Commit 2e62b5e8 authored by Emmanuel Bertin's avatar Emmanuel Bertin
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Doc: updated WIN measurement section.

parent db3e61fd
......@@ -9,14 +9,12 @@ Isophotal
Measurements are made on the isophotal object footprints, which are defined on the filtered detection image. Only pixels with values above the threshold set with ``ANALYSIS_THRESH`` are considered [#thresh]_, which makes the analysis extremely fast, but obviously strongly dependent on the threshold itself. This is an issue particularly when the amplitude of the bakground noise varies over the image. Many of the isophotal measurements (e.g., ``X_IMAGE``, ``Y_IMAGE``, ``FLUX_ISO``) are necessary for the internal operations of |SExtractor| and are therefore executed even if they are not requested.
Full
Measurements have access to all pixels of the image. These measurements are generally more sophisticated, less affected by the detection threshold, and still reasonably fast. They are done at a later stage of the processing, after CLEANing and MASKing.
Measurements have access to all pixels of the image. These measurements are generally more sophisticated, less affected by variable biases induced by the detection threshold, and still reasonably fast. They are done at a later stage of the processing, after CLEANing and MASKing.
Model-fitting
Measurements require PSF models [#psf_models]_. They are often the most accurate and can recover the flux of saturated objects. They are also much slower, allowing typically only a few tens of objects to be processed every second.
.. toctree::
:numbered:
:maxdepth: 2
Position
PositionWin
......
.. File Position.rst
.. _position_iso:
Position and shape parameters derived from the isophotal profile
================================================================
......@@ -89,6 +91,8 @@ These expressions are more subject to roundoff errors than if the
however kept to a negligible value by measuring all positions relative
here again to XMIN and YMIN.
.. _shape_iso:
Basic shape parameters: A, B, THETA
-----------------------------------
......@@ -159,12 +163,12 @@ A and B can be computed directly from the 2nd-order moments, using the
following equations derived from :eq:`varproj` after some algebra:
.. math::
:label: aimage2
:label: aimage_2
\begin{aligned}
{\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},\\
{\tt B}^2 & = & \frac{\overline{x^2}+\overline{y^2}}{2}
{\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}.\end{aligned}
Note that A and B are exactly halves the :math:`a` and :math:`b`
......@@ -174,6 +178,8 @@ as the semi-major and semi-minor axes of an elliptical shape with constant
surface brightness, which would have the same 2nd-order moments as the
analyzed object.
.. _ellipse_iso:
Ellipse parameters: CXX, CYY, CXY
---------------------------------
......@@ -229,6 +235,8 @@ These parameters [#elongation]_ are directly derived from A and B:
{\tt ELONGATION} & = & \frac{\tt A}{\tt B}\ \ \ \ \ \mbox{and}\\
{\tt ELLIPTICITY} & = & 1 - \frac{\tt B}{\tt A}.\end{aligned}
.. _poserr:
Position uncertainties: ERRX2, ERRY2, ERRXY, ERRA, ERRB, ERRTHETA, ERRCXX, ERRCYY, ERRCXY
-----------------------------------------------------------------------------------------
......
......@@ -3,28 +3,25 @@
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 |
+----------------------------------------+-------------------------------------------------+
Measurements performed through a *window* function (an *envelope*) do not have many of the drawbacks of isophotal measurements. |SExtractor| implements “windowed” versions for most
of the measurements described in the :ref:`previous section<position_iso>`:
.. note::
Unless otherwise noted, all parameter names given below are only prefixes. They must be followed by _IMAGE if the results shall be expressed in pixel coordinates or _WORLD, _SKY, _J2000 or _B1950 for |WCS|_ coordinates (see :ref:`coord_suffix`).
+----------------------+-------------------------------+
| Isophotal parameters | Equivalent windowed parameters|
+======================+===============================+
| X, Y | XWIN, YWIN |
+----------------------+-------------------------------+
| ERRA, ERRB, ERRTHETA | ERRAWIN, ERRBWIN, ERRTHETAWIN |
+----------------------+-------------------------------+
| A, B, THETA | AWIN, BWIN, THETAWIN |
+----------------------+-------------------------------+
| X2, Y2, XY | X2WIN, Y2WIN, XYWIN |
+----------------------+-------------------------------+
| CXX, CYY, CXY | CXXWIN, CYYWIN, CXYWIN |
+----------------------+-------------------------------+
The computations involved are roughly the same except that the pixel
values are integrated within a circular Gaussian window as opposed to
......@@ -33,6 +30,8 @@ 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.
.. _xywin:
Windowed centroid: XWIN, YWIN
-----------------------------
......@@ -44,9 +43,9 @@ each iteration :math:`t`, :math:`\overline{x_{\tt WIN}}` and
:math:`\overline{y_{\tt WIN}}` are refined using:
.. math::
:label: xywin
\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)})}
......@@ -59,11 +58,15 @@ each iteration :math:`t`, :math:`\overline{x_{\tt WIN}}` and
where
.. math:: w_i^{(t)} = \exp \left(-\frac{r_i^{(t)^2}}{2s_{\tt WIN}^2} \right),
.. math::
:label: wi_t
w_i^{(t)} = \exp \left(-\frac{r_i^{(t)^2}}{2s_{\tt WIN}^2} \right),
with
.. math::
:label: ri_t
r_i^{(t)} = \sqrt{\left(x_i - \overline{x_{\tt WIN}}^{(t)}\right)^2 + \left(y_i
- \overline{y_{\tt WIN}}^{(t)}\right)^2}
......@@ -81,7 +84,7 @@ 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]_.
the theoretical limit set by image noise [#win_accuracy]_.
.. _fig_xwinprec:
......@@ -99,10 +102,10 @@ the theoretical limit set by image noise [1]_.
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:
Windowed second-order moments are computed on the image data once the :ref:`centering process <xywin>` has converged:
.. math::
:label: x2y2win
\begin{aligned}
{\tt X2WIN} & = \overline{x_{\tt WIN}^2}
......@@ -119,21 +122,54 @@ centering process from §[chap:wincent] has converged:
Windowed second-order moments are typically twice smaller than their
isophotal equivalent.
Windowed shape parameters: AWIN, BWIN, THETAWIN
-----------------------------------------------
They are computed from the windowed 2nd order moments exactly the same
way as in :eq:`theta0_3` and :eq:`aimage_2` from the :ref:`isophotal shape parameter<shape_iso>` section:
.. math::
:label: shapewin
\begin{aligned}
{\tt AWIN}^2 & = & \frac{\overline{x_{\tt WIN}^2}+\overline{y_{\tt WIN}^2}}{2}
+ \sqrt{\left(\frac{\overline{x_{\tt WIN}^2}-\overline{y_{\tt WIN}^2}}{2}\right)^2 + \overline{xy_{\tt WIN}}^2},\\
{\tt BWIN}^2 & = & \frac{\overline{x_{\tt WIN}^2}+\overline{y_{\tt WIN}^2}}{2}
- \sqrt{\left(\frac{\overline{x_{\tt WIN}^2}-\overline{y_{\tt WIN}^2}}{2}\right)^2 + \overline{xy_{\tt WIN}}^2},\\
\tan (2\,{\tt THETAWIN}) & = & 2 \frac{\overline{xy_{\tt WIN}}}{\overline{x_{\tt WIN}^2} - \overline{y_{\tt WIN}^2}}.
\end{aligned}
Windowed ellipse parameters: CXXWIN, CYYWIN, CXYWIN
---------------------------------------------------
They are computed from the windowed 2nd order moments exactly the same
way as in §[chap:cxx].
way as in :eq:`ellipse_2` describing the :ref:`isophotal ellipse parameters<ellipse_iso>`:
.. math::
:label: ellipsewin_2
\begin{aligned}
{\tt CXXWIN} & = & \frac{\cos^2 {\tt THETAWIN}}{{\tt AWIN}^2} + \frac{\sin^2
{\tt THETAWIN}}{{\tt BWIN}^2} =
\frac{\overline{y_{\tt WIN}^2}}{\overline{x_{\tt WIN}^2} \overline{y_{\tt WIN}^2} - \overline{xy_{\tt WIN}}^2}\\
{\tt CYYWIN} & = & \frac{\sin^2 {\tt THETAWIN}}{{\tt
AWIN}^2} + \frac{\cos^2 {\tt THETAWIN}}{{\tt BWIN}^2} =
\frac{\overline{x_{\tt WIN}^2}}{\overline{x_{\tt WIN}^2} \overline{y_{\tt WIN}^2} - \overline{xy_{\tt WIN}}^2}\\
{\tt CXYWIN} & = & 2 \,\cos {\tt THETAWIN}\,\sin {\tt
THETAWIN} \left( \frac{1}{{\tt AWIN}^2} - \frac{1}{{\tt BWIN}^2}\right) = -2\,
\frac{\overline{xy_{\tt WIN}}}{\overline{x_{\tt WIN}^2} \overline{y_{\tt WIN}^2} - \overline{xy_{\tt WIN}}^2}\end{aligned}
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
centering process of the :ref:`windowed centroid <xywin>` has converged. Assuming that
noise is uncorrelated among pixels, standard error propagation applied
to ([eq:xwin]) and ([eq:xwin]) gives us:
to :eq:`xywin` writes:
.. math::
:label: errwin
\begin{aligned}
{\tt ERRX2WIN} & = {\rm var}(\overline{x_{\tt WIN}})
......@@ -153,11 +189,9 @@ 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]).
similarly to the :ref:`isophotal error ellipse <poserr>`.
.. [1]
see http://www.astromatic.net/forum/showthread.php?tid=581
.. [#win_accuracy] See http://www.astromatic.net/forum/showthread.php?tid=581 .
.. include:: keys.rst
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