Extending Capture Range for Piston Error in Segmented Primary Mirror Telescopes Based on Wavelet Support Vector Machine With Improved Particle Swarm Optimization

In the co-phasing techniques applied to the segmented telescope, a Shack-Hartmann wavefront sensor cannot accurately detect the piston error of the segment. Although the phase diversity (PD) algorithm can detect the piston error of each segment, it fails to reconstruct the wavefront quickly, and its dynamic range is small. Other technologies, such as prisms or micro-lens arrays, will significantly increase the complexity and construction cost of the optical system. Moreover, they may also introduce non-common path errors. In this study, we propose an approach to address this challenging problem via curvature sensing. This method uses multi-wavelength to eliminate the influence of $2\pi $ ambiguity and improve the capture range of co-phasing detection. However, curvature sensing is easily influenced by atmospheric seeing. We propose a wavelet support vector machine optimized via particle swarm optimization (PSO-WSVM) method to deal with this problem, and to improve the application scope of curvature sensing. We reshape SVM with a wavelet kernel function, and improve the PSO algorithm. We train the SVM to build a prediction model to distinguish the piston error range of each pair of adjacent segments and surpass $2\pi $ ambiguity. First, we obtain defocused images by means of the convolution technique. Second, we propose a prediction model based on SVM. We select the correlation coefficient between the sampling signal and the template signal at different wavelengths as the input vector, and we choose a wavelet basis function as the kernel function of SVM. Third, we improve the PSO algorithm with the exponential decreasing inertia weight (EDIW) to tune the parameters of SVM. Finally, we perform a simulation experiment on a real optical system model based on the Keck telescope. The results indicate that the performance of this method is better than that of other state-of-the-art SVM-based classifiers, and it works rapidly during the observation.