Doc: Added background modeling section.

doc/src/Background.rst

+.. File Background.rst
+
+.. include:: global.rst
+
+Modeling the background
+=======================
+
+On linear detectors, the value measured at each pixel is the sum of a "background" signal and light coming from the sources of interest.
+To be able to detect the faintest objects and make accurate measurements, |SExtractor| needs first computing a precise estimate of the background level at any position of the image: a *background map*.
+Strictly speaking, there should be one background map per source, that is, what would the image look like if that very source was missing.
+But, at least for detection, one can start by assuming that most discrete sources do not overlap too severely — which is generally the case for high galactic latitude fields —, and that the background varies smoothly across the field.
+
+Background estimation
+---------------------
+
+To compute the background map, |SExtractor| makes a first pass through the pixel data, estimating the local background in each mesh of a rectangular grid that covers the whole frame.
+The background estimator is a combination of :math:`\kappa\,\sigma` clipping and mode estimation, similar to Stetson’s |DAOPHOT|_ program :cite:`1987PASP_99_191S,1992ASPC_23_90D`.
+
+Briefly, the local background histogram is clipped iteratively until convergence at :math:`\pm 3\sigma` around its median. The mode of the histogram is estimated using:
+
+.. math::
+  :label: sexbackmode
+
+  \mbox{Mode} = 2.5 \times \mbox{Median} - 1.5 \times \mbox{Mean}.
+
+Using simulated images, the expression above was found more accurate with clipped distributions :cite:1996AAS_117_393B than the usual approximation (e.g., :cite:`stuart2009kendall`):
+
+.. math::
+  :label: ksbackmode
+
+  \mbox{Mode} = 3 \times \mbox{Median} - 2 \times \mbox{Mean}.
+
+:numref:`fig_modevsmean` shows that the mode estimation in :eq:`sexbackmode` is considerably less affected by source crowding than a simple clipped mean :cite:`1981AJ_86_476J,1987AA_183_177I` but it is :math:`\approx 30\%` noisier. 
+Obviously :eq:`sexbackmode` is not valid for any distribution; |SExtractor| falls back to a simple median for estimating the local background value if the mode and the median disagree by more than 30%.
+
+.. _fig_modevsmean:
+
+.. figure:: figures/modevsmean.*
+   :figwidth: 100%
+   :align: center
+
+   Simulations of :math:`32\times32` pixels background meshes contamined by random Gaussian profiles.
+   The true background lies at 0 ADUs.
+   While being a bit noisier, the clipped "mode" gives a more robust estimate than the clipped mean in crowded regions. 
+
+Background map
+--------------
+
+Once the grid is set up, a median filter can be applied to suppress possible local overestimations due to bright stars.
+The final background map is simply a (natural) bicubic-spline interpolation between the meshes of the grid.
+Median filtering helps reducing possible ringing effects of the bicubic-spline around bright features.
+
+In parallel with the making of the background map, an *RMS background map*, that is, a map of the background noise standard deviation in the image is produced.
+It may be used as an internal weight map if the ``WEIGHT_TYPE`` configuration parameter is to ``BACKGROUND`` (see :ref:`weight-map_format`).
+
+Configuration and tuning
+------------------------
+
+.. note::
+  All background configuration parameters also affect background-RMS maps.
+
+The choice of the mesh size ``BACK_SIZE`` is very important.
+If it is too small, the background estimation is affected by the presence of
+objects and random noise.
+Most importantly, part of the flux of the most extended objects can be absorbed into the background map.
+If the mesh size is too large, it cannot reproduce the small scale variations of the background.
+Therefore a good compromise must be found by the user.
+Typically, for reasonably sampled images, a width [#recmesh]_ of 32 to 512 pixels works well.
+
+The user has some control over background map filtering by specifying the size of the median filter ``BACK_FILTERSIZE``.
+A width and height of 1 means that no filtering is applied to the background grid.
+A :math:`3\times3` filtering is sufficient in most cases.
+Larger dimensions may occasionally be used to compensate for small background mesh sizes, or in the presence of large image artifacts. 
+In some specific cases it is desirable to median-filter only background meshes that have values exceeding some threshold above the filtered value.
+This differential threshold is set by the ``BACK_FILTERTHRESH`` parameter, in ADUs.
+
+By default, the computed background maps are automatically subtracted from input images.
+However there are some situations where it is more appropriate to subtract a *constant* from the image (e.g., images with strongly non-Gaussian background noise |pdf|\ s).
+The ``BACK_TYPE`` configuration parameter (set to ``AUTO`` by default) may be switched to ``MANUAL`` to force the value specified by the ``BACK_VALUE`` parameter to be subtracted from the input image, instead of the background map. ``BACK_VALUE`` is ``0`` by default.
+
+Computing cost
+--------------
+
+The background estimation operation is generally |I/O|\ -bound, unless the image file already resides in the disk cache.
+
+.. [#recmesh]
+   It is possible to specify rectangular background meshes, although it is advised to use square ones, except in special cases (background varying rapidly along the x or y axis).

doc/src/index.rst

@@ -20,6 +20,7 @@ Contents
    Using
    Config
    Param
+   Background
    Weighting
    Flagging
    Measurements

doc/src/keys.rst

 .. |ATLAS| replace:: :program:`ATLAS`
 .. _ATLAS: http://math-atlas.sourceforge.net
 
+.. |DAOPHOT| replace:: :program:`DAOPHOT`
+.. _DAOPHOT: http://ascl.net/phpBB3/viewtopic.php?t=23410
+
 .. |.dmg| replace:: :file:`.dmg`
 .. _.dmg: https://en.wikipedia.org/wiki/Apple_Disk_Image
 
@@ -22,6 +25,9 @@
 .. |Intel| replace:: Intel\ :sup:`®`\ 
 .. _Intel: http://intel.com
 
+.. |I/O| replace:: :abbr:`I/O (Input/Output)`
+.. _I/O: https://en.wikipedia.org/wiki/Input/output
+
 .. |LevMar| replace:: :program:`LevMar`
 .. _LevMar: http://users.ics.forth.gr/~lourakis/levmar
 

doc/src/references.bib

@@ -184,6 +184,20 @@ archivePrefix = "arXiv",
   adsnote = {Provided by the SAO/NASA Astrophysics Data System}
 }
 
+@ARTICLE{1981AJ_86_476J,
+   author = {{Jarvis}, J.~F. and {Tyson}, J.~A.},
+    title = "{FOCAS - Faint Object Classification and Analysis System}",
+  journal = {AJ},
+ keywords = {Astronomical Catalogs, Astronomical Photography, Celestial Bodies, Image Resolution, Systems Analysis, Classifications, Digital Systems, Galaxies, Magnitude, Pattern Recognition, Photographic Plates, Stars},
+     year = 1981,
+    month = mar,
+   volume = 86,
+    pages = {476-495},
+      doi = {10.1086/112907},
+   adsurl = {http://adsabs.harvard.edu/abs/1981AJ.....86..476J},
+  adsnote = {Provided by the SAO/NASA Astrophysics Data System}
+}
+
 @ARTICLE{2012MNRAS_427_2711K,
    author = {{Kacprzak}, T. and {Zuntz}, J. and {Rowe}, B. and {Bridle}, S. and  {Refregier}, A. and {Amara}, A. and {Voigt}, L. and {Hirsch}, M.},
     title = "{Measurement and calibration of noise bias in weak lensing galaxy shape estimation}",
@@ -335,6 +349,20 @@ ISSN={1045-9227},
 month={Sep}
 }
 
+@ARTICLE{1987PASP_99_191S,
+   author = {{Stetson}, P.~B.},
+    title = "{DAOPHOT - A computer program for crowded-field stellar photometry}",
+  journal = {PASP},
+ keywords = {Algorithms, Astronomical Photometry, Computer Programs, Charge Coupled Devices, Color-Magnitude Diagram, Magellanic Clouds, Point Spread Functions, Television Cameras},
+     year = 1987,
+    month = mar,
+   volume = 99,
+    pages = {191-222},
+      doi = {10.1086/131977},
+   adsurl = {http://adsabs.harvard.edu/abs/1987PASP...99..191S},
+  adsnote = {Provided by the SAO/NASA Astrophysics Data System}
+}
+
 @ARTICLE{2001AJ_122_1104Y,
    author = {{Yasuda}, N. and {Fukugita}, M. and {Narayanan}, V.~K. and {Lupton}, R.~H. and 
 	{Strateva}, I. and {Strauss}, M.~A. and {Ivezi{\'c}}, {\v Z}. and 
@@ -405,6 +433,19 @@ booktitle = {Astronomical Data Analysis Software and Systems (ADASS) XIII},
   adsnote = {Provided by the SAO/NASA Astrophysics Data System}
 }
 
+@INPROCEEDINGS{1992ASPC_23_90D,
+   author = {{Da Costa}, G.~S.},
+    title = "{Basic Photometry Techniques}",
+booktitle = {Astronomical CCD Observing and Reduction Techniques},
+     year = 1992,
+   series = {Astronomical Society of the Pacific Conference Series},
+   volume = 23,
+   editor = {{Howell}, S.~B.},
+    pages = {90},
+   adsurl = {http://adsabs.harvard.edu/abs/1992ASPC...23...90D},
+  adsnote = {Provided by the SAO/NASA Astrophysics Data System}
+}
+
 @INPROCEEDINGS{2008ASPC_394_619M,
    author = {{Marmo}, C. and {Bertin}, E.},
     title = "{MissFITS and WeightWatcher: Two Optimised Tools for Managing FITS Data.}",
@@ -452,6 +493,18 @@ booktitle = {Conference on Applications of Digital Image Processing to Astronomy
   adsnote = {Provided by the SAO/NASA Astrophysics Data System}
 }
 
+@book{stuart2009kendall,
+  title={Kendall's Advanced Theory of Statistics: Volume 1: Distribution Theory},
+  author={Stuart, A. and Ord, K.},
+  number={vol.~1~;vol.~1994},
+  isbn={9780340614303},
+  lccn={94188490},
+  series={Kendall's Advanced Theory of Statistics},
+  url={https://books.google.fr/books?id=tW18thQWJQIC},
+  year={2009},
+  publisher={Wiley}
+}
+
 @BOOK{1968adga_book_S,
    author = {{Sérsic}, J.~L.},
     title = "{Atlas de Galaxias Australes}",