| Line 1... |
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\section{Positional parameters derived from the isophotal profile}
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\section{Positional parameters derived from the isophotal profile}
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\label{chap:isoparam}
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\label{chap:isoparam}
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The following parameters are derived from the spatial distribution $\cal S$ of pixels detected
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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)
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detection image}.
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detection \index{image} image}.
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|
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{\bf Note that, unless otherwise noted, all parameter names given
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{\bf Note that, unless otherwise noted, all parameter names given
|
below are only prefixes. They must be followed by "{\tt\_IMAGE}" if
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below are only prefixes. They must be followed by "{\tt\_IMAGE}" if
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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
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"{\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
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THETA\_IMAGE}. In all cases, parameters are first computed in the image
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THETA\_IMAGE}. In all cases, parameters are first computed in the \index{image} image
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coordinate system, and then converted to WCS if requested.
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coordinate system, and then converted to \index{WCS} WCS if requested.
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|
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\subsection{Limits: {\tt XMIN}, {\tt YMIN}, {\tt XMAX}, {\tt YMAX}}
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\subsection{Limits: {\tt XMIN}, {\tt YMIN}, {\tt XMAX}, {\tt YMAX}}
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These coordinates define two corners of a rectangle which encloses the detected object:
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These coordinates define two corners of a rectangle which encloses the detected object:
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\begin{eqnarray}
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\begin{eqnarray}
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{\tt XMIN} & = & \min_{i \in {\cal S}} x_i,\\
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{\tt XMIN} & = & \min_{i \in {\cal S}} x_i,\\
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| Line 24... |
Line 24... |
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\subsection{Barycenter: {\tt X}, {\tt Y}}
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\subsection{Barycenter: {\tt X}, {\tt Y}}
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Barycenter coordinates generally define the position of the ``center'' of a source,
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Barycenter coordinates generally define the position of the ``center'' of a source,
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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}
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{\tt X} & = & \overline{x} = \frac{\displaystyle \sum_{i \in {\cal S}}
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{\tt X} & = & \overline{x} = \frac{\displaystyle \sum_{i \in {\cal S}}
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I_i x_i}{\displaystyle \sum_{i \in {\cal S}} I_i},\\ {\tt Y} & = &
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I_i x_i}{\displaystyle \sum_{i \in {\cal S}} I_i},\\ {\tt Y} & = &
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\overline{y} = \frac{\displaystyle \sum_{i \in {\cal S}} I_i
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\overline{y} = \frac{\displaystyle \sum_{i \in {\cal S}} I_i
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y_i}{\displaystyle \sum_{i \in {\cal S}} I_i}.
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y_i}{\displaystyle \sum_{i \in {\cal S}} I_i}.
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| Line 43... |
Line 43... |
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
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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
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positions and \index{barycenter} barycenters coincide within a fraction of pixel ({\tt
|
XPEAK} and {\tt YPEAK} coordinates are quantized by steps of 1 pixel,
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XPEAK} and {\tt YPEAK} coordinates are quantized by steps of 1 pixel,
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thus {\tt XPEAK\_IMAGE} and {\tt YPEAK\_IMAGE} are integers). This is
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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.
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asymmetrical objects on well-sampled \index{image} images.
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|
|
\subsection{2nd order moments: {\tt X2}, {\tt Y2}, {\tt XY}}
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\subsection{2nd order \index{moments} moments: {\tt X2}, {\tt Y2}, {\tt XY}}
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(Centered) second-order moments are convenient for measuring the spatial spread of a source
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(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
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{\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} = &
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\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},
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\end{eqnarray}
|
\end{eqnarray}
|
These expressions are more subject to roundoff errors than if the 1st-order moments were
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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
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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}
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for smaller roundoff errors.}
|
for smaller roundoff errors.}
|
|
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\subsection{Basic shape parameters: {\tt A}, {\tt B}, {\tt THETA}}
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\subsection{Basic shape parameters: {\tt A}, {\tt B}, {\tt THETA}}
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\label{chap:abtheta}
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\label{chap:abtheta}
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These parameters are intended to describe the detected object as an elliptical
|
These parameters are intended to describe the detected object as an elliptical
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shape. {\tt A} and {\tt B} are its semi-major and semi-minor axis lengths,
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shape. {\tt A} and {\tt B} are its \index{semi-major} semi-major and \index{semi-minor axis} semi-minor axis lengths,
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respectively. More precisely, they represent the maximum and minimum spatial
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respectively. More precisely, they represent the maximum and minimum spatial
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\rms dispersion of the object profile along any direction. {\tt THETA} is the
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\rms dispersion of the object profile along any direction. {\tt THETA} is the
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position-angle between of the {\tt A} axis relative to the {\tt NAXIS1} image axis. It is
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position-angle between of the {\tt A} axis relative to the {\tt NAXIS1} \index{image} image axis. It is
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counted counter-clockwise. Here is how they are computed:
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counted counter-clockwise. Here is how they are computed:
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|
|
2nd-order moments can easily be expressed in a referential rotated from the
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2nd-order \index{moments} moments can easily be expressed in a referential rotated from the
|
$x,y$ image coordinate system
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$x,y$ \index{image} image coordinate system
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by an angle +$\theta$:
|
by an angle +$\theta$:
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\begin{equation}
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\begin{equation}
|
\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}
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\overline{x_{\theta}^2} & = & \cos^2\theta\:\overline{x^2} & +\,\sin^2\theta\:\overline{y^2}
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| Line 110... |
Line 110... |
If $\overline{y^2} \neq \overline{x^2}$, this implies:
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If $\overline{y^2} \neq \overline{x^2}$, this implies:
|
\begin{equation}
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\begin{equation}
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\label{eq:theta0}
|
\label{eq:theta0}
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\tan 2\theta_0 = 2 \frac{\overline{xy}}{\overline{x^2} - \overline{y^2}},
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\tan 2\theta_0 = 2 \frac{\overline{xy}}{\overline{x^2} - \overline{y^2}},
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\end{equation}
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\end{equation}
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a result which can also be obtained by requiring the covariance
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a result which can also be obtained by requiring the \index{covariance} covariance
|
$\overline{xy_{\theta_0}}$ to be null.
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$\overline{xy_{\theta_0}}$ to be null.
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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}).
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(\ref{eq:theta0}).
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By definition, {\tt THETA} is the position angle for which
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By definition, {\tt THETA} is the position angle for which
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$\overline{x_{\theta}^2}$ is {\em max}\,imized.
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$\overline{x_{\theta}^2}$ is {\em max}\,imized.
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{\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
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the covariance $\overline{xy}$.
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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:
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\begin{eqnarray}
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\begin{eqnarray}
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{\tt A}^2 & = & \overline{x^2}_{\tt THETA},\ \ \ {\rm and}\\
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{\tt A}^2 & = & \overline{x^2}_{\tt THETA},\ \ \ {\rm and}\\
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{\tt B}^2 & = & \overline{y^2}_{\tt THETA}.
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{\tt B}^2 & = & \overline{y^2}_{\tt THETA}.
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\end{eqnarray}
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\end{eqnarray}
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{\tt A} and {\tt B} can be computed directly from the 2nd-order moments, using the following
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{\tt A} and {\tt B} can be computed directly from the 2nd-order \index{moments} moments, using the following
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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}
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{\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}
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{\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
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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,
|
\end{equation}
|
\end{equation}
|
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
|
| Line 195... |
Line 195... |
|
|
\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 $i$:
|
\begin{equation}
|
\begin{equation}
|
\sigma^2_i = {\sigma_B}^2_i + \frac{I_i}{g_i},
|
\sigma^2_i = {\sigma_B}^2_i + \frac{I_i}{g_i},
|
\end{equation}
|
\end{equation}
|
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
|
