Doc: Moved FLUX_ISO and CLASS_STAR to isophotal measurement section (which is...

Doc: Moved FLUX_ISO and CLASS_STAR to isophotal measurement section (which is still named Position.rst, for portability reasons).
Doc: Added ISOAREA description in isophotal shape section.

doc/src/Measurements.rst

@@ -25,7 +25,6 @@ Model-fitting
   PositionWin
   Photom
   Model
-  ClassStar
 
 .. [#thresh] For some isophotal measurements pixel values also have to exceed the local analysis threshold set with ``ANALYSIS_THRESH``.
 .. [#psf_models] PSF models be computed using the |PSFEx|_ package.

doc/src/Param.rst

@@ -5,7 +5,7 @@
 The measurement (or catalog) parameter file
 ===========================================
 
-In addition to the configuration file detailed above, |SExtractor| requires a file containing the list of measurements ("catalog parameters") that will be listed in the output catalog for every detection. This allows the software to compute only the measurements that are needed. The name of this catalog parameter file is traditionally suffixed with ``.param``, and must be specified using the :param:`PARAMETERS_NAME` config parameter. The full set of parameters can be queried with the command
+In addition to the configuration file detailed above, |SExtractor| requires a file containing the list of measurements ("catalog parameters") that will be listed in the output catalog for every detection. This allows the software to compute only the measurements that are needed. The name of this catalog parameter file is traditionally suffixed with :file:`.param`, and must be specified using the :param:`PARAMETERS_NAME` config parameter. The full set of parameters can be queried with the command
 
 .. code-block:: console
 
@@ -44,7 +44,7 @@ The ``MAG_ZEROPOINT`` configuration parameter sets the magnitude zero-point of m
 
    {\tt MAG} = \left\{\begin{array}{ll}
    \mathrm{MAG\_ZEROPOINT} - 2.5 \log_{10} {\tt FLUX}\ &\mbox{if } {\tt FLUX} > 0\\
-   99.0\ &\mbox{otherwise},
+   99.0\ &\mbox{otherwise}.
    \end{array}\right.
 
 Flux and magnitude uncertainties
@@ -79,7 +79,7 @@ The conversion to surface brightness relies on the ``MAG_ZEROPOINT`` and the ``P
    99.0\ &\mbox{otherwise}.
    \end{array}\right.
 
-Setting ``PIXEL_SCALE`` to 0 instructs |SExtractor| to compute the pixel scale from the local Jacobian of the astrometric deprojection, based on the celestial |WCS| info in the |FITS| image header, if available.
+Setting ``PIXEL_SCALE`` to 0 instructs |SExtractor| to compute the pixel scale from the local `Jacobian <https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant>`_ of the astrometric deprojection, based on the celestial |WCS|_ info :cite:`2002AA_395_1077C` in the |FITS| image header, if available.
 
 .. _coord_suffix:
 
@@ -96,7 +96,7 @@ Positions, distances and position angles are computed in pixel coordinates. They
 .. _world_coords:
 
 :param:`_WORLD`
-  Measurements are given in so-called “world coordinates”, converted from pixel coordinates using the local Jacobian of the transformation between both systems. This requires World Coordinate System (|WCS|_) metadata :cite:`2002AA_395_1061G` to be present in the FITS image header(s). Position angles are counted from the first world axis, positive towards the second world axis.
+  Measurements are given in so-called “world coordinates”, converted from pixel coordinates using the local Jacobian of the transformation between both systems. This requires |WCS| metadata :cite:`2002AA_395_1061G` to be present in the FITS image header(s). Position angles are counted from the first world axis, positive towards the second world axis.
 
 .. _sky_coords:
 
@@ -106,7 +106,7 @@ Positions, distances and position angles are computed in pixel coordinates. They
 .. _focal_coords:
 
 :param:`_FOCAL`
-  Measurements are given in “focal plane coordinates”, which are actually projected coordinates, in degrees. This requires World Coordinate System (|WCS|_) metadata :cite:`2002AA_395_1061G` to be present in the FITS image header(s). The computation of focal plane coordinates from pixel coordinates is similar to that of :param:`_SKY` coordinates except that they are not de-projected and remain Cartesian. The main purpose of focal plane coordinates is to provide a common system for all the chips in a mosaic camera.
+  Measurements are given in “focal plane coordinates”, which are actually projected coordinates, in degrees. This requires |WCS| metadata :cite:`2002AA_395_1061G` to be present in the FITS image header(s). The computation of focal plane coordinates from pixel coordinates is similar to that of :param:`_SKY` coordinates except that they are not de-projected and remain Cartesian. The main purpose of focal plane coordinates is to provide a common system for all the chips in a mosaic camera.
 
 .. note::
   Conversion to :param:`_FOCAL` coordinates is available only for a limited subset of measurements.
@@ -213,6 +213,8 @@ of their meaning.
   _`CXX_IMAGE`, pixel\ :sup:`-2`, :ref:`Isophotal image Cxx ellipse parameter <ellipse_iso_def>`
   _`CYY_IMAGE`, pixel\ :sup:`-2`, :ref:`Isophotal image Cyy ellipse parameter <ellipse_iso_def>`
   _`CXY_IMAGE`, pixel\ :sup:`-2`, :ref:`Isophotal image Cxy ellipse parameter <ellipse_iso_def>`
+  _`ISOAREAF_IMAGE`, pixel\ :sup:`2`, :ref:`Isophotal area (filtered) above Detection threshold <isoarea_def>`
+  _`ISOAREA_IMAGE`, pixel\ :sup:`2`, :ref:`Isophotal area above Analysis threshold <isoarea_def>`
   _`X2WIN_IMAGE`, pixel\ :sup:`2`, :ref:`Windowed image 2nd order central moment in x <moments_win_def>`
   _`Y2WIN_IMAGE`, pixel\ :sup:`2`, :ref:`Windowed image 2nd order central moment in y <moments_win_def>`
   _`XYWIN_IMAGE`, pixel\ :sup:`2`, :ref:`Windowed image 2nd order central cross-moment in xy <moments_win_def>`

doc/src/Photom.rst

@@ -4,10 +4,10 @@
 
 .. _photometry:
 
-Photometry
-==========
+Aperture photometry
+===================
 
-Besides |PSF| and :ref:`model-fitting <models_def>` flux estimates, |SExtractor| can currently perform four types of flux measurements: :ref:`isophotal <flux_iso_def>`, :ref:`corrected-isophotal <mag_isocor_def>`, :ref:`fixed-aperture <flux_aper_def>` and :ref:`adaptive-aperture <flux_auto_def>`.
+Besides :ref:`isophotal <flux_iso_def>`, |PSF| and :ref:`model-fitting <models_def>` flux estimates, |SExtractor| can currently perform two types of flux measurements: :ref:`fixed-aperture <flux_aper_def>` and :ref:`adaptive-aperture <flux_auto_def>`.
 For every :param:`FLUX_` measurement, an error estimate :param:`FLUXERR_`, a magnitude :param:`MAG_` and a magnitude error estimate :param:`MAGERR_` are also available: see :ref:`fluxes_and_magnitudes`.
 
 An estimate of the error is available for each type of flux.
@@ -18,55 +18,8 @@ For aperture fluxes, the flux uncertainty is computed using
 
   {\tt FLUXERR} = \sqrt{\sum_{i\in{\cal A}}\, (\sigma_i^2 + \frac{p_i}{g_i})}
 
-where :math:`{\cal A}` is the set of pixels defining the photometric aperture, and :math:`\sigma_i`, :math:`p_i`, :math:`g_i` respectively the standard deviation of noise (in ADU) estimated from the local background, :math:`p_i` the measurement image pixel value subtracted from the background, and :math:`g_i` the effective detector gain in :math:`e^- / \mbox{ADU}` at pixel :math:`i`. Note that this error estimate provides a lower limit of the true uncertainty, as it only takes into account photon and detector noise.
-
-.. _flux_iso_def:
-
-Isophotal flux: :param:`FLUX_ISO`
----------------------------------
-
-:param:`FLUX_ISO` is computed simply by integrating the background-subracted pixels values :math:`p_i` from the measurement image 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} = F_{\rm iso} = \sum_{i \in {\cal D}} p_i.
-
-
-.. _mag_isocor_def:
-
-Corrected isophotal magnitude: :param:`MAG_ISOCOR`
---------------------------------------------------
-
-.. note::
-  Corrected isophotal magnitudes are now deprecated; they remain in |SExtractor| v2.x for compatibility with |SExtractor| v1.
-
-:param:`MAG_ISOCOR` magnitudes are a quick-and-dirty way of retrieving the fraction of flux lost by isophotal magnitudes.
-
-If one makes the assumption that the intensity profiles of faint objects recorded in the frame are roughly Gaussian because of atmospheric blurring, then the fraction :math:`\eta = \frac{F_{\rm iso}}{F_{\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}{F_{\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}{F_{\rm iso}} - 0.7512
-   \left( \frac{A\,t}{F_{\rm iso}}\right)^2.
-
-A “total” magnitude :param:`MAG_ISOCOR` estimate is then
-
-.. math::
- :label: magisocor
-
-   {\tt MAG\_ISOCOR} = {\tt MAG\_ISO} + 2.5 \log_{10} \eta.
-
-Clearly the :param:`MAG_ISOCOR` correction works best with stars; and although it gives reasonably accurate results with most disk galaxies, it breaks down for ellipticals because of the broader wings in the profiles.
+where :math:`{\cal A}` is the set of pixels defining the photometric aperture, and :math:`\sigma_i`, :math:`p_i`, :math:`g_i` respectively the standard deviation of noise (in ADU) estimated from the local background, :math:`p_i` the measurement image pixel value subtracted from the background, and :math:`g_i` the effective detector gain in :math:`e^- / \mbox{ADU}` at pixel :math:`i`.
+Note that this error estimate provides a lower limit of the true uncertainty, as it only takes into account photon and detector noise.
 
 .. _flux_aper_def:
 

doc/src/Position.rst

@@ -2,23 +2,26 @@
 
 .. include:: global.rst
 
-Position and shape measurements derived from the isophotal profile
-==================================================================
+Isophotal measurements
+======================
 
-The following quantities are derived from the spatial distribution :math:`\cal S` of pixels detected above the analysis threshold (see :ref:`description<isophotal_measurements>`).
+Position and shape
+------------------
+
+The following quantities are derived from the spatial distribution :math:`\cal S` of pixels detected above the detection threshold (see :ref:`description <isophotal_measurements>`).
 
 .. important::
-  Unless otherwise noted, in this section pixel values :math:`p_i` above the local background are taken from the (filtered) detection image.
+  Unless otherwise noted, the pixel values used for computing "isophotal" positions and shapes are taken from the filtered, background-subtracted detection image.
 
 .. note::
   Unless otherwise noted, all parameter names given below are only prefixes.
-  They must be followed by _IMAGE if the results shall be expressed in pixel coordinates or :param:`_WORLD`, :param:`_SKY, :param:`_J2000` or :param:`_B1950` for |WCS|_ coordinates (see :ref:`coord_suffix`).
+  They must be followed by _IMAGE if the results shall be expressed in pixel coordinates or :param:`_WORLD`, :param:`_SKY`, :param:`_J2000` or :param:`_B1950` for |WCS|_ coordinates (see :ref:`coord_suffix`).
 
 
 .. _xyminmax_def:
 
 Limits: :param:`XMIN`, :param:`YMIN`, :param:`XMAX`, :param:`YMAX`
-------------------------------------------------------------------
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
 These coordinates define two corners of a rectangle which encloses the detected object:
 
@@ -38,7 +41,7 @@ where :math:`x_i` and :math:`y_i` are respectively the x-coordinate and y-coordi
 .. _pos_iso_def:
 
 Barycenter: :param:`X`, :param:`Y`
-----------------------------------
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
 Barycenter coordinates generally define the position of the “center” of a source, although this definition can be inadequate or inaccurate if its spatial profile shows a strong skewness or very large wings. X and Y are simply computed as the first order moments of the profile:
 
@@ -47,18 +50,19 @@ Barycenter coordinates generally define the position of the “center” of a so
 
    \begin{eqnarray}
    {\tt X} & = & \overline{x} = \frac{\displaystyle \sum_{i \in {\cal S}}
-   p_i x_i}{\displaystyle \sum_{i \in {\cal S}} p_i},\\
+   p^{(f)}_i x_i}{\displaystyle \sum_{i \in {\cal S}} p^{(f)}_i},\\
    {\tt Y} & = & \overline{y} = \frac{\displaystyle \sum_{i \in {\cal S}}
-   p_i y_i}{\displaystyle \sum_{i \in {\cal S}} p_i}.
+   p^{(f)}_i y_i}{\displaystyle \sum_{i \in {\cal S}} p^{(f)}_i},
    \end{eqnarray}
 
+where :math:`p^{(f)}_i` is the value of the pixel :math:`i` in the (filtered) detection image.
 In practice, :math:`x_i` and :math:`y_i` are summed relative to :param:`XMIN` and :param:`YMIN` in order to reduce roundoff errors in the summing.
 
 
 .. _pospeak_def:
 
 Position of the peak: :param:`XPEAK`, :param:`YPEAK`
-----------------------------------------------------
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
 It is sometimes useful to have the position :param:`XPEAK`, :param:`YPEAK` of the pixel with maximum intensity in a detected object, for instance when working with likelihood maps, or when searching for artifacts. For better robustness, PEAK coordinates are computed on *filtered* profiles if available. On symmetrical profiles, PEAK positions and barycenters coincide within a fraction of pixel (:param:`XPEAK` and :param:`YPEAK` coordinates are quantized by steps of 1 pixel, hence :param:`XPEAK_IMAGE` and :param:`YPEAK_IMAGE` are integers). This is no longer true for skewed profiles, therefore a simple comparison between PEAK and barycenter coordinates can be used to identify asymmetrical objects on well-sampled images.
 
@@ -66,7 +70,7 @@ It is sometimes useful to have the position :param:`XPEAK`, :param:`YPEAK` of th
 .. _moments_iso_def:
 
 Second-order moments: :param:`X2`, :param:`Y2`, :param:`XY`
------------------------------------------------------------
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
 (Centered) second-order moments are convenient for measuring the spatial spread of a source profile. In |SExtractor| they are computed with:
 
@@ -74,9 +78,9 @@ Second-order moments: :param:`X2`, :param:`Y2`, :param:`XY`
   :label: x2y2
 
    \begin{eqnarray}
-   {\tt X2} & = \overline{x^2} = & \frac{\displaystyle \sum_{i \in {\cal S}} p_i x_i^2}{\displaystyle \sum_{i \in {\cal S}} p_i} - \overline{x}^2,\\
-   {\tt Y2} & = \overline{y^2} = & \frac{\displaystyle \sum_{i \in {\cal S}} p_i y_i^2}{\displaystyle \sum_{i \in {\cal S}} p_i} - \overline{y}^2,\\
-   {\tt XY} & = \overline{xy} = & \frac{\displaystyle \sum_{i \in {\cal S}} p_i x_i y_i}{\displaystyle \sum_{i \in {\cal S}} p_i} - \overline{x}\,\overline{y},
+   {\tt X2} & = \overline{x^2} = & \frac{\displaystyle \sum_{i \in {\cal S}} p^{(f)}_i x_i^2}{\displaystyle \sum_{i \in {\cal S}} p^{(f)}_i} - \overline{x}^2,\\
+   {\tt Y2} & = \overline{y^2} = & \frac{\displaystyle \sum_{i \in {\cal S}} p^{(f)}_i y_i^2}{\displaystyle \sum_{i \in {\cal S}} p^{(f)}_i} - \overline{y}^2,\\
+   {\tt XY} & = \overline{xy} = & \frac{\displaystyle \sum_{i \in {\cal S}} p^{(f)}_i x_i y_i}{\displaystyle \sum_{i \in {\cal S}} p^{(f)}_i} - \overline{x}\,\overline{y},
    \end{eqnarray}
 
 These expressions are more subject to roundoff errors than if the 1st-order moments were subtracted before summing, but allow both 1st and
@@ -87,7 +91,7 @@ Roundoff errors are however kept to a negligible value by measuring all position
 .. _shape_iso_def:
 
 Basic shape parameters: :param:`A`, :param:`B`, :param:`THETA`
---------------------------------------------------------------
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
 These parameters are intended to describe the detected object as an elliptical shape. :param:`A` and :param:`B` are the lengths of the semi-major and semi-minor axes, respectively.
 More precisely, they represent the maximum and minimum spatial dispersion of the object profile along any direction.
@@ -165,7 +169,7 @@ Actually, :math:`a` and :math:`b` are defined in :cite:`1980SPIE_264_208S` as th
 .. _elong_iso_def:
 
 By-products of shape parameters: :param:`ELONGATION` and :param:`ELLIPTICITY`
------------------------------------------------------------------------------
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
 These parameters [#elongation]_ are directly derived from :param:`A` and :param:`B`:
 
@@ -177,11 +181,12 @@ These parameters [#elongation]_ are directly derived from :param:`A` and :param:
    {\tt ELLIPTICITY} & = & 1 - \frac{\tt B}{\tt A}.
    \end{eqnarray}
 
+.. [#elongation] These parameters are dimensionless, and for historical reasons do not accept suffixes such as :param:`_IMAGE` or :param:`_WORLD`.
 
 .. _ellipse_iso_def:
 
 Ellipse parameters: :param:`CXX`, :param:`CYY`, :param:`CXY`
-------------------------------------------------------------
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
 :param:`A`, :param:`B` and :param:`THETA` are not very convenient to use when, for instance, one wants to know if a particular |SExtractor| detection extends over some
 position.
@@ -226,7 +231,7 @@ parameters can be derived from the 2nd order moments:
 .. _poserr_iso_def:
 
 Position uncertainties: :param:`ERRX2`, :param:`ERRY2`, :param:`ERRXY`, :param:`ERRA`, :param:`ERRB`, :param:`ERRTHETA`, :param:`ERRCXX`, :param:`ERRCYY`, :param:`ERRCXY`
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
 Uncertainties on the position of the barycenter can be estimated using
 photon statistics.
@@ -239,18 +244,18 @@ Furthermore, |SExtractor| does not currently take into account possible correlat
    \begin{eqnarray}
    {\tt ERRX2} & = {\rm var}(\overline{x}) =
    & \frac{\displaystyle \sum_{i \in {\cal S}} \sigma^2_i (x_i-\overline{x})^2}
-   {\displaystyle \left(\sum_{i \in {\cal S}} p_i\right)^2},\\
+   {\displaystyle \left(\sum_{i \in {\cal S}} p^{(f)}_i\right)^2},\\
    {\tt ERRY2} & = {\rm var}(\overline{y}) =
    & \frac{\displaystyle \sum_{i \in {\cal S}} \sigma^2_i (y_i-\overline{y})^2}
-   {\displaystyle \left(\sum_{i \in {\cal S}} p_i\right)^2},\\
+   {\displaystyle \left(\sum_{i \in {\cal S}} p^{(f)}_i\right)^2},\\
    {\tt ERRXY} & = {\rm cov}(\overline{x},\overline{y}) =
    & \frac{\displaystyle \sum_{i \in {\cal S}} \sigma^2_i (x_i-\overline{x})(y_i-\overline{y})}
-   {\displaystyle \left(\sum_{i \in {\cal S}} p_i\right)^2}.
+   {\displaystyle \left(\sum_{i \in {\cal S}} p^{(f)}_i\right)^2}.
    \end{eqnarray}
 
 :math:`\sigma_i` is the flux uncertainty estimated for pixel :math:`i`:
 
-.. math:: \sigma^2_i = {\sigma_B}^2_i + \frac{p_i}{g_i},
+.. math:: \sigma^2_i = {\sigma_B}^2_i + \frac{p^{(f)}_i}{g_i},
 
 where :math:`{\sigma_B}_i` is the local background noise and :math:`g_i` the local gain — conversion factor — for pixel :math:`i` (see :ref:`effect_of_weighting` for more details). Semi-major axis :param:`ERRA`, semi-minor axis :param:`ERRB`, and position angle :param:`ERRTHETA` of the :math:`1\sigma` position error ellipse are computed from the covariance matrix exactly like for :ref:`basic shape parameters<shape_iso_def>`:
 
@@ -291,7 +296,7 @@ And the error ellipse parameters are:
 
 
 Handling of “infinitely thin” detections
-----------------------------------------
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
 Apart from the mathematical singularities that can be found in some of the above equations describing shape parameters (and which |SExtractor| is able to handle, of course), some detections with very specific shapes may yield unphysical parameters, namely null values for :param:`B`, :param:`ERRB`, or even :param:`A` and :param:`ERRA`. 
 Such detections include single-pixel objects and horizontal, vertical or diagonal lines which are 1-pixel wide. They will generally originate from glitches; but strongly undersampled and/or low S/N genuine sources may also produce such shapes.
@@ -319,5 +324,158 @@ Positional errors are more difficult to handle, as objects with very high signal
 where :math:`\rho_e` is arbitrarily set to :math:`\left( \sum_{i \in {\cal S}} \sigma^2_i \right) / \left(\sum_{i \in {\cal S}} p_i\right)^2`.
 If :eq:`singutest2` is true, then :math:`\overline{x^2}` and :math:`\overline{y^2}` are incremented by :math:`\rho_e`.
 
-.. [#elongation] These parameters are dimensionless, and for historical reasons do not accept suffixes such as :param:`_IMAGE` or :param:`_WORLD`.
+.. _isoarea_def:
+
+Isophotal areas: :param:`ISOAREA`, :param:`ISOAREAF`
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+An isophotal area is defined as the number of pixels with values exceeding some threshold above the background. Isophotal areas played an important role in the 80's and the 90's by providing,  at a small computing cost,  morphometric quantities complementary to second-order moments. |SExtractor| computes two isophotal areas inside the detection footprint:
+
+* :param:`ISOAREAF` applies to the (filtered) detection image, above the detection threshold.
+* :param:`ISOAREA` applies to the measurement image, with the additional constraint that the background-subtracted value of measurement image pixels must exceed the threshold set with the ``ANALYSIS_THRESH`` configuration parameter.
+
+Photometry
+----------
+
+.. important::
+  Unless otherwise noted, the pixel values used for computing "isophotal" fluxes and surface brightnesses are taken from the background-subtracted measurement image.
+
+.. _flux_iso_def:
+
+Isophotal flux: :param:`FLUX_ISO`
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+:param:`FLUX_ISO` is computed simply by integrating the background-subracted pixels values :math:`p_i` from the *measurement* image within the detection footprint
+ 
+.. math::
+ :label: fluxiso
+
+   {\tt FLUX\_ISO} = F_{\rm iso} = \sum_{i \in {\cal S}} p_i.
+
+|SExtractor| also provides an estimation of the uncertainty :param:`FLUXERR_ISO`, a magnitude :param:`MAG_ISO` and a magnitude error estimate :param:`MAGERR_ISO`: see :ref:`fluxes_and_magnitudes`.
+
+.. _mag_isocor_def:
+
+Corrected isophotal magnitude: :param:`MAG_ISOCOR`
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+.. note::
+  Corrected isophotal magnitudes are now deprecated; they remain in |SExtractor| v2.x for compatibility with |SExtractor| v1.
+
+:param:`MAG_ISOCOR` magnitudes are a quick-and-dirty way of retrieving the fraction of flux lost by isophotal magnitudes.
+
+If one makes the assumption that the intensity profiles of faint objects recorded in the frame are roughly Gaussian because of atmospheric blurring, then the fraction :math:`\eta = \frac{F_{\rm iso}}{F_{\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}{F_{\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}{F_{\rm iso}} - 0.7512
+   \left( \frac{A\,t}{F_{\rm iso}}\right)^2.
+
+A “total” magnitude :param:`MAG_ISOCOR` estimate is then
+
+.. math::
+ :label: magisocor
+
+   {\tt MAG\_ISOCOR} = {\tt MAG\_ISO} + 2.5 \log_{10} \eta.
+
+Clearly the :param:`MAG_ISOCOR` correction works best with stars; and although it gives reasonably accurate results with most disk galaxies, it breaks down for ellipticals because of the broader wings in the profiles.
+
+.. _flux_max_def:
+
+Peak value: :param:`FLUX_MAX`
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+:param:`FLUX_MAX` is the peak pixel value (above the local background) in the measurement image. Note that it may not correspond to the pixel with coordinates given by :param:`XPEAK` and :param:`YPEAK` (see :ref:`Position of the peak <pospeak_def>`) if ``FILTER`` is set to ``Y`` or if the measurement image differs from the detection image.
+
+.. _class_star_def:
+
+:param:`CLASS_STAR` classifier
+------------------------------
+
+.. note::
+  The :param:`CLASS_STAR` classifier has been superseded by the :param:`SPREAD_MODEL` estimator (see :ref:`spread_model_def`), which offers better performance by making explicit use of the full, variable |PSF| model.
+
+A good discrimination between stars and galaxies is essential for both galactic and extragalactic statistical studies.
+The common assumption is that galaxy images look more extended or fuzzier than those of stars (or |QSO|\ s).
+|SExtractor| provides the :param:`CLASS_STAR` catalog parameter for separating both types of sources.
+The  :param:`CLASS_STAR` classifier relies on a `multilayer feed-forward neural network <https://en.wikipedia.org/wiki/Multilayer_perceptron>`_ trained using `supervised learning <https://en.wikipedia.org/wiki/Supervised_learning>`_ to estimate the *a posteriori* probability :cite:`Richard1991,Saerens2002` of a |SExtractor| detection to be a point source or an extended object.
+Below is a shortened description of the estimator, see :cite:`1996AAS_117_393B` for more details.
+
+Inputs and outputs
+~~~~~~~~~~~~~~~~~~
+
+The neural network is a multilayer Perceptron with a single fully connected, hidden layers.
+Of all neural networks it is probably the best-studied, and it has been intensively applied with success for many classification tasks.
+
+The classifier (:numref:`fig_classstarnn`) has 10 inputs:
+
+* 8 isophotal areas :math:`A_0..A_7`, measured at isophotes exponentially spaced between the analysis threshold (which may be modified with the ``ANALYSIS_THRESH`` configuration parameter) and the object's peak pixel value
+* The object's peak pixel value above the local background :math:`I_{\mathrm max}`
+* A |seeing| input, which acts as a tuning button.
+
+The output layer contains only one neuron, as "star" and "galaxy" are two classes mutually exclusive.
+The output value is a "stellarity index", which for images that reasonably match those of the training sample is an estimation of the *a posteriori* probability for the classified object to be a point-source.
+Hence a :param:`CLASS_STAR` close to 0 means that the object is very likely a galaxy, and 1 that it is a star.
+In practice, real data always differ at least slightly from the training sample, and the :param:`CLASS_STAR` output is often a poor approximation of the expected *a posteriori* probabilities.
+Nevertheless, a :param:`CLASS_STAR` value closer to 0 or 1 normally indicates a higher confidence in the classification, and the balance between sample completeness and purity may still be adjusted by tweaking the decision threshold .
+
+.. _fig_classstarnn:
+
+.. figure:: figures/classstarnn.*
+  :figwidth: 100%
+  :align: center
+
+  Architecture of |SExtractor|'s :param:`CLASS_STAR` classifier
+
+The |seeing| input must be set by the user with the ``SEEING_FWHM`` configuration parameter.
+If ``SEEING_FWHM`` is set to 0,  it is automatically measured on the |PSF| model which must be provided (using the ``PSF_NAME`` configuration parameter).
+
+If no |PSF| model is available, the ``SEEING_FWHM`` configuration parameter must be adjusted by the user to match the actual average |PSF| |FWHM| on the image.
+The accuracy with which ``SEEING_FWHM`` must be set for optimal results ranges from :math:`\pm 20\%` for bright sources to about :math:`\pm 5\%` for the faintest (:numref:`fig_classstar_seeing`). ``SEEING_FWHM`` is expressed in arcseconds.
+The ``PIXEL_SCALE`` configuration parameter must therefore also be set by the user if |WCS| information is missing from the |FITS| image header.
+There are several ways to measure, directly or indirectly, the size of point sources in |SExtractor|; they may lead to slightly discordant results, depending on the exact shape of the |PSF|.
+The measurement :param:`FWHM_IMAGE` (although not the most reliable as an image quality estimator) sets the reference when it comes to setting ``SEEING_FWHM``.
+
+One may check that the ``SEEING_FWHM`` is set correctly by making sure that the typical :param:`CLASS_STAR` value of unclassifiable sources at the faint end of the catalog hovers around the 0.5 mark.
+
+.. _fig_classstar_seeing:
+
+.. figure:: figures/classstar_seeing.*
+  :figwidth: 100%
+  :align: center
+
+  Architecture of |SExtractor|'s :param:`CLASS_STAR` classifier
+
+
+Training
+~~~~~~~~
+
+This section gives some insight on how the :param:`CLASS_STAR` classifier has been trained.
+The main issue with supervised machine learning is the labeling of the large training sample.
+Hopefully a big percentage of contemporary astronomical images share a common set of generic features, which can be simulated with sufficient realism to create a large training sample together with the ground truth (labels).
+The :param:`CLASS_STAR` classifier was trained on such a sample of artificial images.
+
+Six hundred :math:`512\times512` simulation images containing stars and galaxies were generated to train the network using an early prototype of the |SkyMaker|_ package :cite:`2009MmSAI_80_422B`.
+They were done in the blue band, where galaxies present very diversified aspects.
+The two parameters governing the shape of the |PSF| (|seeing| |FWHM| and Moffat :math:`\beta` parameter :cite:`1969AA_3_455M`) were chosen randomly with :math:`0.025\le` FWHM :math:`\le 5.5` and :math:`2\le\beta\le4`. Note that the `Moffat function <https://en.wikipedia.org/wiki/Moffat_distribution>`_ used in the simulation is a poor approximation to diffraction-limited images, hence the :param:`CLASS_STAR` classifier is not optimized for space-based images.
+The pixel scale was always taken less than :math:`\approx 0.7` FWHM to warrant correct sampling of the image.
+Bright galaxies are simply too rare in the sky to constitute a significant training sample on such a small number of simulations.
+So, keeping a constant comoving number density, we increased artificially the number of nearby galaxies by making the volume element proportional to :math:`zdz`.
+Stars were given a number-magnitude distribution identical to that of galaxies.
+**Therefore any pattern presented to the network had a 50% chance to correspond to a star or a galaxy, irrespective of magnitude** [#faint]_.
+Crowding in the simulated images was higher than what one sees on real images of high galactic latitude fields, allowing for the presence of many “difficult cases” (close double stars, truncated profiles, etc...) that the neural network classifier had to deal with.
+
+|SExtractor| was run on each image with 8 different extraction thresholds. This led to a catalog with about :math:`10^6` entries with the 10 input parameters plus the class label. Back-propagation learning took about 15 min on a SUN SPARC20 workstation. The final set of synaptic weights was saved to the file :file:`default.nnw` , ready to be used in “feed-forward” mode during source extraction.
+
+.. [#faint]
+   Faint galaxies have less chance being detected than faint stars, but it has little effect because the ones that are lost at a given magnitude are predominantly the most extended and consequently the easiest to classify.