What quantity or property can I use that describes by how much a point spread function distorts/blurs an image?
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Common numbers used are
The easiest thing is probably to convolve the PSF with the characteristic function of a half-plane, and describe the important features of the result (*). For example, if the PSF was symmetric, non-negative and suitably normalized, the distance between the point were the result has the value $0.25$ and the point where the result has the value $0.75$ gives you a reasonable measure for the blur.
One motivation behind this is that quantiles are a convenient and robust way to characterize 1D probability distributions, and using only $0.25$ and $0.75$ is analog to using quartiles.
However, another motivation is the assumption that edges are the most important features in your images. So for image processing related to astronomy or any discipline where "points" are the most important features, other measures are definitively more suitable.
(*) For simpler analytical computations, a poor man's variant is to first integrate over the radial direction, and then describe the important features of the anti-derivative of the resulting function.