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\section{Windowed positional parameters}
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\section{Windowed positional parameters}
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\label{chap:winparam}
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\label{chap:winparam}
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Parameters measured within an object's isophotal limit are sensitive to
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Parameters measured within an object's isophotal limit are sensitive to
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two main factors: 1) changes in the detection threshold, which
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two main factors: 1) changes in the detection \index{threshold} threshold, which
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create a variable bias and 2) irregularities in the object's
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create a variable bias and 2) irregularities in the object's
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isophotal boundaries, which act as additional ``noise'' in the measurements.
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isophotal \index{boundaries} boundaries, which act as additional ``noise'' in the measurements.
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Measurements performed through a {\em window} function (an {\em envelope}) do
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Measurements performed through a {\em \index{window} window} function (an {\em envelope}) do
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not have such drawbacks. {\sc SExtractor} versions 2.4 and above implement
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not have such drawbacks. {\sc SExtractor} versions 2.4 and above implement
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``windowed'' versions for most of the measurements described in
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``windowed'' versions for most of the measurements described in
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\ref{chap:isoparam}:
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\ref{chap:isoparam}:
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{\small
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{\small
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\begin{tabular}{ll}
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\begin{tabular}{ll}
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Isophotal parameters & Equivalent windowed parameters \\
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Isophotal parameters & Equivalent \index{window} \index{windowed} windowed parameters \\
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\hline
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\hline
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{\tt X\_IMAGE}, {\tt Y\_IMAGE} &
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{\tt X\_IMAGE}, {\tt Y\_IMAGE} &
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{\tt XWIN\_IMAGE}, {\tt YWIN\_IMAGE}\\
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{\tt XWIN\_IMAGE}, {\tt YWIN\_IMAGE}\\
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{\tt ERRA\_IMAGE}, {\tt ERRB\_IMAGE}, {\tt ERRTHETA\_IMAGE} &
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{\tt ERRA\_IMAGE}, {\tt ERRB\_IMAGE}, {\tt ERRTHETA\_IMAGE} &
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{\tt ERRAWIN\_IMAGE}, {\tt ERRBWIN\_IMAGE}, {\tt ERRTHETAWIN\_IMAGE}\\
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{\tt ERRAWIN\_IMAGE}, {\tt ERRBWIN\_IMAGE}, {\tt ERRTHETAWIN\_IMAGE}\\
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{\tt CXXWIN\_IMAGE}, {\tt CYYWIN\_IMAGE}, {\tt CXYWIN\_IMAGE}\\
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{\tt CXXWIN\_IMAGE}, {\tt CYYWIN\_IMAGE}, {\tt CXYWIN\_IMAGE}\\
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\hline
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\hline
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\end{tabular}}
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\end{tabular}}
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The computations involved are roughly the same except that the pixel values are
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The computations involved are roughly the same except that the pixel values are
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integrated within a circular Gaussian window as opposed to the object's
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integrated within a circular Gaussian \index{window} window as opposed to the object's
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isophotal footprint.
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isophotal footprint.
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The Gaussian window is scaled to each object; its FWHM is the diameter of the
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The Gaussian \index{window} window is scaled to each object; its \index{FWHM} FWHM is the diameter of the
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disk that contains half of the object flux ($d_{50}$). Note that in double-image
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disk that contains half of the object flux ($d_{50}$). Note that in double-image
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mode (\S\ref{chap:using}) the window is scaled based on the {\em measurement}
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\index{mode} mode (\S\ref{chap:using}) the \index{window} window is scaled based on the {\em measurement}
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image. \gam{Can we provide the precise sub-section for this?}
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\index{image} image. \gam{Can we provide the precise sub-section for this?}
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\subsection{Windowed centroid: {\tt XWIN}, {\tt YWIN}}
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\subsection{Windowed \index{centroid} centroid: {\tt XWIN}, {\tt YWIN}}
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\label{chap:wincent}
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\label{chap:wincent}
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This is an iterative process. The computation starts by initializing the
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This is an iterative process. The computation starts by initializing the
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windowed centroid coordinates $\overline{x_{\tt WIN}}^{(0)}$ and
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\index{window} \index{windowed} windowed \index{centroid} centroid coordinates $\overline{x_{\tt WIN}}^{(0)}$ and
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$\overline{y_{\tt WIN}}^{(0)}$ to their basic $\overline{x}$ and $\overline{y}$
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$\overline{y_{\tt WIN}}^{(0)}$ to their basic $\overline{x}$ and $\overline{y}$
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isophotal equivalents, respectively.
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isophotal equivalents, respectively.
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\gam{I understand that the notation {\tt XXX} describes a {\sc SExtractor}
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\gam{I understand that the notation {\tt XXX} describes a {\sc SExtractor}
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parameter. But then, {\tt XWIN}$^{(t)}$ is abusive, since the output
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parameter. But then, {\tt XWIN}$^{(t)}$ is abusive, since the output
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parameter is only that of the last iteration. Otherwise, it might be
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parameter is only that of the last iteration. Otherwise, it might be
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The process stops when the change in position between two iterations is less
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The process stops when the change in position between two iterations is less
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than $2\times10^{-4}$ pixel, a condition which is generally achieved in about 3 to 5
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than $2\times10^{-4}$ pixel, a condition which is generally achieved in about 3 to 5
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iterations.
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iterations.
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Although the iterative nature of the processing slows down the processing \hide{a bit},
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Although the iterative nature of the processing slows down the processing \hide{a bit},
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it is recommended to use whenever possible windowed parameters instead of their
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it is recommended to use whenever possible \index{window} \index{windowed} windowed parameters instead of their
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isophotal equivalents, since the measurements they provide
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isophotal equivalents, since the measurements they provide
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are much more precise (Fig. \ref{fig:winpres}). The precision in centroiding
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are much more precise (Fig. \ref{fig:winpres}). The precision in \index{centroid} centroiding
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offered by {\tt XWIN\_IMAGE} and {\tt YWIN\_IMAGE} is actually very close to
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offered by {\tt XWIN\_IMAGE} and {\tt YWIN\_IMAGE} is actually very close to
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that of PSF-fitting on focused and properly sampled star images, and can also
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that of \index{PSF} PSF-fitting on focused and properly sampled star \index{image} images, and can also
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be applied to galaxies. It has been verified that for isolated,
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be applied to galaxies. It has been verified that for isolated,
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Gaussian-like PSFs, its accuracy is close to the theoretical limit set by
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Gaussian-like \index{PSF} PSFs, its accuracy is close to the theoretical limit set by
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image noise\footnote{see
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\index{image} image noise\footnote{see
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{\tt http://www.astromatic.net/forum/showthread.php?tid=581}}.
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{\tt http://www.astromatic.net/forum/showthread.php?tid=581}}.
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%------------------------------ Fig. winpres -----------------------------
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%------------------------------ Fig. winpres -----------------------------
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\begin{figure}[htbp]
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\begin{figure}[htbp]
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\centerline{\includegraphics[width=8cm]{ps/sex_xpres.ps}
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\centerline{\includegraphics[width=8cm]{ps/sex_xpres.ps}
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\includegraphics[width=8cm]{ps/sex_xw2pres.ps}}
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\includegraphics[width=8cm]{ps/sex_xw2pres.ps}}
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\caption{Comparison between isophotal and windowed centroid measurement
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\caption{Comparison between isophotal and \index{window} \index{windowed} windowed \index{centroid} centroid measurement
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accuracies on simulated, background noise-limited images.{\em Left}: histogram
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accuracies on simulated, background noise-limited \index{image} images.{\em Left}: histogram
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of the difference between {\tt X\_IMAGE} and the simulation centroid in x.
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of the difference between {\tt X\_IMAGE} and the simulation \index{centroid} centroid in x.
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{\em Right}: histogram of the difference between {\tt XWIN\_IMAGE} and the
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{\em Right}: histogram of the difference between {\tt XWIN\_IMAGE} and the
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simulation centroid in x.}
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simulation \index{centroid} centroid in x.}
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\label{fig:winpres}
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\label{fig:winpres}
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\end{figure}
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\end{figure}
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\subsection{Windowed 2nd order moments: {\tt X2}, {\tt Y2}, {\tt XY}}
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\subsection{Windowed 2nd order \index{moments} moments: {\tt X2}, {\tt Y2}, {\tt XY}}
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Windowed second-order moments are computed on the image data once the centering
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Windowed second-order \index{moments} moments are computed on the \index{image} image data once the centering
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process from \S{\ref{chap:wincent}} has converged:
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process from \S{\ref{chap:wincent}} has converged:
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\begin{eqnarray}
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\begin{eqnarray}
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{\tt X2WIN} & = \overline{x_{\tt WIN}^2}
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{\tt X2WIN} & = \overline{x_{\tt WIN}^2}
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= & \frac{\sum_{r_i < r_{\rm max}} w_i I_i (x_i - \overline{x_{\tt WIN}})^2}
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= & \frac{\sum_{r_i < r_{\rm max}} w_i I_i (x_i - \overline{x_{\tt WIN}})^2}
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{\sum_{r_i < r_{\rm max}} w_i I_i},\\
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{\sum_{r_i < r_{\rm max}} w_i I_i},\\
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| Line 111... |
Line 111... |
{\tt XYWIN} & = \overline{xy_{\tt WIN}}
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{\tt XYWIN} & = \overline{xy_{\tt WIN}}
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= & \frac{\sum_{r_i < r_{\rm max}} w_i I_i (x_i - \overline{x_{\tt WIN}})
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= & \frac{\sum_{r_i < r_{\rm max}} w_i I_i (x_i - \overline{x_{\tt WIN}})
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(y_i - \overline{y_{\tt WIN}})}
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(y_i - \overline{y_{\tt WIN}})}
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{\sum_{r_i < r_{\rm max}} w_i I_i}.
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{\sum_{r_i < r_{\rm max}} w_i I_i}.
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\end{eqnarray}
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\end{eqnarray}
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Windowed second-order moments are typically twice smaller than their isophotal
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Windowed second-order \index{moments} moments are typically twice smaller than their isophotal
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equivalent.
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equivalent.
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\subsection{Windowed ellipse parameters:
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\subsection{Windowed ellipse parameters:
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{\tt CXXWIN}, {\tt CYYWIN}, {\tt CXYWIN}}
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{\tt CXXWIN}, {\tt CYYWIN}, {\tt CXYWIN}}
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They are computed from the windowed 2nd order moments exactly the same way as
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They are computed from the \index{window} \index{windowed} windowed 2nd order \index{moments} moments exactly the same way as
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in \S\ref{chap:cxx}.
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in \S\ref{chap:cxx}.
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\subsection{Windowed position errors: {\tt ERRX2WIN}, {\tt ERRY2WIN},
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\subsection{Windowed position errors: {\tt ERRX2WIN}, {\tt ERRY2WIN},
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{\tt ERRXYWIN}, {\tt ERRAWIN}, {\tt ERRBWIN}, {\tt ERRTHETAWIN},
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{\tt ERRXYWIN}, {\tt ERRAWIN}, {\tt ERRBWIN}, {\tt ERRTHETAWIN},
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{\tt ERRCXXWIN}, {\tt ERRCYYWIN}, {\tt ERRCXYWIN}}
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{\tt ERRCXXWIN}, {\tt ERRCYYWIN}, {\tt ERRCXYWIN}}
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Windowed position errors are computed on the image data once the centering
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Windowed position errors are computed on the \index{image} image data once the centering
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process from \S{\ref{chap:wincent}} has converged. Assuming that noise is
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process from \S{\ref{chap:wincent}} has converged. Assuming that noise is
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uncorrelated among pixels, standard error propagation applied to
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uncorrelated among pixels, standard error propagation applied to
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(\ref{eq:xwin}) and (\ref{eq:xwin}) gives us:
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(\ref{eq:xwin}) and (\ref{eq:xwin}) gives us:
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\begin{eqnarray}
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\begin{eqnarray}
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{\tt ERRX2WIN} & = {\rm var}(\overline{x_{\tt WIN}})
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{\tt ERRX2WIN} & = {\rm var}(\overline{x_{\tt WIN}})
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| Line 139... |
Line 139... |
= & 4\,\frac{\sum_{r_i < r_{\rm max}}
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= & 4\,\frac{\sum_{r_i < r_{\rm max}}
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w_i^2 \sigma^2_i (x_i-\overline{x_{\tt WIN}})(y_i-\overline{y_{\tt WIN}})}
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w_i^2 \sigma^2_i (x_i-\overline{x_{\tt WIN}})(y_i-\overline{y_{\tt WIN}})}
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{\left(\sum_{r_i < r_{\rm max}} w_i I_i\right)^2}.
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{\left(\sum_{r_i < r_{\rm max}} w_i I_i\right)^2}.
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\end{eqnarray}
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\end{eqnarray}
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The semi-major axis {\tt ERRAWIN}, semi-minor
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The \index{semi-major} semi-major axis {\tt ERRAWIN}, semi-minor
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axis {\tt ERRBWIN}, and position angle {\tt ERRTHETAWIN} of the
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axis {\tt ERRBWIN}, and position angle {\tt ERRTHETAWIN} of the
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$1\sigma$ position error ellipse are computed from the covariance
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$1\sigma$ position \index{error ellipse} error ellipse are computed from the \index{covariance} covariance
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matrix elements ${\rm var}(\overline{x_{\tt WIN}})$,
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matrix elements ${\rm var}(\overline{x_{\tt WIN}})$,
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${\rm var}(\overline{y_{\tt WIN}})$,
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${\rm var}(\overline{y_{\tt WIN}})$,
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${\rm cov}(\overline{x_{\tt WIN}},\overline{y_{\tt WIN}})$,
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${\rm cov}(\overline{x_{\tt WIN}},\overline{y_{\tt WIN}})$,
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exactly as in \S\ref{chap:poserr}: see eqs. (\ref{eq:erra}),
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exactly as in \S\ref{chap:poserr}: see eqs. (\ref{eq:erra}),
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(\ref{eq:errb}), (\ref{eq:errtheta}), (\ref{eq:errcxx}), (\ref{eq:errcyy})
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(\ref{eq:errb}), (\ref{eq:errtheta}), (\ref{eq:errcxx}), (\ref{eq:errcyy})
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and (\ref{eq:errcxy}).
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and (\ref{eq:errcxy}).
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%\section{2D-model fitting}
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%\section{2D-model fitting}
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%\subsection{Star/galaxy separation}
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%\subsection{Star/galaxy separation}
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%With the local PSF and a noise model in hand, one can easily derive an optimum
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%With the local \index{PSF} PSF and a noise \index{mode} model in hand, one can easily derive an optimum
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%star/galaxy classifier. The problem was first addressed by \cite{sebok:1979}
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%star/galaxy classifier. The problem was first addressed by \cite{sebok:1979}
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and \cite{valdes:1982}. If detections can be classified as either a star
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and \cite{valdes:1982}. If detections can be classified as either a star
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%(s) or a galaxy (g), then the {\em a posteriori} probability for having a
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%(s) or a galaxy (g), then the {\em a posteriori} probability for having a
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%star, given the observed vector of pixel values $\vec{I}$ is given by the Bayes
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%star, given the observed vector of pixel values $\vec{I}$ is given by the Bayes
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%theorem:
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%theorem:
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| Line 167... |
Line 167... |
%P(s|\vec{I}) = \frac{1}{1+\frac{P(\vecs{I}|g)}{P(\vecs{I}|s)}\frac{P(g)}{P(s)}}.
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%P(s|\vec{I}) = \frac{1}{1+\frac{P(\vecs{I}|g)}{P(\vecs{I}|s)}\frac{P(g)}{P(s)}}.
|
%\end{equation}
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%\end{equation}
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%The probability for the detected object to be a star $p(s|\vec{I})$ depends
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%The probability for the detected object to be a star $p(s|\vec{I})$ depends
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%on both the likelihood ratio $P(\vec{I}|g)/P(\vec{I}|s)$, and the ratio of
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%on both the likelihood ratio $P(\vec{I}|g)/P(\vec{I}|s)$, and the ratio of
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%{\em a priori} $P(g)/P(s)$. If we make the assumption that the measurement
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%{\em a priori} $P(g)/P(s)$. If we make the assumption that the measurement
|
%noise at pixel $i$ is additive, Gaussian with mean 0 and standard deviation
|
%noise at pixel $i$ is additive, Gaussian with \index{mean} mean 0 and \index{standard deviation} standard deviation
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%$\sigma_i$, and statistically independent from its neighbours, then we have
|
%$\sigma_i$, and statistically independent from its \index{neighbour} \index{neighbours} neighbours, then we have
|
%\begin{equation}
|
%\begin{equation}
|
%P(\vec{I}|s) = \prod_i \frac{1}{\sqrt{2\pi}\sigma_i}
|
%P(\vec{I}|s) = \prod_i \frac{1}{\sqrt{2\pi}\sigma_i}
|
% \exp -\frac{(I_i - S_i)^2}{2\sigma^2_i}
|
% \exp -\frac{(I_i - S_i)^2}{2\sigma^2_i}
|
%\end{equation}
|
%\end{equation}
|
%and
|
%and
|
%\begin{equation}
|
%\begin{equation}
|
%P(\vec{I}|g) = \prod_i \frac{1}{\sqrt{2\pi}\sigma_i}
|
%P(\vec{I}|g) = \prod_i \frac{1}{\sqrt{2\pi}\sigma_i}
|
% \exp -\frac{(I_i - G_i)^2}{2\sigma^2_i}
|
% \exp -\frac{(I_i - G_i)^2}{2\sigma^2_i}
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%\end{equation}
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%\end{equation}
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%where the $S_i$ and $G_i$ are the pixel values for the best-fitting galaxy and
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%where the $S_i$ and $G_i$ are the pixel values for the best-fitting galaxy and
|
%star models, respectively.
|
%star \index{mode} models, respectively.
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No newline at end of file
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No newline at end of file
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