# Piston Error Measurement for Segmented Telescopes with an Artificial Neural Network
## 4 Simulation
### Piston Error Detection for the Telescope Composed of Two Segments
This was due to the peak heights of the MTF sidelobes approaching the edges of the capture range not changing with piston error very obviously; namely the gradients of the curve were small in these parts, and we can see this directly from the theoretical relationship curve between the piston errors and peak heights of MTF sidelobes shown in Figure 3a. This results from the insensitiveness of the network to data approaching the ends of the capture range, so the detection accuracy decreased a little bit, but the accuracy  
Figure 6: Distribution of the errors between the targets and the actual outputs of the network in the form of a histogram. at the two ends of the detection range was still high enough to meet the requirement of co-phasing the piston error. We also compared our proposed method with the Gaussian fitting and two-piece quartic polynomial function fitting methods presented in [21; 22], respectively, and the comparison results on detection accuracy are shown in Figure 8. From Figure 8a, we can see that the detection accuracy of our proposed method is much higher than that of the Gaussian fitting method. Figure 8b shows the detection accuracy curve of our method compared with that of the two-piece quartic polynomial function fitting method. Our method produced larger differences (-5 nm) for small piston errors, but with the increase of the piston error, especially after 125 \(\upmu\)m, the two-piece quartic polynomial function fitting method generated much larger differences. From the error analysis, the difference of our method was 1.3 nm RMS, which is smaller than the 1.8 nm RMS presented in [22]. This demonstrates that our method also has higher detection accuracy than the two-piece quartic polynomial function fitting method, while the implementation complexity is lower since there is no need to build different computation models in different intervals of the capture range. Figure 8: The comparison results on detection accuracy of a BP artificial neural network approach with **(a)** the Gaussian fitting method **(b)** and the two-piece quartic polynomial function fitting method. Figure 7: The piston error detection results in bar chart form for two segments system with a BP artificial neural network. To further evaluate the ability of the trained network, we designed some other simulations to discuss the effect of image noise and the tip-tilt aberrations of the segments on the detection accuracy of the proposed method. To approximate the practical imaging environment, different intensity Gaussian distribution noises were introduced in the simulated PSF images. Figure 9a gives the detection accuracy under three different image noise intensities by the trained network, and the simulation result showed that PSF image noise had little effect on the piston error detection accuracy. This was due to the fact that the Gaussian distribution noise contributed little to the peak heights of MTF sidelobes, so the proposed piston error detection algorithm had a very good noise immunity. Additionally, since the tip-tilt aberrations could not be corrected entirely before measuring the piston error, slight tip-tilt distortions were introduced to the optics system while generating the simulated PSF images. Here we introduced two different tip-tilt aberrations: the RMS errors were \(0.01\lambda\) and \(0.05\lambda\) (\(\lambda=632.8\ nm\)), respectively. Since the introduced tip-tilt aberrations were quite small, we assumed that the aberration distribution for each wavelength was the same. The effect on the piston error detection accuracy is shown in Figure 9b, which confirms that the existence of the tip-tilt error had a great influence on the detection accuracy. This was because that the tip-tilt error greatly changed the amplitudes of MTF sidelobes. So, in order to guarantee the detection accuracy, the tip-tilt aberrations should be corrected well enough (at least less than \(0.01\lambda\) RMS) before piston error measurement.