Accurate shear measurement is a key topic in weak lensing community. Point Spread Function (PSF), which smears the observed galaxy image, plays one of the main roles in the systematic errors in shear measurement and must be treated carefully to avoid bias and errors in cosmological parameters. In this paper, we present new PSF measurement methods, Smooth-PCA (SPCA) and Improved-SPCA (iSPCA), which can reconstruct smooth PSFs with high efficiency. Our methods decompose the star images into smooth principal components by using the Expectation-Maximization-PCA (EMPCA) method, and the smooth principal components are composed by Moffatlets basis functions, which are derived from the Moffat function. We demonstrate our approaches based on simulated Moffat PSFs and PhoSim star images. The constructed smooth principal components show flexible and efficient as the same as EMPCA, and have more stable patterns than EMPCA under noises contamination. We then check the reconstruction accuracy on the shape of PSFs. We find that our methods are able to reconstruct the PSFs at the same precision as the EMPCA method which indicates and iSPCA are promising for weak lensing shear measurement.

Inferring the Point Spread Function (PSF) at galaxy positions is one of the crucial steps of the shear measurement. We introduce a novel method to estimate the PSFs at the galaxy positions by using the galaxy images, which could provide additional constrains for the PSF field variations. We construct the PSF for each star image by using Principal-Components-Analysis (PCA) method, which can capture the most significant characteristics of the data. Our method utilises the image difference of the same object between multi-exposures to probe the coefficients of the principal components, in which the differences are mainly caused by PSFs. We apply our method to the observed data. The results show that the corresponding PSFs can be properly estimated from multiple images of different exposures. We then use the obtained principal components from the observations to mock multi-exposure images, where the PSFs field of each exposure is constructed by bivariate polynomial on coefficients. We find that our method can reproduce the PSFs consistently with mocked data. Our results show that the multi-exposed galaxy images could provide us additional constraints for the PSF fields in PCA scenario. It offers a promising prospect for combing the information of stars and galaxies together to construct the PSF field when the point sources are sparsely sampled.