  Point-spread function (PSF) estimation in spatially undersampled images is
challenging because large pixels average fine-scale spatial information. This
is problematic when fine-resolution details are necessary, as in optimal
photometry where knowledge of the illumination pattern beyond the native
spatial resolution of the image may be required. Here, we introduce a method of
PSF reconstruction where point sources are artificially sampled beyond the
native resolution of an image and combined together via stacking to return a
finely sampled estimate of the PSF. This estimate is then deconvolved from the
pixel-gridding function to return a superresolution kernel that can be used for
optimally weighted photometry. We benchmark against the < 1% photometric error
requirement of the upcoming SPHEREx mission to assess performance in a concrete
example. We find that standard methods like Richardson--Lucy deconvolution are
not sufficient to achieve this stringent requirement. We investigate a more
advanced method with significant heritage in image analysis called iterative
back-projection (IBP) and demonstrate it using idealized Gaussian cases and
simulated SPHEREx images. In testing this method on real images recorded by the
LORRI instrument on New Horizons, we are able to identify systematic pointing
drift. Our IBP-derived PSF kernels allow photometric accuracy significantly
better than the requirement in individual SPHEREx exposures. This PSF
reconstruction method is broadly applicable to a variety of problems and
combines computationally simple techniques in a way that is robust to
complicating factors such as severe undersampling, spatially complex PSFs,
noise, crowded fields, or limited source numbers.
