I will review the mathematical challenges facing the astronomer who wishes to do state-of-the-art image analysis using Point Spread Function modeling and extraction algorithms. Accurate and precise image reconstruction of
space-based undersampled astronomical images can be quite challenging when the observations are corrupted by internal instrumentation effects like readout noise, charge diffusion and transfer effects, and external
high spatial-frequency noise due to the detection of energetic particles from
the Sun by the camera. Precision stellar photometry and astrometry is possible with such corrupted imaging data provided that accurate physical
knowledge of the detection process within the instrument is folded into the
mathematical modeling of the astronomical image observation data. A critical requirement of many astronomical image analysis problems is the conservation of photon flux; many deconvolution techniques can produce sharp images that are aesthetically pleasing yet may be of little use for many scientific applications because photon flux is not conserved. Many large ground-based
telescopes are being planned which will use adaptive optic techniques which
can remove much of the blurring caused by the Earth's atmosphere but at the
cost of having Point Spread Functions which may be both spatially and
temporally variable. Space-based infrared and optical telescopes missions
are currently being designed with imagers that reduce mission cost by having
complicated optical designs that have Point Spread Functions with much higher
spatial frequencies than are typically present in ground-based astronomical
imagers. Innovative solutions will be required to overcome the many
mathematical challenges facing the astronomer wishing to extract the best
science possible from the images that these new telescope projects will start
producing in the next decade.
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