Positional parameters derived from the isophotal profile
========================================================

The following parameters are derived from the spatial distribution
:math:`\cal S` of pixels detected above the extraction threshold. *The
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.

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

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

.. math::

   \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`.

Barycenter: X, 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::

   \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 XMIN
and YMIN in order to reduce roundoff errors in the summing.

Position of the peak: XPEAK, YPEAK
----------------------------------

It is sometimes useful to have the position XPEAK,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 (XPEAK and YPEAK coordinates are
quantized by steps of 1 pixel, thus XPEAK\_IMAGE and 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.

2nd order moments: X2, Y2, XY
-----------------------------

(Centered) second-order moments are convenient for measuring the spatial
spread of a source profile. In SExtractor they are computed with:

.. math::

   \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 XMIN and YMIN.

Basic shape parameters: A, B, THETA
-----------------------------------

These parameters are intended to describe the detected object as an
elliptical shape. A and B are its semi-major and semi-minor axis
lengths, respectively. More precisely, they represent the maximum and
minimum spatial dispersion of the object profile along any direction.
THETA is the position-angle of the A axis relative to the first image
axis. It is counted positive in the direction of the second axis. Here
is how they 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{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},\\
   \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 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,

which leads to

.. math::

   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{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
which :math:`\overline{x_{\theta}^2}` is *max*\ imized. THETA is
therefore the solution to ([eq:theta0]) that has the same sign as the
covariance :math:`\overline{xy}`. A and B can now simply be expressed
as:

.. math::

   \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:

.. math::

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

Note that A and B are exactly halves the :math:`a` and :math:`b`
parameters computed by the COSMOS image analyser (Stobie 1980,1986).
Actually, :math:`a` and :math:`b` are defined by Stobie 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
analysed object.

Ellipse parameters: CXX, CYY, CXY
---------------------------------

A, B and 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; CXX, CYY and 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::

   {\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 A
(or B). Generally, the isophotal limit of a detected object is well
represented by :math:`R\approx 3` (Fig. [fig:ellipse]). Ellipse
parameters can be derived from the 2nd order moments:

.. math::

   \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}

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

These parameters are directly derived from A and B:

.. math::

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

Position errors: ERRX2, ERRY2, ERRXY, ERRA, ERRB, ERRTHETA, ERRCXX, ERRCYY, ERRCXY
----------------------------------------------------------------------------------

Uncertainties on the position of the barycenter can be estimated using
photon statistics. Of course, this kind of estimate has to be considered
as a lower-value of the real error since it does not include, for
instance, the contribution of detection biases or the contamination by
neighbours. As SExtractor does not currently take into account possible
correlations between pixels, the variances simply write:

.. math::

   \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 §[chap:weight] 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::

   \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::

   \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
   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
handles, of course), some detections with very specific shapes may yield
quite unphysical parameters, namely null values for B, ERRB, or even A
and 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 very 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{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
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{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
: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