ISPY -- NaCo Imaging Survey for Planets around Young stars. CenteR: the impact of centering and frame selection

Abridged: Direct imaging has made significant progress over the past decade leading to the detection of several giant planets. Observing strategies and data rates vary from instrument to instrument and wavelength, and can result in tens of thousands of images to be combined. We here present a new approach, tailored for VLT/NaCo observations performed with the Annular Groove Phase Mask (AGPM) coronagraph at $L'$ filter. Our pipeline aims at improving the post-processing of the observations on two fronts: identifying the location of the star behind the AGPM to better align the science frames and performing frame selection. Our method relies on finding the position of the AGPM in the sky frame observations, and correlate it with the circular aperture of the coronagraphic mask. This relationship allows us to retrieve the location of the AGPM in the science frames, in turn allowing us to estimate the position of the star. In the process we also gather additional information useful for our frame selection approach. We tested our pipeline on several targets, and find that we improve the S/N of companions around $β$ Pictoris and R CrA by $24\pm3$ \% and $117\pm11$ \% respectively, compared to other state-of-the-art reductions. The astrometry of the point sources is slightly different but remains compatible within $3σ$ compared to published values. Finally, we find that even for NaCo observations with tens of thousands of frames, frame selection yields just marginal improvement for point sources but may improve the final images for objects with extended emission such as disks. We proposed a novel approach to identify the location of the star behind a coronagraph even when it cannot easily be determined by other methods, leading to better S/N for nearby point sources, and led a thorough study on the importance of frame selection, concluding that the improvements are marginal in most case.