Photom.rst

.. File Photom.rst

.. include:: global.rst

Photometry
==========

Besides PSF and model-fitting flux estimates, |SExtractor| can currently perform four types of flux measurements: isophotal, *corrected-isophotal*, fixed-aperture and *adaptive-aperture*.
For every :param:`FLUX_` measurement, an error estimate :param:`FLUXERR_`, a magnitude :param:`MAG_` and a magnitude error estimate :param:`MAGERR_` are also available.
The ``MAG_ZEROPOINT`` configuration parameter sets the magnitude zero-point
of magnitudes:

.. math::
 :label: mag

   {\tt MAG} = \mathrm{MAG\_ZEROPOINT} - 2.5 \log_{10} {\tt FLUX}

Magnitude uncertainties (error estimates) are computed using

.. math::
 :label: magerr

   {\tt MAGERR} = \frac{2.5}{\ln 10}\frac{\tt FLUXERR}{\tt FLUX}

.. _flux_iso_def:

Isophotal flux
--------------

:param:`FLUX_ISO` is computed simply by integrating pixels values within the detection footprint, with the additional constraint that the background-subtracted, filtered value of detection image pixels must exceed the threshold set with the ``ANALYSIS_THRESH`` configuration parameter.
 
.. math::
 :label: fluxiso

   {\tt FLUX\_ISO} = \sum_{i \in {\cal S}} I_i


.. _mag_isocor_def:

Corrected isophotal magnitude
-----------------------------

:param:`MAG_ISOCOR` can be considered as a quick-and-dirty way for retrieving the fraction of flux lost by isophotal magnitudes.
Although their use is now deprecated, they have been kept in |SExtractor| v2.x and above for compatibility with |SExtractor| v1.
If we make the assumption that the intensity profiles of the faint objects recorded in the frame are roughly Gaussian because of atmospheric blurring, then the fraction :math:`\eta = \frac{I_{\rm iso}}{I_{\rm tot}}` of the total flux enclosed within a particular isophote reads :cite:`1990MNRAS_246_433M`:

.. math::
  :label: isocor

   \left(1-\frac{1}{\eta}\right ) \ln (1-\eta) = \frac{A\,t}{I_{\rm iso}}

where :math:`A` is the area and :math:`t` the threshold related to this isophote.
:eq:isocor is not analytically invertible, but a good approximation to :math:`\eta` (error :math:`< 10^{-2}` for :math:`\eta > 0.4`) can be done with the second-order polynomial fit:

.. math::
 :label: isocor2

   \eta \approx 1 - 0.1961 \frac{A\,t}{I_{\rm iso}} - 0.7512
   \left( \frac{A\,t}{I_{\rm iso}}\right)^2 \label{eq:isocor}

A “total” magnitude :param:`MAG_ISOCOR` estimate is then

.. math::
 :label: magisocor

   {\tt MAG\_ISOCOR} = {\tt MAG\_ISO} + 2.5 \log_{10} \eta

Clearly this cheap correction works best with stars; and although it is shown to give tolerably accurate results with most disk galaxies, it fails with ellipticals because of the broader wings of their profiles.

.. flux_aper_def:

Fixed-aperture flux
-------------------

:param:`FLUX_APER` estimates the flux above the background within a circular
aperture. The diameter of the aperture in pixels is defined by the
``PHOTOM_APERTURES`` configuration parameter. It does not have to be an
integer: each “normal” pixel is subdivided in :math:`5\times 5` sub-pixels
before measuring the flux within the aperture. If :param:`FLUX_APER` is provided as a
vector :param:`FLUX_APER[n]`, at least :math:`n` apertures must be
specified with ``PHOTOM_APERTURES``.

Automatic aperture magnitudes
-----------------------------

:param:`MAG_AUTO` provides an estimate of the “total magnitude” by integrating the source flux within an adaptively scaled aperture.
|SExtractor|’s automatic aperture photometry routine is inspired by Kron’s “first moment” algorithm :cite:`1980ApJS_43_305K`.

#. An elliptical aperture is defined whose elongation :math:`\epsilon` and position angle :math:`\theta` are defined by second order moments of the object’s light distribution. The ellipse is scaled to :math:`R_{\rm max}.\sigma_{\rm iso}`
(:math:`6 \sigma_{\rm iso}`, which corresponds roughly to 2 isophotal
“radii”).
#. Within this aperture the “first moment” is computed:

.. math:: r_1 = \frac{\sum r\,I(r)}{\sum I(r)}

:cite:`1980ApJS_43_305K` and :cite:`1987AA_183_177I` have shown that for stars and galaxy profiles convolved with Gaussian seeing, :math:`\ge 90\%` of the flux is expected to lie within a circular aperture of radius :math:`k r_1` if :math:`k = 2`, almost independently of their magnitude.
This picture remains unchanged if one considers an ellipse with :math:`\epsilon\, k r_1` and :math:`k r_1 / \epsilon` as principal axes.
:math:`k = 2` defines a sort of balance between systematic and random errors.
By choosing a larger :math:`k = 2.5`, the mean fraction of flux lost drops from about 10% to 6%.
When Signal to Noise is low, it may appear that an erroneously small aperture is taken by the algorithm.
That’s why we have to bound the smallest accessible aperture to :math:`R_{\rm min}` (typically :math:`R_{\rm min} = 3 - 4\,\sigma_{\rm iso}`).
The user has full control over the parameters :math:`k` and :math:`R_{\rm min}` through the configuration parameters ``PHOT_AUTOPARAMS``; by default, ``PHOT_AUTOPARAMS`` is set to ``2.5,3.5``.

.. figure:: ps/simlostflux.ps
   :alt:  Flux lost (expressed as a mean magnitude difference) with different faint-object photometry techniques as a function of total magnitude (see text). Only isolated galaxies (no blends) of the simulations have been considered.
   :width: 15.00000cm

   Flux lost (expressed as a mean magnitude difference) with different
   faint-object photometry techniques as a function of total magnitude
   (see text). Only isolated galaxies (no blends) of the simulations
   have been considered. 

Aperture magnitudes are sensitive to crowding.
In |SExtractor| v1, :param:`MAG_AUTO` measurements were not very robust in that respect.
It was therefore suggested to replace the aperture magnitude by the corrected-isophotal one when an object is too close to its neighbors (2 isophotal radii for instance).
This was done automatically when using the :param:`MAG_BEST` magnitude: :math:`{\tt MAG\_BEST} = {\tt MAG\_AUTO}` when it is sure that no neighbor can bias :param:`MAG_AUTO` by more than 10%, and :math:`{\tt MAG\_BEST} = {\tt MAG\_ISOCOR}` otherwise.
Experience showed that the :param:`MAG_ISOCOR` and :param:`MAG_AUTO` magnitude would loose about the same fraction of flux on stars or compact galaxy profiles: around 0.06 % for default extraction parameters.
The use of :param:`MAG_BEST` is now deprecated as :param:`MAG_AUTO` measurements are much more robust in versions 2.x of |SExtractor|.
The first improvement is a crude subtraction of all the neighbors that have been detected around the measured source (``MASK_TYPE BLANK`` option).
The second improvement is an automatic correction of parts of the aperture that are suspected to be contaminated by a neighbor.
This is done by mirroring the opposite, cleaner side of the measurement ellipse if available (``MASK_TYPE CORRECT`` option, which is also the default).
Figure [figphot] shows the mean loss of flux measured with isophotal (threshold :math:`= 24.4\ \mbox{\rm magnitude\,arsec}^{-2}`), corrected isophotal and automatic aperture photometries for simulated galaxy :math:`B_J` on a typical Schmidt-survey plate image. 
The automatic adaptive aperture photometry leads to the lowest loss of flux.

Photographic photometry
-----------------------

In ``DETECT_TYPE PHOTO`` mode, SExtractor assumes that the response of the detector, over the dynamic range of the image, is logarithmic.
This is generally a good approximation for photographic density on deep exposures. 
Photometric procedures described above remain unchanged, except that for each pixel we apply first the transformation

.. math::

   I = I_0\,10^{D/\gamma} \ ,
   \label{eq:dtoi}

where :math:`\gamma` (``MAG_GAMMA``) is the contrast index of the emulsion, :math:`D` the original pixel value from the background-subtracted image, and :math:`I_0` is computed from the magnitude zero-point :math:`m_0`:

.. math:: I_0 = \frac{\gamma}{\ln 10} \,10^{-0.4\, m_0} \ .

One advantage of using a density-to-intensity transformation relative to the local sky background is that it corrects (to some extent) large-scale inhomogeneities in sensitivity (see :cite:`1996PhDT_68B` for details).

Magnitude uncertainties
-----------------------

An estimate of the error [#error]_ is available for each type of magnitude.
It is computed through

.. math:: \Delta m = 1.0857\, \frac{\sqrt{A\,\sigma^2 + F/g}}{F}

where :math:`A` is the area (in pixels) over which the total flux
:math:`F` (in ADU) is summed, :math:`\sigma` the standard deviation of
noise (in ADU) estimated from the background, and g the detector gain
(GAIN parameter [#gain]_ , in :math:`e^- / \mbox{ADU}`).
For corrected-isophotal magnitudes, a term, derived from Eq. [eq:isocor] is quadratically added to take into account the error on the correction itself.

In ``DETECT_TYPE PHOTO`` mode, things are slightly more complex.
Making the assumption that plate-noise is the major contributor to photometric errors, and that it is roughly constant in density, one can write:

.. math::

   \Delta m = 1.0857 \,\ln 10\, {\sigma\over \gamma}\,
     \frac{\sqrt{\sum_{x,y}{I^2(x,y)}}}{\sum_{x,y}I(x,y)}
   =2.5\,{\sigma\over \gamma}\,
   \frac{\sqrt{\sum_{x,y}{I^2(x,y)}}}{\sum_{x,y}I(x,y)}

where :math:`I(x,y)` is the contribution of pixel :math:`(x,y)` to the
total flux (Eq. [eq:dtoi]). ``GAIN`` is ignored in ``PHOTO`` mode.

Background
----------

Background is the last point relative to photometry. The assumption made in §[chap:backest] — that the "local" background associated to an object can be interpolated from the global background map — is no longer valid in crowded regions.
An example is a globular cluster superimposed to a bulge of galaxy.
SExtractor offers the possibility to estimate locally the background used to compute magnitudes.
When this option is switched on (``BACKPHOTO_TYPE LOCAL`` instead of ``GLOBAL``), the "photometric" background is estimated within a "rectangular annulus" around the isophotal limits of the object.
The thickness of the annulus (in pixels) can be specified by the user with ``BACKPHOTO_SIZE``. A typical value is ``BACKPHOTO_SIZE``=``24``.

.. [#error]
   It is important to note that this error provides a lower limit, since
   it does not take into account the (complex) uncertainty on the local
   background estimate.

.. [#gain]
   Setting GAIN to 0 in the configuration file is equivalent to
   :math:`g = +\infty`