# public documents.sextractor_doc

## [/] [measure_astrom.tex] - Diff between revs 19 and 25

Rev 19 Rev 25
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\section{Positional parameters derived from the isophotal profile}
\section{Positional parameters derived from the isophotal profile}
\label{chap:isoparam}
\label{chap:isoparam}
The following parameters are derived from the spatial distribution $\cal S$ of pixels detected
The following parameters are derived from the spatial distribution $\cal S$ of pixels detected
above the extraction threshold. {\em The pixel values $I_i$ are taken from the (filtered)
above the extraction \index{threshold} threshold. {\em The pixel values $I_i$ are taken from the (filtered)
detection image}.
detection \index{image} image}.
 
 
{\bf Note that, unless otherwise noted, all parameter names given
{\bf Note that, unless otherwise noted, all parameter names given
below are only prefixes. They must be followed by "{\tt\_IMAGE}" if
below are only prefixes. They must be followed by "{\tt\_IMAGE}" if
the results shall be expressed in pixel units (see \S..), or
the results shall be expressed in pixel units (see \S..), or
"{\tt\_WORLD}" for World Coordinate System (WCS) units (see
"{\tt\_WORLD}" for World Coordinate System (WCS) units (see
\S\ref{astrom})}. For example: {\tt THETA} $\rightarrow$ {\tt
\S\ref{astrom})}. For example: {\tt THETA} $\rightarrow$ {\tt
THETA\_IMAGE}. In all cases, parameters are first computed in the image
THETA\_IMAGE}. In all cases, parameters are first computed in the \index{image} image
coordinate system, and then converted to WCS if requested.
coordinate system, and then converted to \index{WCS} WCS if requested.
 
 
\subsection{Limits: {\tt XMIN}, {\tt YMIN}, {\tt XMAX}, {\tt YMAX}}
\subsection{Limits: {\tt XMIN}, {\tt YMIN}, {\tt XMAX}, {\tt YMAX}}
These coordinates define two corners of a rectangle which encloses the detected object:
These coordinates define two corners of a rectangle which encloses the detected object:
\begin{eqnarray}
\begin{eqnarray}
{\tt XMIN} & = & \min_{i \in {\cal S}} x_i,\\
{\tt XMIN} & = & \min_{i \in {\cal S}} x_i,\\
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\subsection{Barycenter: {\tt X}, {\tt Y}}
\subsection{Barycenter: {\tt X}, {\tt Y}}
Barycenter coordinates generally define the position of the center'' of a source,
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
although this definition can be inadequate or inaccurate if its spatial profile shows
a strong skewness or very large wings. {\tt X} and {\tt Y} are simply computed
a strong skewness or very large wings. {\tt X} and {\tt Y} are simply computed
as the first order moments of the profile:
as the first order \index{moments} moments of the profile:
\begin{eqnarray}
\begin{eqnarray}
{\tt X} & = & \overline{x} = \frac{\displaystyle \sum_{i \in {\cal S}}
{\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} & = &
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
\overline{y} = \frac{\displaystyle \sum_{i \in {\cal S}} I_i
y_i}{\displaystyle \sum_{i \in {\cal S}} I_i}.
y_i}{\displaystyle \sum_{i \in {\cal S}} I_i}.
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It is sometimes useful to have the position {\tt XPEAK},{\tt YPEAK} of
It is sometimes useful to have the position {\tt XPEAK},{\tt YPEAK} of
the pixel with maximum intensity in a detected object, for instance
the pixel with maximum intensity in a detected object, for instance
when working with likelihood maps, or when searching for artifacts.
when working with likelihood maps, or when searching for artifacts.
For better robustness, {\tt PEAK} coordinates are computed on {\em
For better robustness, {\tt PEAK} coordinates are computed on {\em
filtered} profiles if available. On symmetrical profiles, {\tt PEAK}
filtered} profiles if available. On symmetrical profiles, {\tt PEAK}
positions and barycenters coincide within a fraction of pixel ({\tt
positions and \index{barycenter} barycenters coincide within a fraction of pixel ({\tt
XPEAK} and {\tt YPEAK} coordinates are quantized by steps of 1 pixel,
XPEAK} and {\tt YPEAK} coordinates are quantized by steps of 1 pixel,
thus {\tt XPEAK\_IMAGE} and {\tt YPEAK\_IMAGE} are integers). This is
thus {\tt XPEAK\_IMAGE} and {\tt YPEAK\_IMAGE} are integers). This is
no longer true for skewed profiles, therefore a simple comparison
no longer true for skewed profiles, therefore a simple comparison
between {\tt PEAK} and barycenter coordinates can be used to identify
between {\tt PEAK} and \index{barycenter} barycenter coordinates can be used to identify
asymmetrical objects on well-sampled images.
asymmetrical objects on well-sampled \index{image} images.
 
 
\subsection{2nd order moments: {\tt X2}, {\tt Y2}, {\tt XY}}
\subsection{2nd order \index{moments} moments: {\tt X2}, {\tt Y2}, {\tt XY}}
(Centered) second-order moments are convenient for measuring the spatial spread of a source
(Centered) second-order \index{moments} moments are convenient for measuring the spatial spread of a source
profile. In {\sc SExtractor} they are computed with:
profile. In {\sc SExtractor} they are computed with:
\begin{eqnarray}
\begin{eqnarray}
{\tt X2} & = \overline{x^2} = & \frac{\displaystyle \sum_{i \in {\cal
{\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} -
S}} I_i x_i^2}{\displaystyle \sum_{i \in {\cal S}} I_i} -
\overline{x}^2,\\ {\tt Y2} & = \overline{y^2} = & \frac{\displaystyle
\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}}
\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} = &
I_i} - \overline{y}^2,\\ {\tt XY} & = \overline{xy} = &
\frac{\displaystyle \sum_{i \in {\cal S}} I_i x_i y_i}{\displaystyle
\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},
\sum_{i \in {\cal S}} I_i} - \overline{x}\,\overline{y},
\end{eqnarray}
\end{eqnarray}
These expressions are more subject to roundoff errors than if the 1st-order moments were
These expressions are more subject to roundoff errors than if the 1st-order \index{moments} moments were
subtracted before summing, but allow both 1st and 2nd order moments to be computed in one
subtracted before summing, but allow both 1st and 2nd order \index{moments} moments to be computed in one
pass. Roundoff errors are however kept to a negligible value by measuring all positions
pass. Roundoff errors are however kept to a negligible value by measuring all positions
relative here again to {\tt XMIN} and {\tt YMIN}.
relative here again to {\tt XMIN} and {\tt YMIN}.
\gam{Could be summed relative to
\gam{Could be summed relative to
{\tt XPEAK} and {\tt YPEAK} or
{\tt XPEAK} and {\tt YPEAK} or
{\tt X} and {\tt Y}
{\tt X} and {\tt Y}
for smaller roundoff errors.}
for smaller roundoff errors.}
 
 
\subsection{Basic shape parameters: {\tt A}, {\tt B}, {\tt THETA}}
\subsection{Basic shape parameters: {\tt A}, {\tt B}, {\tt THETA}}
\label{chap:abtheta}
\label{chap:abtheta}
These parameters are intended to describe the detected object as an elliptical
These parameters are intended to describe the detected object as an elliptical
shape. {\tt A} and {\tt B} are its semi-major and semi-minor axis lengths,
shape. {\tt A} and {\tt B} are its \index{semi-major} semi-major and \index{semi-minor axis} semi-minor axis lengths,
respectively. More precisely, they represent the maximum and minimum spatial
respectively. More precisely, they represent the maximum and minimum spatial
\rms dispersion of the object profile along any direction. {\tt THETA} is the
\rms dispersion of the object profile along any direction. {\tt THETA} is the
position-angle between of the {\tt A} axis relative to the {\tt NAXIS1} image axis. It is
position-angle between of the {\tt A} axis relative to the {\tt NAXIS1} \index{image} image axis. It is
counted counter-clockwise. Here is how they are computed:
counted counter-clockwise. Here is how they are computed:
 
 
2nd-order moments can easily be expressed in a referential rotated from the
2nd-order \index{moments} moments can easily be expressed in a referential rotated from the
$x,y$ image coordinate system
$x,y$ \index{image} image coordinate system
by an angle +$\theta$:
by an angle +$\theta$:
\label{eq:varproj}
\label{eq:varproj}
\begin{array}{lcrrr}
\begin{array}{lcrrr}
\overline{x_{\theta}^2} & = & \cos^2\theta\:\overline{x^2} & +\,\sin^2\theta\:\overline{y^2}
\overline{x_{\theta}^2} & = & \cos^2\theta\:\overline{x^2} & +\,\sin^2\theta\:\overline{y^2}
Line 110... Line 110...
If $\overline{y^2} \neq \overline{x^2}$, this implies:
If $\overline{y^2} \neq \overline{x^2}$, this implies:
\label{eq:theta0}
\label{eq:theta0}
\tan 2\theta_0 = 2 \frac{\overline{xy}}{\overline{x^2} - \overline{y^2}},
\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
a result which can also be obtained by requiring the \index{covariance} covariance
$\overline{xy_{\theta_0}}$ to be null.
$\overline{xy_{\theta_0}}$ to be null.
Over the domain $[-\pi/2, +\pi/2[$, two different angles --- with opposite signs --- satisfy
Over the domain $[-\pi/2, +\pi/2[$, two different angles --- with opposite signs --- satisfy
(\ref{eq:theta0}).
(\ref{eq:theta0}).
By definition, {\tt THETA} is the position angle for which
By definition, {\tt THETA} is the position angle for which
$\overline{x_{\theta}^2}$ is {\em max}\,imized.
$\overline{x_{\theta}^2}$ is {\em max}\,imized.
{\tt THETA} is therefore the solution to (\ref{eq:theta0}) that has the same sign as
{\tt THETA} is therefore the solution to (\ref{eq:theta0}) that has the same sign as
the covariance $\overline{xy}$.
the \index{covariance} covariance $\overline{xy}$.
{\tt A} and {\tt B} can now simply be expressed as:
{\tt A} and {\tt B} can now simply be expressed as:
\begin{eqnarray}
\begin{eqnarray}
{\tt A}^2 & = & \overline{x^2}_{\tt THETA},\ \ \ {\rm and}\\
{\tt A}^2 & = & \overline{x^2}_{\tt THETA},\ \ \ {\rm and}\\
{\tt B}^2 & = & \overline{y^2}_{\tt THETA}.
{\tt B}^2 & = & \overline{y^2}_{\tt THETA}.
\end{eqnarray}
\end{eqnarray}
{\tt A} and {\tt B} can be computed directly from the 2nd-order moments, using the following
{\tt A} and {\tt B} can be computed directly from the 2nd-order \index{moments} moments, using the following
equations derived from (\ref{eq:varproj}) after some algebra:
equations derived from (\ref{eq:varproj}) after some algebra:
\begin{eqnarray}
\begin{eqnarray}
\label{eq:aimage}
\label{eq:aimage}
{\tt A}^2 & = & \frac{\overline{x^2}+\overline{y^2}}{2}
{\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},\\
        + \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}
{\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}.
        - \sqrt{\left(\frac{\overline{x^2}-\overline{y^2}}{2}\right)^2 + \overline{xy}^2}.
\end{eqnarray}
\end{eqnarray}
Note that {\tt A} and {\tt B} are exactly halves the $a$ and $b$
Note that {\tt A} and {\tt B} are exactly halves the $a$ and $b$
parameters computed by the COSMOS image analyser (Stobie 1980,1986).
parameters computed by the \index{COSMOS} COSMOS \index{image} image analyser (Stobie 1980,1986).
Actually, $a$ and $b$ are defined by Stobie as the semi-major and
Actually, $a$ and $b$ are defined by Stobie as the \index{semi-major} semi-major and
semi-minor axes of an elliptical shape with constant surface
semi-minor axes of an elliptical shape with constant surface
brightness, which would have the same 2nd-order moments as the
brightness, which would have the same 2nd-order \index{moments} moments as the
analysed object.
analysed object.
 
 
\subsection{Ellipse parameters: {\tt CXX}, {\tt CYY}, {\tt CXY}}
\subsection{Ellipse parameters: {\tt CXX}, {\tt CYY}, {\tt CXY}}
\label{chap:cxx}
\label{chap:cxx}
{\tt A}, {\tt B} and {\tt THETA} are not very convenient to use when,
{\tt A}, {\tt B} and {\tt THETA} are not very convenient to use when,
Line 155... Line 155...
        + {\tt CXY} (x-\overline{x})(y-\overline{y}) = R^2,
        + {\tt CXY} (x-\overline{x})(y-\overline{y}) = R^2,
where $R$ is a parameter which scales the ellipse, in units of {\tt A}
where $R$ is a parameter which scales the ellipse, in units of {\tt A}
(or {\tt B}). Generally, the isophotal limit of a detected object is
(or {\tt B}). Generally, the isophotal limit of a detected object is
well represented by $R\approx 3$ (Fig. \ref{fig:ellipse}). Ellipse
well represented by $R\approx 3$ (Fig. \ref{fig:ellipse}). Ellipse
parameters can be derived from the 2nd order moments:
parameters can be derived from the 2nd order \index{moments} moments:
\begin{eqnarray}
\begin{eqnarray}
{\tt CXX} & = & \frac{\cos^2 {\tt THETA}}{{\tt A}^2} + \frac{\sin^2
{\tt CXX} & = & \frac{\cos^2 {\tt THETA}}{{\tt A}^2} + \frac{\sin^2
{\tt THETA}}{{\tt B}^2} =
{\tt THETA}}{{\tt B}^2} =
\frac{\overline{y^2}}{\sqrt{\left(\frac{\overline{x^2}-\overline{y^2}}{2}\right)^2
\frac{\overline{y^2}}{\sqrt{\left(\frac{\overline{x^2}-\overline{y^2}}{2}\right)^2
+ \overline{xy}^2}}\\ {\tt CYY} & = & \frac{\sin^2 {\tt THETA}}{{\tt
+ \overline{xy}^2}}\\ {\tt CYY} & = & \frac{\sin^2 {\tt THETA}}{{\tt
Line 174... Line 174...
 
 
%------------------------------ Fig. phot -----------------------------
%------------------------------ Fig. phot -----------------------------
   \begin{figure}[htbp]
   \begin{figure}[htbp]
      \centerline{\includegraphics[width=16cm]{ps/ellipse.ps}}
      \centerline{\includegraphics[width=16cm]{ps/ellipse.ps}}
      \caption{
      \caption{
              The meaning of basic shape parameters.
              The \index{mean} meaning of basic shape parameters.
              }
              }
      \label{fig:ellipse}
      \label{fig:ellipse}
   \end{figure}
   \end{figure}
 
 
\subsection{By-products of shape parameters: {\tt ELONGATION} and
\subsection{By-products of shape parameters: {\tt ELONGATION} and
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\subsection{Position errors: {\tt ERRX2}, {\tt ERRY2}, {\tt ERRXY},
\subsection{Position errors: {\tt ERRX2}, {\tt ERRY2}, {\tt ERRXY},
{\tt ERRA}, {\tt ERRB}, {\tt ERRTHETA}, {\tt ERRCXX}, {\tt ERRCYY},
{\tt ERRA}, {\tt ERRB}, {\tt ERRTHETA}, {\tt ERRCXX}, {\tt ERRCYY},
{\tt ERRCXY}}
{\tt ERRCXY}}
\label{chap:poserr}
\label{chap:poserr}
Uncertainties on the position of the barycenter can be estimated using
Uncertainties on the position of the \index{barycenter} barycenter can be estimated using
photon statistics. Of course, this kind of estimate has to be
photon statistics. Of course, this kind of estimate has to be
considered as a lower-value of the real error since it does not
considered as a lower-value of the real error since it does not
include, for instance, the contribution of detection biases or the
include, for instance, the contribution of detection biases or the
contamination by neighbours. As {\sc SExtractor} does not currently
contamination by \index{neighbour} \index{neighbours} neighbours. As {\sc SExtractor} does not currently
take into account possible correlations between pixels, the variances
take into account possible correlations between pixels, the variances
simply write:
simply write:
\begin{eqnarray}
\begin{eqnarray}
{\tt ERRX2} & = {\rm var}(\overline{x}) = & \frac{\displaystyle
{\tt ERRX2} & = {\rm var}(\overline{x}) = & \frac{\displaystyle
\sum_{i \in {\cal S}} \sigma^2_i (x_i-\overline{x})^2} {\displaystyle
\sum_{i \in {\cal S}} \sigma^2_i (x_i-\overline{x})^2} {\displaystyle
Line 217... Line 217...
\end{eqnarray}
\end{eqnarray}
$\sigma_i$ is the flux uncertainty estimated for pixel $i$:
$\sigma_i$ is the flux uncertainty estimated for pixel $i$:
\sigma^2_i = {\sigma_B}^2_i + \frac{I_i}{g_i},
\sigma^2_i = {\sigma_B}^2_i + \frac{I_i}{g_i},
where ${\sigma_B}_i$ is the local background noise and $g_i$ the local
where ${\sigma_B}_i$ is the \index{local background} local background noise and $g_i$ the local
gain --- conversion factor --- for pixel $i$ (see
\index{gain} gain --- conversion factor --- for pixel $i$ (see
\S\ref{chap:weight} for more details). Semi-major axis {\tt ERRA}, semi-minor
\S\ref{chap:weight} for more details). Semi-major axis {\tt ERRA}, semi-minor
axis {\tt ERRB}, and position angle {\tt ERRTHETA} of the
axis {\tt ERRB}, and position angle {\tt ERRTHETA} of the
$1\sigma$ position error ellipse are computed from the covariance
$1\sigma$ position \index{error ellipse} error ellipse are computed from the \index{covariance} covariance
matrix exactly like in \ref{chap:abtheta} for shape parameters:
matrix exactly like in \ref{chap:abtheta} for shape parameters:
\begin{eqnarray}
\begin{eqnarray}
\label{eq:erra}
\label{eq:erra}
{\tt ERRA}^2 & = & \frac{{\rm var}(\overline{x})+{\rm var}(\overline{y})}{2}
{\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
        + \sqrt{\left(\frac{{\rm var}(\overline{x})-{\rm var}(\overline{y})}{2}\right)^2
Line 267... Line 267...
SExtractor} handles, of course), some detections with very specific
SExtractor} handles, of course), some detections with very specific
shapes may yield quite unphysical parameters, namely null values for
shapes may yield quite unphysical parameters, namely null values for
{\tt B}, {\tt ERRB}, or even {\tt A} and {\tt ERRA}. Such detections
{\tt B}, {\tt ERRB}, or even {\tt A} and {\tt ERRA}. Such detections
include single-pixel objects and horizontal, vertical or diagonal
include single-pixel objects and horizontal, vertical or diagonal
lines which are 1-pixel wide. They will generally originate from
lines which are 1-pixel wide. They will generally originate from
glitches; but very undersampled and/or low S/N genuine sources may
\index{glitch} \index{glitches} glitches; but very undersampled and/or low S/N genuine sources may
also produce such shapes. \hide{How to handle them?}
also produce such shapes. \hide{How to handle them?}
 
 
For basic shape parameters, the following convention was adopted: if
For basic shape parameters, the following convention was adopted: if
the light distribution of the object falls on one single pixel, or
the light distribution of the object falls on one single pixel, or
lies on a sufficiently thin line of pixels, which we translate
lies on a sufficiently thin line of pixels, which we translate