@@ -281,9 +281,24 @@ where :math:`\boldsymbol{p}` is the image vector centered on the source.
The definition of :param:`SPREAD_MODEL` above differs from the one given in previous papers, which was incorrect, although in practice both estimators give very similar results.
By construction, :param:`SPREAD_MODEL` is close to zero for point sources, positive for extended sources (galaxies), and negative for detections smaller than the |PSF|, such as cosmic ray hits.
On images taken with linear detectors, :param:`SPREAD_MODEL` should not depend on the source flux or |SNR|. This property may be used to identify bad exposures or |PSF| modeling issues (:numref:`fig_spread`).
On images taken with linear detectors, the average :param:`SPREAD_MODEL` of point sources should not depend on flux nor |SNR|.
This property may be used to identify bad exposures or |PSF| modeling issues (:numref:`fig_spread`).
More importantly, this makes :param:`SPREAD_MODEL` a very convenient estimator for star-galaxy classification, using a positive threshold to identify extended sources.
The |RMS| error on :param:`SPREAD_MODEL` is estimated by propagating the uncertainties on individual pixel values:
.. _fig_spread:
.. figure:: figures/spread.png
:figwidth: 100%
:align: center
|SNR| *vs* :param:`SPREAD_MODEL` for three exposures from :cite:`2013AA_554A_101B`.
*Left plot*: "good" exposure; extended sources (galaxies and nebulous features) are on the right handside of the stellar locus, and electronic glitches create a small cloud of points on the left handside.
*Middle*: exposure with an unusual amount of electronic glitches.
*Right*: exposure with tracking/guiding issues; the |PSF| is too complex for individual sources to be identified as a single objects.
The |pdf| of :param:`SPREAD_MODEL` is close to Gaussian for isolated point sources at a given |SNR|; it gets larger for the fainter sources because of image noise.
In order to maintain a certain level of purity or completeness across the whole magnitude range, it is therefore necessary to take into account the uncertainty on :param:`SPREAD_MODEL`, which can be estimated by propagating the uncertainties on individual pixel values:
.. math::
:label: spreaderr_model
...
...
@@ -295,15 +310,7 @@ The |RMS| error on :param:`SPREAD_MODEL` is estimated by propagating the uncerta
\end{eqnarray}
where :math:`{\bf V}` is the noise covariance matrix, which we assume to be diagonal.
.. _fig_spread:
.. figure:: figures/spread.png
:figwidth: 100%
:align: center
:param:`SPREAD_MODEL`.
In practice, one may for instance adopt a threshold for star-galaxy separation which is a combination of a fixed and a variable components, such as :math:`\sqrt{\epsilon^2 + (\kappa \times {\tt SPREADERR\_MODEL})^2}`, with :math:`\epsilon \approx 5.10^{-3}` and :math:`\kappa \approx 4`.
..
Models are measured according to the following table.
