$β$-SGP: Scaled Gradient Projection with $β$-divergence for astronomical image restoration

Abstract: Image restoration in astronomy has been considered a vital step in many ground-based observational programs that often suffer from sub-optimal seeing due to atmospheric turbulence, distortion of stellar shapes due to instrumental aberrations, trailing, and other issues. It holds importance for various tasks: improved astrometry, deblending of overlapping sources, faint source detection, and identification of point sources near bright extended objects, such as galaxies, to name a few. We conduct an empirical study by applying the Scaled Gradient Projection (SGP) iterative image deconvolution algorithm to restore distorted stellar shapes in our observed data. We investigate using a more flexible divergence measure, the $\beta$-divergence, which contains the commonly-used Kullback-Leibler (KL) divergence as a special case and allows automatic adaptation of the parameter $\beta$ to the data. An extensive set of experiments comparing the performance of SGP and its $\beta$-divergence variant ($\beta$-SGP) is carried out on extracted star stamps and on images containing multiple stars (both crowded and relatively sparser fields). We show a consistent enhancement in the flux conservation across all considered scenarios using $\beta$-SGP compared to SGP. Using a few quantifiable metrics such as the Full-Width-at-Half-Maximum (FWHM) and ellipticity of stars, we observe that $\beta$-SGP improves restoration quality, compared to the SGP, in many cases and still preserves restoration quality in others. We conclude that generalized versions of image restoration algorithms are more robust due to their enhanced flexibility and could be a promising modification for astronomical image restoration.