Added photometry section (not yet finalized).

doc/src/Measurements.rst

@@ -18,6 +18,7 @@ processing (after CLEANing and MASKing).
 
   Position
   PositionWin
+  Photom
 
 .. include:: keys.rst
 

doc/src/Photom.rst

+.. File Photom.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 ``FLUX`` measurement,
+an error estimate ``FLUXERR``, a magnitude ``MAG`` and a magnitude
+error estimate ``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}
+
+Isophotal flux
+--------------
+
+``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
+
+
+Corrected isophotal magnitudes
+------------------------------
+
+``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 ``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.
+
+Fixed-aperture flux
+-------------------
+
+``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 ``FLUX_APER`` is provided as a
+vector ``FLUX_APER[n]``, at least :math:`n` apertures must be
+specified with ``PHOTOM_APERTURES``.
+
+Automatic aperture magnitudes
+-----------------------------
+
+(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 (1980). (1) We define an elliptical aperture 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”). (2) Within this aperture we compute the “first moment”:
+
+.. math:: r_1 = \frac{\sum r\,I(r)}{\sum I(r)}
+
+Kron (1980) and Infante (1987) 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 we consider 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 1,
+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 neighbours (2
+isophotal radii for instance). This was done automatically when using
+the MAG\_BEST magnitude: :math:`{\tt MAG\_BEST} = {\tt MAG\_AUTO}` when
+it is sure that no neighbour can bias MAG\_AUTO by more than 10%, or
+:math:`{\tt MAG\_BEST} = {\tt MAG\_ISOCOR}` otherwise. Experience showed
+that the MAG\_ISOCOR and 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 MAG\_BEST is now deprecated as
+MAG\_AUTO measurements are much more robust in versions 2.x of
+SExtractor. The first improvement is a crude subtraction of all the
+neighbours which have been detected around the measured source (the
+MASK\_TYPE BLANK option). The second improvement is an automatic
+correction of parts of the aperture that are suspected to be
+contaminated by a neighbour. This is done by mirroring the opposite,
+cleaner side of the measurement ellipse if available (the 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 Bertin 1996 for
+details).
+
+Errors on magnitude
+-------------------
+
+An estimate of the error [1]_ 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 [2]_ , 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, we 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]). The GAIN is ignored in PHOTO mode.
+
+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 on 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.
+
+.. [1]
+   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.
+
+.. [2]
+   Setting GAIN to 0 in the configuration file is equivalent to
+   :math:`g = +\infty`
+
+.. include:: keys.rst
+

doc/src/Position.rst

@@ -45,9 +45,9 @@ are simply computed as the first order moments of the profile:
 
    \begin{aligned}
    {\tt X} & = & \overline{x} = \frac{\displaystyle \sum_{i \in {\cal S}}
-   I_i x_i}{\displaystyle \sum_{i \in {\cal S}} I_i},\\ {\tt Y} & = &
-   \overline{y} = \frac{\displaystyle \sum_{i \in {\cal S}} I_i
-   y_i}{\displaystyle \sum_{i \in {\cal S}} I_i}.
+   I_i x_i}{\displaystyle \sum_{i \in {\cal S}} I_i},\\
+   {\tt Y} & = & \overline{y} = \frac{\displaystyle \sum_{i \in {\cal S}}
+   I_i y_i}{\displaystyle \sum_{i \in {\cal S}} I_i}.
    \end{aligned}
 
 In practice, :math:`x_i` and :math:`y_i` are summed relative to XMIN

doc/src/references.bib

@@ -42,6 +42,19 @@
   adsnote = {Provided by the SAO/NASA Astrophysics Data System}
 }
 
+@ARTICLE{1990MNRAS.246..433M,
+   author = {{Maddox}, S.~J. and {Efstathiou}, G. and {Sutherland}, W.~J.
+	},
+    title = "{The APM Galaxy Survey - Part Two - Photometric Corrections}",
+  journal = {\mnras},
+     year = 1990,
+    month = oct,
+   volume = 246,
+    pages = {433},
+   adsurl = {http://adsabs.harvard.edu/abs/1990MNRAS.246..433M},
+  adsnote = {Provided by the SAO/NASA Astrophysics Data System}
+}
+
 @ARTICLE{1856MNRAS..17...12P,
    author = {{Pogson}, N.},
     title = "{Magnitudes of Thirty-six of the Minor Planets for the first day of each month