trained network. The results show that our network has quite good generalization ability, since the datasets used to train the network are the same when the optical systems have the same number of sub-pupils and working wavelength.