Point spread function¶
Introduction¶
The point spread function (PSF) (Wikipedia - PSF) represents the spatial probability distribution of reconstructed event positions for a point source. So far we’re only considering radially symmetric PSFs here.
Probability distributions¶
- \(dP/d\Omega(r)\), where \(dP\) is the probability to find an event in a solid angle \(d\Omega\) at an offset \(r\) from the point source. This is the canonical form we use and the values we store in files.
- Often, when comparing observered event distributions with a PSF model, the \(dP/dr^2\) distributions in equal-width bins in \(r^2\) is used. The relation is \(d\Omega = \pi dr^2\), i.e. \(dP/dr^2=(1/\pi)(dP/d\Omega)\).
- Sometimes, the distribution \(dP/dr(r)\) is used. The relation is \(dP/dr = 2 \pi r dP/d\Omega\).
TODO: explain “encircled energy” = “encircled counts” = “cumulative” representation of PSF and define containment fraction and containment radius.
Normalisation¶
PSFs must be normalised to integrate to total probability 1, i.e.
This implies that the PSF producer is responsible for choosing the Theta range and normalising. I.e. it’s OK to choose a theta range that contains only 95% of the PSF, and then the integral will be 0.95.
We recommend everyone store PSFs so that truncation is completely negligible, i.e. the containment should be 99% or better for all of parameter space.
Comments¶
- Usually the PSF is derived from Monte Carlo simulations, but in principle it can be estimated from bright point sources (AGN) as well.
- Tools should assume the PSF is well-sampled and noise-free. I.e. if limited event statistics in the PSF computation is an issue, it is up to the PSF producer to denoise it to an acceptable level.