.. File PositionWin.rst Windowed positional parameters ============================== Parameters measured within an object’s isophotal limit are sensitive to two main factors: 1) changes in the detection threshold, which create a variable bias and 2) irregularities in the object’s isophotal boundaries, which act as additional “noise” in the measurements. Measurements performed through a *window* function (an *envelope*) do not have such drawbacks. |SExtractor| implements “windowed” versions for most of the measurements described in [chap:isoparam]: +----------------------------------------+-------------------------------------------------+ | Isophotal parameters | Equivalent windowed parameters | +========================================+=================================================+ | X_IMAGE, Y_IMAGE | XWIN_IMAGE, YWIN_IMAGE | +----------------------------------------+-------------------------------------------------+ | ERRA_IMAGE, ERRB_IMAGE, ERRTHETA_IMAGE | ERRAWIN_IMAGE, ERRBWIN_IMAGE, ERRTHETAWIN_IMAGE | +----------------------------------------+-------------------------------------------------+ | A_IMAGE, B_IMAGE, THETA_IMAGE | AWIN_IMAGE, BWIN_IMAGE, THETAWIN_IMAGE | +----------------------------------------+-------------------------------------------------+ | X2_IMAGE, Y2_IMAGE, XY_IMAGE | X2WIN_IMAGE, Y2WIN_IMAGE, XYWIN_IMAGE | +----------------------------------------+-------------------------------------------------+ | CXX_IMAGE, CYY_IMAGE, CXY_IMAGE | CXXWIN_IMAGE, CYYWIN_IMAGE, CXYWIN_IMAGE | +----------------------------------------+-------------------------------------------------+ The computations involved are roughly the same except that the pixel values are integrated within a circular Gaussian window as opposed to the object’s isophotal footprint. The Gaussian window is scaled to each object; its FWHM is the diameter of the disk that contains half of the object flux (:math:`d_{50}`). Note that in double-image mode (§[chap:using]) the window is scaled based on the *measurement* image. Windowed centroid: XWIN, YWIN ----------------------------- This is an iterative process. The computation starts by initializing the windowed centroid coordinates :math:`\overline{x_{\tt WIN}}^{(0)}` and :math:`\overline{y_{\tt WIN}}^{(0)}` to their basic :math:`\overline{x}` and :math:`\overline{y}` isophotal equivalents, respectively. Then at each iteration :math:`t`, :math:`\overline{x_{\tt WIN}}` and :math:`\overline{y_{\tt WIN}}` are refined using: .. math:: \begin{aligned} \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{aligned} where .. math:: w_i^{(t)} = \exp \left(-\frac{r_i^{(t)^2}}{2s_{\tt WIN}^2} \right), with .. math:: r_i^{(t)} = \sqrt{\left(x_i - \overline{x_{\tt WIN}}^{(t)}\right)^2 + \left(y_i - \overline{y_{\tt WIN}}^{(t)}\right)^2} and :math:`s_{\tt WIN} = d_{50} / \sqrt{8 \ln 2}`. The process stops when the change in position between two iterations is less than :math:`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 , it is recommended to use whenever possible windowed parameters instead of their isophotal equivalents, since the measurements they provide are much more precise (:numref:`fig_xwinprec`). The precision in centroiding offered by XWIN_IMAGE and YWIN_IMAGE is actually very close to that of PSF-fitting on focused and properly sampled star images, and can also be applied to galaxies. It has been verified that for isolated, Gaussian-like PSFs, its accuracy is close to the theoretical limit set by image noise [1]_. .. _fig_xwinprec: .. figure:: figures/xwinprec.* :figwidth: 100% :align: center Comparison between isophotal and windowed centroid measurement residuals on simulated, background noise-limited images. *Left*: histogram of the difference between X_IMAGE and the true centroid in x. *Right*: histogram of the difference between XWIN_IMAGE and the true centroid in x. Windowed 2nd order moments: X2, Y2, XY -------------------------------------- Windowed second-order moments are computed on the image data once the centering process from §[chap:wincent] has converged: .. math:: \begin{aligned} {\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{aligned} Windowed second-order moments are typically twice smaller than their isophotal equivalent. Windowed ellipse parameters: CXXWIN, CYYWIN, CXYWIN --------------------------------------------------- They are computed from the windowed 2nd order moments exactly the same way as in §[chap:cxx]. Windowed position uncertainties: ERRX2WIN, ERRY2WIN, ERRXYWIN, ERRAWIN, ERRBWIN, ERRTHETAWIN, ERRCXXWIN, ERRCYYWIN, ERRCXYWIN ----------------------------------------------------------------------------------------------------------------------------- Windowed position uncertainties are computed on the image data once the centering process from §[chap:wincent] has converged. Assuming that noise is uncorrelated among pixels, standard error propagation applied to ([eq:xwin]) and ([eq:xwin]) gives us: .. math:: \begin{aligned} {\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{aligned} The semi-major axis ERRAWIN, semi-minor axis ERRBWIN, and position angle ERRTHETAWIN of the :math:`1\sigma` position error ellipse are computed from the covariance matrix elements :math:`{\rm var}(\overline{x_{\tt WIN}})`, :math:`{\rm var}(\overline{y_{\tt WIN}})`, :math:`{\rm cov}(\overline{x_{\tt WIN}},\overline{y_{\tt WIN}})`, exactly as in §[chap:poserr]: see eqs. ([eq:erra]), ([eq:errb]), ([eq:errtheta]), ([eq:errcxx]), ([eq:errcyy]) and ([eq:errcxy]). .. [1] see http://www.astromatic.net/forum/showthread.php?tid=581 .. include:: keys.rst