Doc: added SPREAD_MODEL sub-section in model-fitting section.

doc/src/Model.rst

@@ -258,6 +258,42 @@ ellipticities greater than unity, SExtractor avoids dichotomies of
 position angle when the ellipticity is very low. The Sérsic index is
 allowed values between 1 and 10.
 
+.. _spread_model_def:
+
+Star-Galaxy separation
+----------------------
+
+The :param:`SPREAD_MODEL` estimator has been developed as a star/galaxy classifier for the DESDM pipeline :cite:`2012SPIE_8451E_0DM`, and has also been used in other surveys :cite:`2012ApJ_757_83D,2013AA_554A_101B`.
+:param:`SPREAD_MODEL` indicates which of the best fitting local PSF model resampled at the current position :math:`\tilde{\boldsymbol{\phi}}` (representing a point source) or a slightly ``fuzzier'' resampled model :math:`\tilde{\boldsymbol{G}}` (representing a galaxy) matches best the image data.
+:math:`\tilde{\boldsymbol{G}}` is obtained by convolving the local PSF model with a circular exponential model with scalelength = 1/16 |FWHM|, and resampling the result at the current position on the pixel grid. :param:`SPREAD_MODEL` is normalized to allow comparing sources with different PSFs throughout the field:
+
+.. math::
+  :label: spread_model
+
+  {\tt SPREAD\_MODEL} = \frac{\tilde{\boldsymbol{G}}^\mathsf{T} {\bf W}\,\boldsymbol{p}}{\tilde{\boldsymbol{\phi}}^\mathsf{T} {\bf W}\,\boldsymbol{p}}
+    - \frac{\tilde{\boldsymbol{G}}^\mathsf{T} {\bf W}\,\tilde{\boldsymbol{\phi}}}{\tilde{\boldsymbol{\phi}}^\mathsf{T} {\bf W}\,\tilde{\boldsymbol{\phi}}},
+
+where :math:`\boldsymbol{p}` is the image vector centered on the source.
+:math:`{\bf W}` is a weight matrix constant along the diagonal except for bad pixels where the weight is 0.
+By construction, :param:`SPREAD_MODEL` is close to zero for point sources, positive for extended sources (galaxies), and negative for detections smaller than the PSF, such as cosmic ray hits.
+
+.. note::
+  The definition of :param:`SPREAD_MODEL` above differs from the one given in previous papers, which was incorrect, although in practice both estimators give very similar results.
+
+The |RMS| error on :param:`SPREAD_MODEL` is estimated by propagating the uncertainties on individual pixel values:
+
+.. math::
+  :label: spreaderr_model
+
+  \begin{eqnarray}
+  {\tt SPREADERR\_MODEL} & = & \frac{1}{(\tilde{\boldsymbol{\phi}}^\mathsf{T} {\bf W}\,\boldsymbol{p})^2} \left((\tilde{\boldsymbol{G}}^\mathsf{T} {\bf V}\,\tilde{\boldsymbol{G}})\,(\tilde{\boldsymbol{\phi}}^\mathsf{T} {\bf W}\,\boldsymbol{p})^2\right.\nonumber \\
+  & & + (\tilde{\boldsymbol{\phi}}^\mathsf{T} {\bf V}\,\tilde{\boldsymbol{\phi}})\,(\tilde{\boldsymbol{G}}^\mathsf{T} {\bf W}\,\boldsymbol{p})^2\nonumber \\
+  & & \left. - 2\,(\tilde{\boldsymbol{G}}^\mathsf{T} {\bf V}\,\tilde{\boldsymbol{\phi}})\,(\tilde{\boldsymbol{G}}^\mathsf{T} {\bf W}\,\boldsymbol{p})\,(\tilde{\boldsymbol{\phi}}^\mathsf{T} {\bf W}\,\boldsymbol{p}) \right)^{1/2},
+  \end{eqnarray}
+
+where :math:`{\bf V}` is the noise covariance matrix, which we assume to be diagonal.
+
+
 ..
    Models are measured according to the following table.
 

doc/src/keys.rst

@@ -7,6 +7,9 @@
 .. |FFTw| replace:: :program:`FFTw`
 .. _FFTw: http://www.fftw.org
 
+.. |FWHM| replace:: :abbr:`FWHM (Full Width at Half Maximum)`
+.. _FWHM: https://en.wikipedia.org/wiki/Full_width_at_half_maximum
+
 .. |FITS| replace:: :abbr:`FITS (Flexible Image Transport System)`
 .. _FITS: http://fits.gsfc.nasa.gov
 
@@ -37,6 +40,9 @@
 .. |RPM| replace:: :program:`RPM`
 .. _RPM: http://www.rpm.org
 
+.. |RMS| replace:: :abbr:`RMS (Root Mean Square)`
+.. _RMS: https://en.wikipedia.org/wiki/Root_mean_square
+
 .. |SCAMP| replace:: :program:`SCAMP`
 .. _SCAMP: http://astromatic.net/software/scamp
 

doc/src/references.bib

@@ -12,13 +12,31 @@
   adsnote = {Provided by the SAO/NASA Astrophysics Data System}
 }
 
+@ARTICLE{2013AA_554A_101B,
+   author = {{Bouy}, H. and {Bertin}, E. and {Moraux}, E. and {Cuillandre}, J.-C. and {Bouvier}, J. and {Barrado}, D. and {Solano}, E. and {Bayo}, A.},
+    title = "{Dynamical analysis of nearby clusters. Automated astrometry from the ground: precision proper motions over a wide field}",
+  journal = {A&A},
+archivePrefix = "arXiv",
+   eprint = {1306.4446},
+ primaryClass = "astro-ph.IM",
+ keywords = {astrometry, proper motions, stars: kinematics and dynamics},
+     year = 2013,
+    month = jun,
+   volume = 554,
+      eid = {A101},
+    pages = {A101},
+      doi = {10.1051/0004-6361/201220748},
+   adsurl = {http://adsabs.harvard.edu/abs/2013A&A...554A.101B},
+  adsnote = {Provided by the SAO/NASA Astrophysics Data System}
+}
+
 @ARTICLE{Broyden1965ACo,
-       AUTHOR = "C. Broyden",
-       TITLE = "A Class of Methods for Solving Nonlinear Simultaneous Equations",
-       JOURNAL = "Mathematics of Computation",
-       VOLUME = "19",
-       YEAR = "1965",
-       PAGES = "577-593"
+   AUTHOR = "C. Broyden",
+   TITLE = "A Class of Methods for Solving Nonlinear Simultaneous Equations",
+   JOURNAL = "Mathematics of Computation",
+   VOLUME = "19",
+   YEAR = "1965",
+   PAGES = "577-593"
 }
 
 @ARTICLE{2002AA_395_1077C,
@@ -50,6 +68,24 @@
   adsnote = {Provided by the SAO/NASA Astrophysics Data System}
 }
 
+@ARTICLE{2012ApJ_757_83D,
+   author = {{Desai}, S. and {Armstrong}, R. and {Mohr}, J.~J. and {Semler}, D.~R. and {Liu}, J. and {Bertin}, E. and {Allam}, S.~S. and {Barkhouse}, W.~A. and {Bazin}, G. and {Buckley-Geer}, E.~J. and {Cooper}, M.~C. and {Hansen}, S.~M. and {High}, F.~W. and {Lin}, H. and {Lin}, Y.-T. and {Ngeow}, C.-C. and {Rest}, A. and {Song}, J. and {Tucker}, D. and {Zenteno}, A.},
+    title = "{The Blanco Cosmology Survey: Data Acquisition, Processing, Calibration, Quality Diagnostics, and Data Release}",
+  journal = {ApJ},
+archivePrefix = "arXiv",
+   eprint = {1204.1210},
+ primaryClass = "astro-ph.CO",
+ keywords = {cosmology: observations, galaxies: clusters: general, methods: data analysis, surveys, techniques: image processing},
+     year = 2012,
+    month = sep,
+   volume = 757,
+      eid = {83},
+    pages = {83},
+      doi = {10.1088/0004-637X/757/1/83},
+   adsurl = {http://adsabs.harvard.edu/abs/2012ApJ...757...83D},
+  adsnote = {Provided by the SAO/NASA Astrophysics Data System}
+}
+
 @ARTICLE{duchon1979,
    author = {{Duchon}, C.~E.},
     title = "{Lanczos Filtering in One and Two Dimensions}",
@@ -250,6 +286,24 @@ booktitle = {Astronomical Data Analysis Software and Systems (ADASS) XIII},
   adsnote = {Provided by the SAO/NASA Astrophysics Data System}
 }
 
+@INPROCEEDINGS{2012SPIE_8451E_0DM,
+   author = {{Mohr}, J.~J. and {Armstrong}, R. and {Bertin}, E. and {Daues}, G. and {Desai}, S. and {Gower}, M. and {Gruendl}, R. and {Hanlon}, W. and {Kuropatkin}, N. and {Lin}, H. and {Marriner}, J. and {Petravic}, D. and {Sevilla}, I. and {Swanson}, M. and {Tomashek}, T. and {Tucker}, D. and {Yanny}, B.},
+    title = "{The Dark Energy Survey data processing and calibration system}",
+booktitle = {Software and Cyberinfrastructure for Astronomy II},
+     year = 2012,
+   series = {Proc. SPIE},
+   volume = 8451,
+archivePrefix = "arXiv",
+   eprint = {1207.3189},
+ primaryClass = "astro-ph.IM",
+    month = sep,
+      eid = {84510D},
+    pages = {84510D},
+      doi = {10.1117/12.926785},
+   adsurl = {http://adsabs.harvard.edu/abs/2012SPIE.8451E..0DM},
+  adsnote = {Provided by the SAO/NASA Astrophysics Data System}
+}
+
 @INPROCEEDINGS{1980SPIE_264_208S,
    author = {{Stobie}, R.~S.},
     title = "{Application of moments to the analysis of panoramic astronomical photographs}",