Mathematics in the Estimation of Point-spread Functions in Ground-based Astronomy through Turbulence

Stars are “twinkling” when observed on Earth. The twinkling is the effect of atmospheric turbulence. The air is constantly in turbulent motion and light from space objects is bent randomly in many ways just before reaching the surface of Earth, crumpling the wavefront of arriving light at ground-based telescopes. This atmospheric blurring distorts the view of astronomical objects. Our project aims at getting clear images of objects in space from their images obtained on the telescopes. In order to do that, we need to estimate the blurring effect of the atmospheric turbulence—the point-spread function (PSF). Once this is known, a clearer image of the object can be obtained by solving a deblurring problem. Back in the 90’s, the PI and the first co-I were one of the first research teams to work on deblurring problems for satellite images based on a “guide-star” technology where the image of a very distant isolated star or a bright spot in the sodium layer of the atmosphere created by a laser beam is used to estimate the PSF. In this proposal, we use a different approach—the Fourier optics theory. It estimates the PSF by constructing the phase from the arriving wavefront. The phase is the deviation of wavefront from the planarity caused by atmospheric turbulence. It is not directly measurable; however its horizontal and vertical gradients can be roughly measured by low-resolution (LR) wavefront sensors on the telescope. One way to find an accurate phase is to first reconstruct the high-resolution (HR) phase gradients by using a sequence of overlapping LR images that are obtained by moving the telescope across the sky when imaging. By modeling the relationship between the HR phase gradients and LR phase gradients and adding a suitable regularization, one can solve the model to obtain the HR phase gradients and hence the phase and the PSF.The goals of this project are three-fold: (1) Develop efficient and accurate models for the problem by exploring the intrinsic properties of the phase and its gradients and by adopting super-resolution image reconstruction techniques; (2) Design new customized and efficient minimization algorithms for these new models; (3) Extend our models and algorithms to more complex situations, such as the multi-layered atmospheric turbulence case and surveillance of objects, such as satellites, orbiting around the Earth. We will carry out a comprehensive comparison of our models and algorithms in terms of speed, accuracy, and robustness.