public documents.sextractor_doc

\section{Windowed positional parameters}
\label{chap:winparam}
Parameters measured within an object's isophotal limit are sensitive to
two main factors: 1) changes in the detection \index{threshold} threshold, which
create a variable bias and 2) irregularities in the object's
isophotal \index{boundaries} boundaries, which act as additional ``noise'' in the measurements.
 
Measurements performed through a {\em \index{window} window} function (an {\em envelope}) do
not have such drawbacks. {\sc SExtractor} versions 2.4 and above implement
``windowed'' versions for most of the measurements described in
\ref{chap:isoparam}:
 
{\small
\begin{tabular}{ll}
Isophotal parameters & Equivalent \index{window} \index{windowed} windowed parameters \\
\hline
{\tt X\_IMAGE}, {\tt Y\_IMAGE} &
{\tt XWIN\_IMAGE}, {\tt YWIN\_IMAGE}\\
{\tt ERRA\_IMAGE}, {\tt ERRB\_IMAGE}, {\tt ERRTHETA\_IMAGE} &
{\tt ERRAWIN\_IMAGE}, {\tt ERRBWIN\_IMAGE}, {\tt ERRTHETAWIN\_IMAGE}\\
{\tt A\_IMAGE}, {\tt B\_IMAGE}, {\tt THETA\_IMAGE} &
{\tt AWIN\_IMAGE}, {\tt BWIN\_IMAGE}, {\tt THETAWIN\_IMAGE}\\
{\tt X2\_IMAGE}, {\tt Y2\_IMAGE}, {\tt XY\_IMAGE} &
{\tt X2WIN\_IMAGE}, {\tt Y2WIN\_IMAGE}, {\tt XYWIN\_IMAGE}\\
{\tt CXX\_IMAGE}, {\tt CYY\_IMAGE}, {\tt CXY\_IMAGE} &
{\tt CXXWIN\_IMAGE}, {\tt CYYWIN\_IMAGE}, {\tt CXYWIN\_IMAGE}\\
\hline
\end{tabular}}
 
The computations involved are roughly  the same except that the pixel values are 
integrated within a circular Gaussian \index{window} window as opposed to the object's
isophotal footprint.
The Gaussian \index{window} window is scaled to each object; its \index{FWHM} FWHM is the diameter of the
disk that contains half of the object flux ($d_{50}$). Note that in double-image
\index{mode} mode (\S\ref{chap:using}) the \index{window} window is scaled based on the {\em measurement}
\index{image} image. \gam{Can we provide the precise sub-section for this?}
 
\subsection{Windowed \index{centroid} centroid: {\tt XWIN}, {\tt YWIN}}
\label{chap:wincent}
This is an iterative process. The computation starts by initializing the
\index{window} \index{windowed} windowed \index{centroid} centroid coordinates $\overline{x_{\tt WIN}}^{(0)}$ and
 $\overline{y_{\tt WIN}}^{(0)}$  to their basic $\overline{x}$ and $\overline{y}$
isophotal equivalents, respectively. 
\gam{I understand that the notation {\tt XXX} describes a {\sc SExtractor}
  parameter. But then, {\tt XWIN}$^{(t)}$ is abusive, since the output
  parameter is only that of the last iteration. Otherwise, it might be
  simpler to replace all the $\overline{x_{\tt WIN}^{(i)}}$ by {\tt
    XWIN}$^{(i)}$, everywhere in this subsection.}
Then at each iteration $t$,
$\overline{x_{\tt WIN}}$ and $\overline{y_{\tt WIN}}$ are refined using:
\begin{eqnarray}
\label{eq:xwin}
{\tt XWIN}^{(t+1)} & = & \overline{x_{\tt WIN}}^{(t+1)}
= \overline{x_{\tt WIN}}^{(t)} + 2\,\frac{\sum_{r_i^{(t)} < r_{\rm max}}
w_i^{(t)} I_i \ (x_i  - \overline{x_{\tt WIN}}^{(t)})}
{\sum_{r_i^{(t)} < r_{\rm max}} w_i^{(t)} I_i},\\
\label{eq:ywin}
{\tt YWIN}^{(t+1)} & = & \overline{y_{\tt WIN}}^{(t+1)}
= \overline{y_{\tt WIN}}^{(t)} + 2\,\frac{\sum_{r_i^{(t)} < r_{\rm max}}
w_i^{(t)} I_i\ (y_i - \overline{y_{\tt WIN}}^{(t)})}
{\sum_{r_i^{(t)} < r_{\rm max}} w_i^{(t)} I_i},
\end{eqnarray}
where
\begin{equation}
w_i^{(t)} = \exp \left(-\frac{r_i^{(t)^2}}{2s_{\tt WIN}^2} \right),
\end{equation}
with
\begin{equation}
r_i^{(t)} = \sqrt{\left(x_i - \overline{x_{\tt WIN}}^{(t)}\right)^2 + \left(y_i
- \overline{y_{\tt WIN}}^{(t)}\right)^2}
\end{equation}
and
$s_{\tt WIN} = d_{50} / \sqrt{8 \ln 2}$.
The process stops when the change in position between two iterations is less
than $2\times10^{-4}$ pixel, a condition which is generally achieved in about 3 to 5
iterations.
 
Although the iterative nature of the processing slows down the processing \hide{a bit},
it is recommended to use whenever possible \index{window} \index{windowed} windowed parameters instead of their
isophotal equivalents, since the measurements they provide
are much more precise (Fig. \ref{fig:winpres}). The precision in \index{centroid} centroiding
offered by {\tt XWIN\_IMAGE} and {\tt YWIN\_IMAGE} is actually very close to
that of \index{PSF} PSF-fitting on focused and properly sampled star \index{image} images, and can also
be applied to galaxies. It has been verified that for isolated,
Gaussian-like \index{PSF} PSFs, its accuracy is close to the theoretical limit set by
\index{image} image noise\footnote{see
{\tt http://www.astromatic.net/forum/showthread.php?tid=581}}.
 
%------------------------------ Fig. winpres -----------------------------
\begin{figure}[htbp]
\centerline{\includegraphics[width=8cm]{ps/sex_xpres.ps}
\includegraphics[width=8cm]{ps/sex_xw2pres.ps}}
\caption{Comparison between isophotal and \index{window} \index{windowed} windowed \index{centroid} centroid measurement
accuracies on simulated, background noise-limited \index{image} images.{\em Left}: histogram
of the difference between {\tt X\_IMAGE} and the simulation \index{centroid} centroid in x. 
{\em Right}: histogram of the difference between {\tt XWIN\_IMAGE} and the
simulation \index{centroid} centroid in x.}
\label{fig:winpres}
\end{figure}
 
\subsection{Windowed 2nd order \index{moments} moments: {\tt X2}, {\tt Y2}, {\tt XY}}
Windowed second-order \index{moments} moments are computed on the \index{image} image data once the centering
process from \S{\ref{chap:wincent}} has converged:
\begin{eqnarray}
{\tt X2WIN} & = \overline{x_{\tt WIN}^2}
= & \frac{\sum_{r_i < r_{\rm max}} w_i I_i (x_i - \overline{x_{\tt WIN}})^2}
{\sum_{r_i < r_{\rm max}} w_i I_i},\\
{\tt Y2WIN} & = \overline{y_{\tt WIN}^2}
= & \frac{\sum_{r_i < r_{\rm max}} w_i I_i (y_i - \overline{y_{\tt WIN}})^2}
{\sum_{r_i < r_{\rm max}} w_i I_i},\\
{\tt XYWIN} & = \overline{xy_{\tt WIN}}
= & \frac{\sum_{r_i < r_{\rm max}} w_i I_i (x_i - \overline{x_{\tt WIN}})
(y_i - \overline{y_{\tt WIN}})}
{\sum_{r_i < r_{\rm max}} w_i I_i}.
\end{eqnarray}
Windowed second-order \index{moments} moments are typically twice smaller than their isophotal
equivalent.
 
\subsection{Windowed ellipse parameters:
{\tt CXXWIN}, {\tt CYYWIN}, {\tt CXYWIN}}
They are computed from the \index{window} \index{windowed} windowed 2nd order \index{moments} moments exactly the same way as
in \S\ref{chap:cxx}.
 
\subsection{Windowed position errors: {\tt ERRX2WIN}, {\tt ERRY2WIN},
{\tt ERRXYWIN}, {\tt ERRAWIN}, {\tt ERRBWIN}, {\tt ERRTHETAWIN},
{\tt ERRCXXWIN}, {\tt ERRCYYWIN}, {\tt ERRCXYWIN}}
Windowed position errors are computed on the \index{image} image data once the centering
process from \S{\ref{chap:wincent}} has converged. Assuming that noise is
uncorrelated among pixels, standard error propagation applied to
(\ref{eq:xwin}) and (\ref{eq:xwin}) gives us:
\begin{eqnarray}
{\tt ERRX2WIN} & = {\rm var}(\overline{x_{\tt WIN}})
= & 4\,\frac{\sum_{r_i < r_{\rm max}} w_i^2 \sigma^2_i (x_i-\overline{x})^2}
{\left(\sum_{r_i < r_{\rm max}} w_i I_i\right)^2},\\
{\tt ERRY2WIN} & = {\rm var}(\overline{y_{\tt WIN}})
= & 4\,\frac{\sum_{r_i < r_{\rm max}} w_i^2 \sigma^2_i (y_i-\overline{y})^2}
{\left(\sum_{r_i < r_{\rm max}} w_i I_i\right)^2},\\
{\tt ERRXYWIN} & = {\rm cov}(\overline{x_{\tt WIN}},\overline{y_{\tt WIN}})
= & 4\,\frac{\sum_{r_i < r_{\rm max}}
w_i^2 \sigma^2_i (x_i-\overline{x_{\tt WIN}})(y_i-\overline{y_{\tt WIN}})}
{\left(\sum_{r_i < r_{\rm max}} w_i I_i\right)^2}.
\end{eqnarray}
 
The \index{semi-major} semi-major axis {\tt ERRAWIN}, semi-minor
axis {\tt ERRBWIN}, and position angle {\tt ERRTHETAWIN} of the
$1\sigma$ position \index{error ellipse} error ellipse are computed from the \index{covariance} covariance
matrix elements ${\rm var}(\overline{x_{\tt WIN}})$,
${\rm var}(\overline{y_{\tt WIN}})$,
${\rm cov}(\overline{x_{\tt WIN}},\overline{y_{\tt WIN}})$,
exactly as in \S\ref{chap:poserr}: see eqs. (\ref{eq:erra}),
(\ref{eq:errb}), (\ref{eq:errtheta}), (\ref{eq:errcxx}), (\ref{eq:errcyy})
and (\ref{eq:errcxy}).
 
%\section{2D-model fitting}
%\subsection{Star/galaxy separation}
%With the local \index{PSF} PSF and a noise \index{mode} model in hand, one can easily derive an optimum
%star/galaxy classifier. The problem was first addressed by \cite{sebok:1979}
and \cite{valdes:1982}. If detections can be classified as either a star
%(s) or a galaxy (g), then the {\em a posteriori} probability for having a
%star, given the observed vector of pixel values $\vec{I}$ is given by the Bayes
%theorem:
%\begin{equation}
%P(s|\vec{I}) = \frac{P(\vec{I}|s)P(s)}{P(\vec{I}|s)P(s)+p(\vec{I}|g)P(g)},
%\end{equation}
%that is,
%\begin{equation}
%P(s|\vec{I}) = \frac{1}{1+\frac{P(\vecs{I}|g)}{P(\vecs{I}|s)}\frac{P(g)}{P(s)}}.
%\end{equation}
%The probability for the detected object to be a star $p(s|\vec{I})$ depends
%on both the likelihood ratio $P(\vec{I}|g)/P(\vec{I}|s)$, and the ratio of
%{\em a priori} $P(g)/P(s)$. If we make the assumption that the measurement
%noise at pixel $i$ is additive, Gaussian with \index{mean} mean 0 and \index{standard deviation} standard deviation
%$\sigma_i$, and statistically independent from its \index{neighbour} \index{neighbours} neighbours, then we have
%\begin{equation}
%P(\vec{I}|s) = \prod_i \frac{1}{\sqrt{2\pi}\sigma_i}
%		\exp -\frac{(I_i - S_i)^2}{2\sigma^2_i}
%\end{equation}
%and
%\begin{equation}
%P(\vec{I}|g) = \prod_i \frac{1}{\sqrt{2\pi}\sigma_i}
%		\exp -\frac{(I_i - G_i)^2}{2\sigma^2_i}
%\end{equation}
%where the $S_i$ and $G_i$ are the pixel values for the best-fitting galaxy and
%star \index{mode} models, respectively.