# Piston Error Measurement for Segmented Telescopes with an Artificial Neural Network
## 5 Conclusions  
In this paper, we put forward a method to simultaneously detect the multi-piston errors between the segments based on the broadband intensity distribution on the image plane by a BP artificial neural network. A mask with a sparse sub-pupil configuration was set on the exit-pupil plane to sample the wave from the segments. Based on the Fourier optics principle, the accurate theoretical relation between peak heights of MTF sidelobes and piston errors was obtained. Instead of fitting the relation by Gaussian function or two piecewise quartic polynomials, we utilized a BP artificial neural network to establish the nonlinear mapping relationship between the piston errors and the amplitudes of the MTF's surrounding peaks. By introducing different piston errors into the segmented optics system, the corresponding peak heights of MTF sidelobes could be directly calculated by the established theoretical formula, while the corresponding amplitudes of MTF sidelobes and the introduced piston errors were used as the inputs and outputs, respectively, to train the network. Once well trained, it could estimate the piston error to a good precision with high efficiency and robustness using the PSF images collected from the optics imaging system as inputs. Since the MTF sidelobes of all the sub-pupils can be simultaneously obtained by one detection of a CCD image, multiple piston errors of the entire aperture can be retrieved at one time. Adequate simulation experiments were implemented to demonstrate the effectiveness and accuracy of the proposed approach. We established the models of a segmented telescope composed of two hexagonal segments and four hexagonal segments successively in MATLAB and implemented the PSF images collected from these simulated optical systems to test the performance of the corresponding trained networks, respectively. The piston sensing simulation results showed that the average values of RMS errors on the two-segment imaging system and the four-segment imaging system could achieve 1.3 nm and 1.4 nm, respectively, and the method's capture range was the operating light's coherence length. Compared to Gaussian fitting and two-piece quartic polynomial function fitting methods, our method has higher detection accuracy and is easier to implement. The influence of the CCD camera noise and the existence of tip-tilt aberrations on the accuracy of piston error detection were also discussed. The results indicated that the proposed  
Figure 14: The piston error detection accuracy comparison between our method and [PERSON]’s method (**a**) for the 2-pupil imaging system and (**b**) the 4-pupil imaging system. piston error detection algorithm has a very good noise immunity, but the tip-tilt aberration should be corrected well enough before the piston error measurements. Besides, different intensity images from the imaging system with different F\({}^{\theta}\) values were generated and fed into the original trained network to test the network generalization ability. The piston error detection accuracy was the same as that of the original imaging system, which demonstrates that our network has quite good generalization ability. The hardware cost of our method is quite small; only a mask with a sparse multi-subaperture configuration is needed to attach in the exit pupil plane of the segmented telescope, and this mask should ensure the MTF sidelobes' non-redundant distribution. Thus, with this method, multi-piston measurements of the whole aperture can be implemented simultaneously, and piston detection no longer need be divided into coarse and fine regimes, which involves separate dedicated hardware solutions. In view of the efficiency and superiority, it is expected that the piston sensing method based on the BP artificial neural network proposed in this paper can be adapted to any segmented and deployable primary mirror telescope, no matter the shape of the segmented mirror and the number of the segments. Methodology, D.Y. ; Software, D.Y. ; Writing--original draft, D.Y. ; Data curation, Y.H.