From 2d837eeab8a46abd6e64a3cfd070a3373ade78b9 Mon Sep 17 00:00:00 2001 From: Emmanuel Bertin Date: Tue, 31 Oct 2017 23:29:58 +0100 Subject: [PATCH] Added photometry section (not yet finalized). --- doc/src/Measurements.rst | 1 + doc/src/Photom.rst | 241 +++++++++++++++++++++++++++++++++++++++ doc/src/Position.rst | 6 +- doc/src/references.bib | 13 +++ 4 files changed, 258 insertions(+), 3 deletions(-) create mode 100644 doc/src/Photom.rst diff --git a/doc/src/Measurements.rst b/doc/src/Measurements.rst index ec22d91..781db59 100644 --- a/doc/src/Measurements.rst +++ b/doc/src/Measurements.rst @@ -18,6 +18,7 @@ processing (after CLEANing and MASKing). Position PositionWin + Photom .. include:: keys.rst diff --git a/doc/src/Photom.rst b/doc/src/Photom.rst new file mode 100644 index 0000000..af20268 --- /dev/null +++ b/doc/src/Photom.rst @@ -0,0 +1,241 @@ +.. File Photom.rst + +Photometry +========== + +Besides PSF and model-fitting flux estimates, |SExtractor| can currently +perform four types of flux measurements: isophotal, *corrected-isophotal*, +fixed-aperture and *adaptive-aperture*. For every ``FLUX`` measurement, +an error estimate ``FLUXERR``, a magnitude ``MAG`` and a magnitude +error estimate ``MAGERR`` are also available. +The ``MAG_ZEROPOINT`` configuration parameter sets the magnitude zero-point +of magnitudes: + +.. math:: + :label: mag + + {\tt MAG} = \mathrm{MAG\_ZEROPOINT} - 2.5 \log_{10} {\tt FLUX} + +Magnitude uncertainties (error estimates) are computed using + +.. math:: + :label: magerr + + {\tt MAGERR} = \frac{2.5}{\ln 10}\frac{\tt FLUXERR}{\tt FLUX} + +Isophotal flux +-------------- + +``FLUX_ISO`` is computed simply by integrating pixels values within the +detection footprint, with the additional constraint that the +background-subtracted, filtered value of detection image pixels must exceed +the threshold set with the ``ANALYSIS_THRESH`` configuration parameter. + +.. math:: + :label: fluxiso + + {\tt FLUX\_ISO} = \sum_{i \in {\cal S}} I_i + + +Corrected isophotal magnitudes +------------------------------ + +``MAG_ISOCOR`` can be considered as a quick-and-dirty way for retrieving +the fraction of flux lost by isophotal magnitudes. Although their use is +now deprecated, they have been kept in |SExtractor| v2.x and above for +compatibility with |SExtractor| v1. If we make the assumption that the +intensity profiles of the faint objects recorded in the frame are +roughly Gaussian because of atmospheric blurring, then the fraction +:math:`\eta = \frac{I_{\rm iso}}{I_{\rm tot}}` of the total flux enclosed +within a particular isophote reads :cite:`1990MNRAS.246..433M`: + +.. math:: + :label: isocor + + \left(1-\frac{1}{\eta}\right ) \ln (1-\eta) = \frac{A\,t}{I_{\rm iso}} + +where :math:`A` is the area and :math:`t` the threshold related to this +isophote. :eq:isocor is not analytically invertible, but a good +approximation to :math:`\eta` (error :math:`< 10^{-2}` for :math:`\eta > +0.4`) can be done with the second-order polynomial fit: + +.. math:: + :label: isocor2 + + \eta \approx 1 - 0.1961 \frac{A\,t}{I_{\rm iso}} - 0.7512 + \left( \frac{A\,t}{I_{\rm iso}}\right)^2 \label{eq:isocor} + +A “total” magnitude ``MAG_ISOCOR`` estimate is then + +.. math:: + :label: magisocor + + {\tt MAG\_ISOCOR} = {\tt MAG\_ISO} + 2.5 \log_{10} \eta + +Clearly this cheap correction works best with stars; and although it is +shown to give tolerably accurate results with most disk galaxies, it +fails with ellipticals because of the broader wings of their profiles. + +Fixed-aperture flux +------------------- + +``FLUX_APER`` estimates the flux above the background within a circular +aperture. The diameter of the aperture in pixels is defined by the +``PHOTOM_APERTURES`` configuration parameter. It does not have to be an +integer: each “normal” pixel is subdivided in :math:`5\times 5` sub-pixels +before measuring the flux within the aperture. If ``FLUX_APER`` is provided as a +vector ``FLUX_APER[n]``, at least :math:`n` apertures must be +specified with ``PHOTOM_APERTURES``. + +Automatic aperture magnitudes +----------------------------- + +(MAG\_AUTO) provides an estimate of the “total magnitude” by integrating +the source flux within an adaptively scaled aperture. SExtractor’s +automatic aperture photometry routine is inspired by Kron’s “first +moment” algorithm (1980). (1) We define an elliptical aperture whose +elongation :math:`\epsilon` and position angle :math:`\theta` are +defined by second order moments of the object’s light distribution. The +ellipse is scaled to :math:`R_{\rm max}.\sigma_{\rm iso}` +(:math:`6 \sigma_{\rm iso}`, which corresponds roughly to 2 isophotal +“radii”). (2) Within this aperture we compute the “first moment”: + +.. math:: r_1 = \frac{\sum r\,I(r)}{\sum I(r)} + +Kron (1980) and Infante (1987) have shown that for stars and galaxy +profiles convolved with Gaussian seeing, :math:`\ge 90\%` of the flux is +expected to lie within a circular aperture of radius :math:`k r_1` if +:math:`k = +2`, almost independently of their magnitude. This picture remains +unchanged if we consider an ellipse with :math:`\epsilon\, k r_1` and +:math:`k r_1 / +\epsilon` as principal axes. :math:`k = 2` defines a sort of balance +between systematic and random errors. By choosing a larger +:math:`k = 2.5`, the mean fraction of flux lost drops from about 10% to +6%. When Signal to Noise is low, it may appear that an erroneously small +aperture is taken by the algorithm. That’s why we have to bound the +smallest accessible aperture to :math:`R_{\rm min}` (typically +:math:`R_{\rm min} = 3 - 4\, +\sigma_{\rm iso}`). The user has full control over the parameters +:math:`k` and :math:`R_{\rm min}` through the configuration parameters +PHOT\_AUTOPARAMS; by default, PHOT\_AUTOPARAMS is set to 2.5,3.5. + +.. figure:: ps/simlostflux.ps + :alt: Flux lost (expressed as a mean magnitude difference) with + different faint-object photometry techniques as a function of total + magnitude (see text). Only isolated galaxies (no blends) of the + simulations have been considered. + :width: 15.00000cm + + Flux lost (expressed as a mean magnitude difference) with different + faint-object photometry techniques as a function of total magnitude + (see text). Only isolated galaxies (no blends) of the simulations + have been considered. + +Aperture magnitudes are sensitive to crowding. In SExtractor 1, +MAG\_AUTO measurements were not very robust in that respect. It was +therefore suggested to replace the aperture magnitude by the +corrected-isophotal one when an object is too close to its neighbours (2 +isophotal radii for instance). This was done automatically when using +the MAG\_BEST magnitude: :math:`{\tt MAG\_BEST} = {\tt MAG\_AUTO}` when +it is sure that no neighbour can bias MAG\_AUTO by more than 10%, or +:math:`{\tt MAG\_BEST} = {\tt MAG\_ISOCOR}` otherwise. Experience showed +that the MAG\_ISOCOR and MAG\_AUTO magnitude would loose about the same +fraction of flux on stars or compact galaxy profiles: around 0.06 % for +default extraction parameters. The use of MAG\_BEST is now deprecated as +MAG\_AUTO measurements are much more robust in versions 2.x of +SExtractor. The first improvement is a crude subtraction of all the +neighbours which have been detected around the measured source (the +MASK\_TYPE BLANK option). The second improvement is an automatic +correction of parts of the aperture that are suspected to be +contaminated by a neighbour. This is done by mirroring the opposite, +cleaner side of the measurement ellipse if available (the MASK\_TYPE +CORRECT option, which is also the default). Figure [figphot] shows the +mean loss of flux measured with isophotal (threshold +:math:`= 24.4\ \mbox{\rm magnitude\,arsec}^{-2}`), corrected isophotal +and automatic aperture photometries for simulated galaxy :math:`B_J` on +a typical Schmidt-survey plate image. The automatic adaptive aperture +photometry leads to the lowest loss of flux. + +Photographic photometry +----------------------- + +In DETECT\_TYPE PHOTO mode, SExtractor assumes that the response of the +detector, over the dynamic range of the image, is logarithmic. This is +generally a good approximation for photographic density on deep +exposures. Photometric procedures described above remain unchanged, +except that for each pixel we apply first the transformation + +.. math:: + + I = I_0\,10^{D/\gamma} \ , + \label{eq:dtoi} + +where :math:`\gamma` (MAG\_GAMMA) is the contrast index of the +emulsion, :math:`D` the original pixel value from the +background-subtracted image, and :math:`I_0` is computed from the +magnitude zero-point :math:`m_0`: + +.. math:: I_0 = \frac{\gamma}{\ln 10} \,10^{-0.4\, m_0} \ . + + One advantage of using a density-to-intensity transformation relative +to the local sky background is that it corrects (to some extent) +large-scale inhomogeneities in sensitivity (see Bertin 1996 for +details). + +Errors on magnitude +------------------- + +An estimate of the error [1]_ is available for each type of magnitude. +It is computed through + +.. math:: \Delta m = 1.0857\, \frac{\sqrt{A\,\sigma^2 + F/g}}{F} + +where :math:`A` is the area (in pixels) over which the total flux +:math:`F` (in ADU) is summed, :math:`\sigma` the standard deviation of +noise (in ADU) estimated from the background, and g the detector gain +(GAIN parameter [2]_ , in :math:`e^- / \mbox{ADU}`). For +corrected-isophotal magnitudes, a term, derived from Eq. [eq:isocor] is +quadratically added to take into account the error on the correction +itself. + +In DETECT\_TYPE PHOTO mode, things are slightly more complex. Making the +assumption that plate-noise is the major contributor to photometric +errors, and that it is roughly constant in density, we can write: + +.. math:: + + \Delta m = 1.0857 \,\ln 10\, {\sigma\over \gamma}\, + \frac{\sqrt{\sum_{x,y}{I^2(x,y)}}}{\sum_{x,y}I(x,y)} + =2.5\,{\sigma\over \gamma}\, + \frac{\sqrt{\sum_{x,y}{I^2(x,y)}}}{\sum_{x,y}I(x,y)} + +where :math:`I(x,y)` is the contribution of pixel :math:`(x,y)` to the +total flux (Eq. [eq:dtoi]). The GAIN is ignored in PHOTO mode. + +Background +---------- + +is the last point relative to photometry. The assumption made in +§[chap:backest] — that the “local” background associated to an object +can be interpolated from the global background map — is no longer valid +in crowded regions. An example is a globular cluster superimposed on a +bulge of galaxy. SExtractor offers the possibility to estimate locally +the background used to compute magnitudes. When this option is switched +on (BACKPHOTO\_TYPE LOCAL instead of GLOBAL), the “photometric” +background is estimated within a “rectangular annulus” around the +isophotal limits of the object. The thickness of the annulus (in pixels) +can be specified by the user with BACKPHOTO\_SIZE. A typical value is +BACKPHOTO\_SIZE=24. + +.. [1] + It is important to note that this error provides a lower limit, since + it does not take into account the (complex) uncertainty on the local + background estimate. + +.. [2] + Setting GAIN to 0 in the configuration file is equivalent to + :math:`g = +\infty` + +.. include:: keys.rst + diff --git a/doc/src/Position.rst b/doc/src/Position.rst index 4c92918..7f73a51 100644 --- a/doc/src/Position.rst +++ b/doc/src/Position.rst @@ -45,9 +45,9 @@ are simply computed as the first order moments of the profile: \begin{aligned} {\tt X} & = & \overline{x} = \frac{\displaystyle \sum_{i \in {\cal S}} - I_i x_i}{\displaystyle \sum_{i \in {\cal S}} I_i},\\ {\tt Y} & = & - \overline{y} = \frac{\displaystyle \sum_{i \in {\cal S}} I_i - y_i}{\displaystyle \sum_{i \in {\cal S}} I_i}. + I_i x_i}{\displaystyle \sum_{i \in {\cal S}} I_i},\\ + {\tt Y} & = & \overline{y} = \frac{\displaystyle \sum_{i \in {\cal S}} + I_i y_i}{\displaystyle \sum_{i \in {\cal S}} I_i}. \end{aligned} In practice, :math:`x_i` and :math:`y_i` are summed relative to XMIN diff --git a/doc/src/references.bib b/doc/src/references.bib index f14f1f1..fc0759f 100644 --- a/doc/src/references.bib +++ b/doc/src/references.bib @@ -42,6 +42,19 @@ adsnote = {Provided by the SAO/NASA Astrophysics Data System} } +@ARTICLE{1990MNRAS.246..433M, + author = {{Maddox}, S.~J. and {Efstathiou}, G. and {Sutherland}, W.~J. + }, + title = "{The APM Galaxy Survey - Part Two - Photometric Corrections}", + journal = {\mnras}, + year = 1990, + month = oct, + volume = 246, + pages = {433}, + adsurl = {http://adsabs.harvard.edu/abs/1990MNRAS.246..433M}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + @ARTICLE{1856MNRAS..17...12P, author = {{Pogson}, N.}, title = "{Magnitudes of Thirty-six of the Minor Planets for the first day of each month -- GitLab