Furthering the understanding of dark matter is one of the most important problems in modern cosmology. One way to estimate dark matter density is by measuring the shear of galaxies due to dark matter—a very challenging problem not only because galaxies are intrinsically elliptical, but also because of the presence of systematic error resulting from the data collection process. Mathematically, the galaxy data, given as pixelized images, can be thought of as the convolution of the noiseless, spatially-continuous sheared galaxy with the point spread function (PSF) of the recording instruments. The systematic error introduced by the PSF has a much larger effect on galaxy distortions than the shear, biasing the estimates of the shear so the PSF needs to be accurately estimated and accounted for to reliably detect the effect of dark matter, a challenge that is further compounded by the spatial and temporal variation of the shear and PSF.
Methods
To a first approximation, stars are point sources, or two-dimensional delta functions, which makes them the ideal candidate to use for PSF estimation: a star image is a pixelized, noisy representation of the PSF at that particular location. However, star images are not available at galaxy locations, so we need a model that both estimates the observed PSF at the star locations, and interpolates it at galaxy locations. We propose a new approach that is motivated by the fact that the original, continuous-space, noiseless PSF is nearly circularly-symmetric. Thus, after applying subpixel shift to star data such that the peaks are centered on a pixel, the star images are approximately circularly-symmetric in expectation. We take advantage of this property by constructing circularly-symmetric, two-dimensional, binary basis functions, after we demonstrate that any discrete, two-dimensional circularly-symmetric function can be written as the linear combination of fundamental, circularly-symmetric basis functions.
Results
We apply our approach to both a simulation study and real, space-based data collected by the Hubble Space Telescope. In the simulation study, we vary the sample size and the noise level, and compare the efficacy of our method to the current best approach available, PCA. We demonstrate that even with very large sample size, the PCA-based PSF estimate is less accurate than ours, with the discrepancy in accuracy increasing roughly linearly with the noise level. Furthermore, PCA overfits the data while our approach provides the closest fit to the true PSF, rather than its noisy representation. We then demonstrate how this method can be applied on real data.
Conclusions
We propose a new approach for PSF estimation that is much more computationally-efficient than PCA and outperforms its accuracy without overfitting, as demonstrated both theoretically and using simulation studies, and show how it can be applied to real data.