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.