.. File Position.rst

.. include:: global.rst

Position and shape parameters derived from the isophotal profile
================================================================

The following parameters are derived from the spatial distribution :math:`\cal S` of pixels detected above the analysis threshold (see :ref:`description<isophotal_measurements>`).

.. important::
  Unless otherwise noted, pixel values :math:`I_i` are taken from the (filtered) 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`).


.. _xyminmax_def:

Limits: :param:`XMIN`, :param:`YMIN`, :param:`XMAX`, :param:`YMAX`
------------------------------------------------------------------

These coordinates define two corners of a rectangle which encloses the detected object:

.. math::
  :label: xminymax

   \begin{aligned}
   {\tt XMIN} & = & \min_{i \in {\cal S}} x_i,\\
   {\tt YMIN} & = & \min_{i \in {\cal S}} y_i,\\
   {\tt XMAX} & = & \max_{i \in {\cal S}} x_i,\\
   {\tt YMAX} & = & \max_{i \in {\cal S}} y_i,
   \end{aligned}

where :math:`x_i` and :math:`y_i` are respectively the x-coordinate and y-coordinate of pixel :math:`i`.


.. _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:

.. math::
  :label: xy

   \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}.
   \end{aligned}

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.


.. _moments_iso_def:

2nd 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:

.. math::
  :label: x2y2

   \begin{aligned}
   {\tt X2} & = \overline{x^2} = & \frac{\displaystyle \sum_{i \in {\cal
   S}} I_i x_i^2}{\displaystyle \sum_{i \in {\cal S}} I_i} -
   \overline{x}^2,\\ {\tt Y2} & = \overline{y^2} = & \frac{\displaystyle
   \sum_{i \in {\cal S}} I_i y_i^2}{\displaystyle \sum_{i \in {\cal S}}
   I_i} - \overline{y}^2,\\ {\tt XY} & = \overline{xy} = &
   \frac{\displaystyle \sum_{i \in {\cal S}} I_i x_i y_i}{\displaystyle
   \sum_{i \in {\cal S}} I_i} - \overline{x}\,\overline{y},
   \end{aligned}

These expressions are more subject to roundoff errors than if the 1st-order moments were subtracted before summing, but allow both 1st and
2nd order moments to be computed in one pass.
Roundoff errors are however kept to a negligible value by measuring all positions relative here again to :param:`XMIN` and :param:`YMIN`.


.. _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.
:param:`THETA` is the position-angle of the :param:`A` axis relative to the first image axis.
It is counted positive in the direction of the second axis.

Here is how shape parameters are computed. 2nd-order moments can easily be expressed in a referential rotated from the :math:`x,y` image coordinate system by an angle +\ :math:`\theta`:

.. math::
  :label: 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},\\
   \overline{y_{\theta}^2} & = & \sin^2\theta\:\overline{x^2} & +\,\cos^2\theta\:\overline{y^2}
               & +\,2 \cos\theta \sin\theta\:\overline{xy},\\
   \overline{xy_{\theta}} & = & \cos\theta \sin\theta\:\overline{x^2} &
   -\,\cos\theta \sin\theta\:\overline{y^2} & +\,(\cos^2\theta -
   \sin^2\theta)\:\overline{xy}.
   \end{array}

One can find the angle(s) :math:`\theta_0` for which the variance is minimized (or maximized) along :math:`x_{\theta}`:

.. 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

   \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_3`.
By definition, :param:`THETA` is the position angle for which :math:`\overline{x_{\theta}^2}` is maximized.
:param:`THETA` is therefore the solution to :eq:`theta0_3` that has the same sign as the
covariance :math:`\overline{xy}`.
:param:`A` and :param:`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}

:param:`A` and :param:`B` can be computed directly from the 2nd-order moments, using the following equations derived from :eq:`varproj` after some algebra:

.. math::
  :label: aimage_2

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

Note that :param:`A` and :param:`B` are exactly halves the :math:`a` and :math:`b` parameters computed by the COSMOS image analyser :cite:`1980SPIE_264_208S`.
Actually, :math:`a` and :math:`b` are defined in :cite:`1980SPIE_264_208S` as the semi-major and semi-minor axes of an elliptical shape with constant surface brightness, which would have the same 2nd-order moments as the analyzed object.


.. _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`:

.. math::
  :label: elongation

   \begin{aligned}
   {\tt ELONGATION} & = & \frac{\tt A}{\tt B}\ \ \ \ \ \mbox{and}\\
   {\tt ELLIPTICITY} & = & 1 - \frac{\tt B}{\tt A}.
   \end{aligned}


.. _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.
For this kind of application, three other ellipse parameters are provided; :param:`CXX`, :param:`CYY`, and :param:`CXY`.
They do nothing more than describing the same ellipse, but in a different way: the elliptical shape associated to 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,

where :math:`R` is a parameter which scales the ellipse, in units of :param:`A`
(or :param:`B`). Generally, the isophotal limit of a detected object is well
represented by :math:`R\approx 3` (:numref:`fig_ellipse`). Ellipse
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
   {\tt THETA}}{{\tt B}^2} =
   \frac{\overline{y^2}}{\overline{x^2} \overline{y^2} - \overline{xy}^2}\\
   {\tt CYY} & = & \frac{\sin^2 {\tt THETA}}{{\tt
   A}^2} + \frac{\cos^2 {\tt THETA}}{{\tt B}^2} =
   \frac{\overline{x^2}}{\overline{x^2} \overline{y^2} - \overline{xy}^2}\\
   {\tt CXY} & = & 2 \,\cos {\tt THETA}\,\sin {\tt
   THETA} \left( \frac{1}{{\tt A}^2} - \frac{1}{{\tt B}^2}\right) = -2\,
   \frac{\overline{xy}}{\overline{x^2} \overline{y^2} - \overline{xy}^2}.
   \end{aligned}

.. _fig_ellipse:

.. figure:: figures/ellipse.*
   :figwidth: 100%
   :align: center

   Meaning of basic shape parameters.


.. _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.
In practice, such estimates are a lower-value of the full uncertainties because they do not include, for instance, the contribution of detection biases or contamination by neighbors.
Furthermore, |SExtractor| does not currently take into account possible correlations of the noise between adjacent pixels. Hence variances simply write:

.. math::
  :label: errxy

   \begin{aligned}
   {\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}} I_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}} I_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}} I_i\right)^2}.
   \end{aligned}

:math:`\sigma_i` is the flux uncertainty estimated for pixel :math:`i`:

.. math:: \sigma^2_i = {\sigma_B}^2_i + \frac{I_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>`:

.. math::
  :label: errabtheta

   \begin{aligned}
   {\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})},\\
   {\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})},\\
   \tan (2{\tt ERRTHETA}) & = & 2 \,\frac{{\rm cov}(\overline{x},\overline{y})}
                       {{\rm var}(\overline{x}) - {\rm var}(\overline{y})}.
   \end{aligned}

And the error ellipse parameters are:

.. math::
  :label: errellipse

   \begin{aligned}
   {\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})},\\
   {\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})},\\
   {\tt ERRCXY} & = & 2 \cos {\tt
   ERRTHETA}\sin {\tt ERRTHETA} \left( \frac{1}{{\tt ERRA}^2} -
   \frac{1}{{\tt ERRB}^2}\right)\\ & = & -2 \frac{{\rm
   cov}(\overline{x},\overline{y})}{{\rm var}(\overline{x}) {\rm var}(\overline{y}) -
   {\rm cov}^2(\overline{x},\overline{y})}.
   \end{aligned}


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.

For basic shape parameters, the following convention was adopted: if the
light distribution of the object falls on one single pixel, or lies on a
sufficiently thin line of pixels, which we translate mathematically by

.. math::
  :label: singutest

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

then :math:`\overline{x^2}` and :math:`\overline{y^2}` are incremented by :math:`\rho`. |SExtractor| sets :math:`\rho=1/12`, which is the variance of a 1-dimensional top-hat distribution with unit width.
Therefore :math:`1/\sqrt{12}` represents the typical minor-axis values assigned (in pixels units) to undersampled sources in |SExtractor|.

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:

.. math::
  :label: 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 :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`.

