# Piston Error Measurement for Segmented Telescopes Based on a Hybrid Artificial Neural Network
###### Abstract  
To address the difficulty and complexity of detecting piston errors for segmented telescopes, this paper proposes a new piston error measurement method based on a hybrid artificial neural network. First, we use the Resnet network to learn the mapping relationship between the focal plane degradation image and signs of the piston error. Then, based on the established theoretical relationship between the modulation transfer function and the piston error, a BP neural network is used to learn the mapping relationship between the \(MTF\) and the absolute value of the piston error. After the training of the hybrid network is completed, a wide-range and high-precision detection of the piston error of the sub-mirrors can be achieved using the combined output of the two networks, where only a focal plane image of the point source with broadband illumination is used as the input. The detection range can reach the entire coherent length of the input broadband light, and the detection accuracy can reach 10 nm. The method proposed in this paper has the advantages of high detection accuracy, a wide detection range, low hardware cost, a small network scale, and low training difficulty. piston error; segmented telescope; artificial neural network +
Footnote †: journal: Sensors  
0000-0001-5000-0000-0000-0000-0000-0000-0000-0000-0000-00000-00000-00000-00000-00000-00000-00000-00000-00000-00000-00000-00000-000000-0000  
Recently, many piston error detection algorithms have been proposed and applied to segmented telescopes. For example, the improved wideband/narrowband Shack-Hartmann method proposed by [PERSON] has been successfully applied to the Keck series of telescopes; the piston error detection range of the broadband Shack-Hartmann algorithm [4] is \(\pm 10\lambda\), and the accuracy is \(\lambda/3\) RMS. The piston error detection range of the narrowband Shack-Hartmann algorithm [5] is \(\pm \lambda/4\) and the accuracy is \(\lambda/140\) RMS. [PERSON] et al. used a quadrangular mirror to detect the piston error [6; 7], where the detection accuracy could reach 10 nm. A Mach-Zehnder interference method [8] was proposed by [PERSON], where a multi-wavelength light is used for detection; the detection range can be extended to several microns and the detection accuracy can reach 30 nm. When developing next-generation space telescope technology, the United States proposed a white light interferon-based dispersive fringe sensor [9; 10], which is composed of a broadband light source, a splitter prism, and a grating, with a piston error detection range of \(\pm 200\) um and a detection accuracy of 20 nm. However, these abovementioned wavefront detection techniques need to introduce new optical hardware equipment to the original optical system, which heavily increases the complexity of the optical path. The wavefront detection technology based on the focal plane image mainly includes phase retrieval [11; 12] and the phase diversity algorithm [13; 14], and has low hardware costs, since no other hardware equipment is needed. These methods have high precision but narrow detection ranges, which are not suitable for detecting large piston errors. In 2016, [PERSON] et al. [15; 16] found that, for the point source observation target with broadband illumination, the modulation transfer function (_MTF_) of the system optical transfer function (_OTF_) has a clear mathematical relationship with the piston error of the sub-mirror, and the piecework polynomial is adapted to fit the theoretical relationship. The detection range can reach a \(1/2\) coherent wavelength of input broadband light and the detection accuracy is 0.026A RMS (\(\lambda=633\) nm). However, the theoretical relationship between the _MTF_ and the piston error of the system is an even function, which means that the _MTF_ of a system is the same when the piston error has the same absolute value while the sign of the piston error is opposite.