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Once sources have been detected and deblended, they enter the measurement phase. |SExtractor| performs three categories of measurements: isophotal, full, and model-fitting.

.. _isophotal_measurements:

Isophotal

  Measurements are made on the isophotal object footprints. Only pixels with values above the threshold set with ``ANALYSIS_THRESH`` are considered [#thresh]_, which makes the analysis extremely fast, but obviously strongly dependent on the threshold itself. This is an issue particularly when the amplitude of the bakground noise varies over the image. Many of the isophotal measurements (e.g., ``X_IMAGE``, ``Y_IMAGE``, ``FLUX_ISO``) are necessary for the internal operations of |SExtractor| and are therefore executed even if they are not requested.

  Measurements are made on the isophotal object footprints, which are defined on the filtered detection image. Only pixels with values above the threshold set with ``ANALYSIS_THRESH`` are considered [#thresh]_, which makes the analysis extremely fast, but obviously strongly dependent on the threshold itself. This is an issue particularly when the amplitude of the bakground noise varies over the image. Many of the isophotal measurements (e.g., ``X_IMAGE``, ``Y_IMAGE``, ``FLUX_ISO``) are necessary for the internal operations of |SExtractor| and are therefore executed even if they are not requested.

Full

  Measurements have access to all pixels of the image. These measurements are generally more sophisticated, less affected by the detection threshold, and still reasonably fast. They are done at a later stage of the processing, after CLEANing and MASKing.

  PositionWin

  Photom

.. [#thresh] Pixel values also have to exceed the local detection threshold set with ``DETECT_THRESH``

.. [#thresh] Pixel values also have to exceed the local detection threshold set with ``DETECT_THRESH``.

.. [#psf_models] PSF models be computed using the |PSFEx|_ package.

.. include:: keys.rst

   99.0 &\mbox{otherwise}.

   \end{array}\right.

Positions

~~~~~~~~~

.. _coord_suffix:

Positions, distances and position angles can be expressed in image pixels, world coordinates, or in celestial coordinates, depending on the suffix:

Positions and shapes

~~~~~~~~~~~~~~~~~~~~

Positions, distances and position angles are computed in pixel coordinates. They may be expressed in image pixels, world coordinates, or in celestial coordinates, depending on the suffix:

_IMAGE

  Measurements are given in pixel coordinates, in units of pixels. For example: ``Y_IMAGE``, ``ERRAWIN_IMAGE``, ``THETA_IMAGE`` etc. Following the FITS convention, in |SExtractor| the center of the first image pixel has coordinates (1.0,1.0). Position angles are counted from the *x* axis (axis 1), positive towards the *y* axis (axis 2)

_WORLD

  Measurements are given in so-called “world coordinates”. 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 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.

_SKY, _J2000, _B1950

  Measurements are given in celestial (equatorial) coordinates, in units of degrees. This requires celestial |WCS| metadata :cite:`2002AA_395_1077C` to be present in the FITS image header(s). _SKY measurements are given in the native world coordinate system. _J2000 and _B1950 measurements are automatically converted from the native |WCS|, taking into account the change of reference frame. In all cases, positions angles are counted East-of-North.

  Measurements are given in celestial (equatorial) coordinates, converted from pixel coordinates using the local Jacobian of the transformation between both systems. Positions and distances are in units of degrees. This requires celestial |WCS| metadata :cite:`2002AA_395_1077C` to be present in the FITS image header(s). _SKY measurements are given in the native world coordinate system. _J2000 and _B1950 measurements are automatically converted from the native |WCS|, taking into account the change of reference frame. In all cases, positions angles are counted East-of-North.

.. include:: keys.rst

================================================================

The following parameters are derived from the spatial distribution

:math:`\cal S` of pixels detected above the extraction threshold. *The

:math:`\cal S` of pixels detected above the analysis threshold (see :ref:`description<isophotal_measurements>`). *Unless otherwise noted,

pixel values* :math:`I_i` *are taken from the (filtered) detection image*.

**Note that, 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 units (see §..), or “_WORLD” for World Coordinate

System (WCS) units (see §[astrom])**. For example: THETA

:math:`\rightarrow` THETA\_IMAGE. In all cases, parameters are first

computed in the image coordinate system, and then converted to WCS if

requested.

.. 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 _WORLD, _SKY, _J2000 or _B1950 for |WCS|_ coordinates (see :ref:`coord_suffix`).

Limits: XMIN, YMIN, XMAX, YMAX

------------------------------

detected object:

.. math::

  :label: xminymax

   \begin{aligned}

   {\tt XMIN} & = & \min_{i \in {\cal S}} x_i,\\

are simply computed as the first order moments of the profile:

.. math::

  :label: xy

   \begin{aligned}

   {\tt X} & = & \overline{x} = \frac{\displaystyle \sum_{i \in {\cal S}}

spread of a source profile. In |SExtractor| they are computed with:

.. math::

  :label: x2y2

   \begin{aligned}

   {\tt X2} & = \overline{x^2} = & \frac{\displaystyle \sum_{i \in {\cal

the :math:`x,y` image coordinate system by an angle +\ :math:`\theta`:

.. math::

  :label: varproj

   \label{eq:varproj}

   \begin{array}{lcrrr}

   \overline{x_{\theta}^2} & = & \cos^2\theta\:\overline{x^2} & +\,\sin^2\theta\:\overline{y^2}

               & -\,2 \cos\theta \sin\theta\:\overline{xy},\\

One can find interesting angles :math:`\theta_0` for which the variance

is minimized (or maximized) along :math:`x_{\theta}`:

.. math:: {\left.\frac{\partial \overline{x_{\theta}^2}}{\partial \theta} \right|}_{\theta_0} = 0,

.. math::

  :label: theta0

  {\left.\frac{\partial \overline{x_{\theta}^2}}{\partial \theta} \right|}_{\theta_0} = 0,

which leads to

.. math::

  :label: theta0_2

   2 \cos\theta \sin\theta_0\:(\overline{y^2} - \overline{x^2})

       + 2 (\cos^2\theta_0 - \sin^2\theta_0)\:\overline{xy} = 0.

If :math:`\overline{y^2} \neq \overline{x^2}`, this implies:

.. math::

   :label: theta0_3

   \label{eq:theta0}

   \tan 2\theta_0 = 2 \frac{\overline{xy}}{\overline{x^2} - \overline{y^2}},

a result which can also be obtained by requiring the covariance

:math:`\overline{xy_{\theta_0}}` to be null. Over the domain

:math:`[-\pi/2, +\pi/2[`, two different angles — with opposite signs —

satisfy ([eq:theta0]). By definition, THETA is the position angle for

satisfy :eq:`theta0_3`. By definition, THETA is the position angle for

which :math:`\overline{x_{\theta}^2}` is *max*\ imized. THETA is

therefore the solution to ([eq:theta0]) that has the same sign as the

therefore the solution to :eq:`theta0_3`. that has the same sign as the

covariance :math:`\overline{xy}`. A and B can now simply be expressed

as:

.. math::

  :label: aimage

   \begin{aligned}

   {\tt A}^2 & = & \overline{x^2}_{\tt THETA},\ \ \ {\rm and}\\

   {\tt B}^2 & = & \overline{y^2}_{\tt THETA}.\end{aligned}

A and B can be computed directly from the 2nd-order moments, using the

following equations derived from ([eq:varproj]) after some algebra:

following equations derived from :eq:`varproj` after some algebra:

.. math::

  :label: aimage2

   \begin{aligned}

   \label{eq:aimage}

   {\tt A}^2 & = & \frac{\overline{x^2}+\overline{y^2}}{2}

       + \sqrt{\left(\frac{\overline{x^2}-\overline{y^2}}{2}\right)^2 + \overline{xy}^2},\\

   {\tt B}^2 & = & \frac{\overline{x^2}+\overline{y^2}}{2}

a detection is now parameterized as

.. math::

  :label: ellipse

   {\tt CXX} (x-\overline{x})^2 + {\tt CYY} (y-\overline{y})^2

       + {\tt CXY} (x-\overline{x})(y-\overline{y}) = R^2,

parameters can be derived from the 2nd order moments:

.. math::

  :label: ellipse_2

   \begin{aligned}

   {\tt CXX} & = & \frac{\cos^2 {\tt THETA}}{{\tt A}^2} + \frac{\sin^2

   Meaning of shape parameters.

By-products of shape parameters: ELONGATION and ELLIPTICITY [1]_

----------------------------------------------------------------

By-products of shape parameters: ELONGATION and ELLIPTICITY

-----------------------------------------------------------

These parameters are directly derived from A and B:

These parameters [#elongation]_ are directly derived from A and B:

.. math::

  :label: elongation

   \begin{aligned}

   {\tt ELONGATION} & = & \frac{\tt A}{\tt B}\ \ \ \ \ \mbox{and}\\

pixels. Hence variances simply write:

.. math::

  :label: errxy

   \begin{aligned}

   {\tt ERRX2} & = {\rm var}(\overline{x}) = & \frac{\displaystyle

where :math:`{\sigma_B}_i` is the local background noise and

:math:`g_i` the local gain — conversion factor — for pixel :math:`i`

(see §[chap:weight] for more details). Semi-major axis ERRA, semi-minor

(see :ref:`effect_of_weighting` for more details). Semi-major axis ERRA, semi-minor

axis ERRB, and position angle ERRTHETA of the :math:`1\sigma` position

error ellipse are computed from the covariance matrix exactly like in

[chap:abtheta] for shape parameters:

.. math::

  :label: errabtheta

   \begin{aligned}

   \label{eq:erra}

   {\tt ERRA}^2 & = & \frac{{\rm var}(\overline{x})+{\rm var}(\overline{y})}{2}

       + \sqrt{\left(\frac{{\rm var}(\overline{x})-{\rm var}(\overline{y})}{2}\right)^2

       + {\rm cov}^2(\overline{x},\overline{y})},\\

   \label{eq:errb}

   {\tt ERRB}^2 & = & \frac{{\rm var}(\overline{x})+{\rm var}(\overline{y})}{2}

       - \sqrt{\left(\frac{{\rm var}(\overline{x})-{\rm var}(\overline{y})}{2}\right)^2

       + {\rm cov}^2(\overline{x},\overline{y})},\\

   \label{eq:errtheta}

   \tan (2{\tt ERRTHETA}) & = & 2 \,\frac{{\rm cov}(\overline{x},\overline{y})}

                       {{\rm var}(\overline{x}) - {\rm var}(\overline{y})}.\end{aligned}

And the ellipse parameters are:

.. math::

  :label: errellipse

   \begin{aligned}

   \label{eq:errcxx}

   {\tt ERRCXX} & = & \frac{\cos^2 {\tt ERRTHETA}}{{\tt ERRA}^2} +

   \frac{\sin^2 {\tt ERRTHETA}}{{\tt ERRB}^2} = \frac{{\rm

   var}(\overline{y})}{{\rm var}(\overline{x}) {\rm var}(\overline{y}) -

   {\rm cov}^2(\overline{x},\overline{y})},\\

   \label{eq:errcyy}

   {\tt ERRCYY} & = & \frac{\sin^2 {\tt ERRTHETA}}{{\tt ERRA}^2} +

   \frac{\cos^2 {\tt ERRTHETA}}{{\tt ERRB}^2} =

   \frac{{\rm var}(\overline{x})}{{\rm var}(\overline{x}) {\rm var}(\overline{y}) -

   {\rm cov}^2(\overline{x},\overline{y})},\\

   \label{eq:errcxy}

   {\tt ERRCXY} & = & 2 \cos {\tt

   ERRTHETA}\sin {\tt ERRTHETA} \left( \frac{1}{{\tt ERRA}^2} -

   \frac{1}{{\tt ERRB}^2}\right)\\ & = & -2 \frac{{\rm

sufficiently thin line of pixels, which we translate mathematically by

.. math::

  :label: singutest

   \label{eq:singutest}

   \overline{x^2}\,\overline{y^2} - \overline{xy}^2 < \rho^2,

then :math:`\overline{x^2}` and :math:`\overline{y^2}` are incremented

Positional errors are more difficult to handle, as objects with very

high signal-to-noise can yield extremely small position uncertainties,

just like singular profiles do. Therefore |SExtractor| first checks that

([eq:singutest]) is true. If this is the case, a new test is conducted:

:eq:`singutest` is true. If this is the case, a new test is conducted:

.. math::

  :label: singutest2

   \label{eq:singutest2}

   {\rm var}(\overline{x})\,{\rm var}(\overline{y}) - {\rm

   covar}^2(\overline{x}, \overline{y}) < \rho^2_e,

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}} I_i\right)^2`. If

([eq:singutest2]) is true, then :math:`\overline{x^2}` and

:eq:`singutest2` is true, then :math:`\overline{x^2}` and

:math:`\overline{y^2}` are incremented by :math:`\rho_e`.

.. [1]

   Such parameters are dimensionless and therefore do not accept any

   _IMAGE or _WORLD suffix

.. [#elongation] These parameters are dimensionless, and for historical reasons do not accept suffixes such as _IMAGE or _WORLD.

.. include:: keys.rst

    ``MAP_WEIGHT`` is the most commonly used type of weight-map: a flat-field, for example, is generally

    a good approximation to a perfect weight-map.

.. _effect_of_weighting:

Effect of weighting

-------------------

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