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.

\[\int_{0}^{\infty} 2 \pi r dP/dr(r) dr = 1, where dP/dr = 2 \pi r dP/d\Omega\]

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.