We briefly review here the concept of the "effective number of noise pixels"
*N _{p}* and present a general expression that can be used
to derive

In a nutshell, *N _{p}* is the
effective number of pixels contributing
to the flux-variance of a point
source. Sometimes this is referred to as the number of pixels
per beam. The bigger this number, the broader
the PRF (hence the "poorer" the image quality
or resolution), and the lower the sensitivity overall, i.e.,
fluxes will exhibit a larger variance on repeated observation.
This should become clearer from the
derivation below. The final result is given by
Eq. 10 (boxed).
In the end, think of

The signal from a source with true flux *F* as measured in
detector pixel *i* is given by:

(Eq. 1) |

where *r _{i}*
is the PRF volume normalized to unity: ∑

(Eq. 2) |

The true source flux *F* can be estimated from an unweighted linear
least-squares by minimizing the cost function:

(Eq. 3) |

Differentiating Eq. 3 with respect to *F* and setting to zero,
the least-squares solution is given by:

(Eq. 4) |

The noise-variance in *F* can be derived by adding errors
to the true values in Eq. 4:

(Eq. 5) |

Subtracting true values, squaring, and taking expectation values, we have:

(Eq. 6) |

where we assumed the detector pixels are *uncorrelated*:
<*ε _{i}ε_{j}*> = 0
for

(Eq. 7) |

We now assume that the pixel variance *σ ^{2}_{i}*
across a source is approximately constant and there exists some effective
variance

(Eq. 8) |

A constant variance is only
true when the counts are dominated by a spatially uniform
*background* and not when the counts are source-photon dominated.
For the latter case, the *σ ^{2}_{eff}*
corresponds to some average or intermediate value
of the pixel variance across the source.
From Eq. 8, the number of noise pixels

(Eq. 9) |

so that *σ ^{2}_{F}* ≈

We include two
generalizations to Eq. 9: first, the PRF values *r _{i}*
may only be available in un-normalized form, e.g., as the

(Eq. 10) |

where *s* accounts for differences between the PRF
(*R _{i}*) pixel size and the native detector pixel size.
In general, this is the ratio of the native detector pixel
area to PRF pixel area:

(Eq. 11) |

where the CDELT1, CDELT2 refer to standard FITS header keywords
for pixel scales along the *X* and *Y* axes respectively.

As a simple illustration, let's assume
*s* = 1 and we have an un-normalized *top-hat* PRF spread
over *N* pixels equal to constant value *c*.
Equation 10 then gives:

(Eq. 12) |

Therefore, for a top-hat PRF, the number of noise pixels is exactly equal to the number of pixels it spans. For a PRF with tails that decay fast enough, Eq. 10 will converge to some effective value characteristic of the PRF. For a Gaussian PRF, it is not difficult to show that

(Eq. 13) |

Equation 10 was used to estimate the number of noise pixels for
the WISE PRFs derived from survey data.
Given the PRF for each band-array is
non-isoplanatic, we have used the array center PRF from a
9 x 9 grid characterization. Results are shown in Table 1
where units are in #native detector pixels.
The *N _{p}* estimates are in very good agreement
with pre-flight values from optical modelling at SDL.

W1 | W2 | W3 | W4 |
---|---|---|---|

13.772 | 17.636 | 35.476 | 24.462 |

Last update: 2011 July 5

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