Doc: added further paragraphs to the model-fitting section.

Model fitting

=============

Fitting procedure

-----------------

SExtractor can fit models to the images of detected objects since version 2.8. The fit is performed by minimizing the loss function

.. math::

  :label: loss_func

  \lambda(\boldsymbol{q}) = \sum_i \left(g\left(\frac{p_i - \hat{m}_i(\boldsymbol{q})}{\sigma_i}\right)\right)^2 + \sum_j \frac{q_j - \mu_j}{}

  \lambda(\boldsymbol{q}) = \sum_i \left(g\left(\frac{p_i - \tilde{m}_i(\boldsymbol{q})}{\sigma_i}\right)\right)^2 + \sum_j \frac{q_j - \mu_j}{}

with respect to components of the model parameter vector :math:`\boldsymbol{q}`. :math:`\boldsymbol{q}` comprises parameters describing the shape of the model and the model pixel coordinates :math:`\boldsymbol{x}`.

The first term in :eq:`loss_func` is a modified `weighted sum of squares <http://en.wikipedia.org/wiki/Least_squares#Weighted_least_squares>`_ that aims at minimizing the residuals of the fit. :math:`p_i`, :math:`\hat{m}_i(\boldsymbol{q})` and :math:`\sigma_i` are respectively the pixel value above the background, the value of the resampled model, and the pixel value uncertainty at image pixel :math:`i`.

:math:`g(u)` is a derivable monotonous function that reduces the influence of large deviations from the model (e.g., contamination by neighbors):

Modified least squares

~~~~~~~~~~~~~~~~~~~~~~

The first term in :eq:`loss_func` is a modified `weighted sum of squares <http://en.wikipedia.org/wiki/Least_squares#Weighted_least_squares>`_ that aims at minimizing the residuals of the fit. :math:`p_i`, :math:`\tilde{m}_i(\boldsymbol{q})` and :math:`\sigma_i` are respectively the pixel value above the background, the value of the resampled model, and the pixel value uncertainty at image pixel :math:`i`.

:math:`g(u)` is a derivable monotonous function that reduces the influence of large deviations from the model, such as the contamination by neighbors (:numref:`fig_robustgalfit`):

.. math::

  :label: loss_func

  :label: modified_lsq

  g(u) = \left\{

    \begin{array}{rl}

       u_0 \log \left(1 + \frac{u}{u_0}\right) & \mbox{if } u \ge 0,\\

      -u_0 \log \left(1 - \frac{u}{u_0}\right) & \mbox{otherwise.}\\

      -u_0 \log \left(1 - \frac{u}{u_0}\right) & \mbox{otherwise}.\\

    \end{array}

  \right.

The vector :math:`\hat{\boldsymbol{m}}(\boldsymbol{q})` is obtained by convolving the high resolution model :math:`\boldsymbol{m}(\boldsymbol{q})` with the local PSF model :math:`\boldsymbol{\phi}` and applying a resampling operator :math:`\mathbf{R}(\boldsymbol{x})` to generate the final model raster at position :math:`\boldsymbol{x}` at the nominal image resolution:

:math:`u_0` sets the level below which :math:`g(u)\approx u`.

In practice, choosing :math:`u_0 = \kappa \sigma_i` with :math:`\kappa = 10` makes the first term in :eq:`loss_func` behave like a traditional weighted sum of squares for residuals close to the noise level.

.. math::

  :label: model_resampling

.. _fig_robustgalfit:

  \hat{\boldsymbol{m}}(\boldsymbol{q}) = \mathbf{R}(\boldsymbol{x}) (\boldsymbol{m}(\boldsymbol{q})*\boldsymbol{\phi}).

.. figure:: figures/robustgalfit.*

   :figwidth: 100%

   :align: center

Levenberg-Marquardt minimization, inside a disk which diameter is scaled to include the isophotal footprint plus a 20 % margin, plus the size of the PSF model image.

   Effect of the modified least squares loss function on fitting a model to a galaxy with a bright neighbor. *Left*: the original image; *Middle*: residuals of the model fitting with a regular least squares (:math:`\kappa = +\infty`); *Right*: modified least squares with :math:`\kappa = 10`.

The models that can be fit are:

- Exponential disk

The vector :math:`\tilde{\boldsymbol{m}}(\boldsymbol{q})` is obtained by convolving the high resolution model :math:`\boldsymbol{m}(\boldsymbol{q})` with the local PSF model :math:`\boldsymbol{\phi}` and applying a resampling operator :math:`\mathbf{R}(\boldsymbol{x})` to generate the final model raster at position :math:`\boldsymbol{x}` at the nominal image resolution:

.. math::

  :label: model_convolution

      \Sigma_{\tt ExpDisk}(R) = \Sigma(0) \exp \left (- {R\over h}\right ) 

      \label{expdisk}

  \tilde{\boldsymbol{m}}(\boldsymbol{q}) = \mathbf{R}(\boldsymbol{x}) (\boldsymbol{m}(\boldsymbol{q})*\boldsymbol{\phi}).

- Sérsic (:math:`R^{1/n}`) spheroid (bulg)

:math:`\mathbf{R}(\boldsymbol{x})` depends on the pixel coordinates :math:`\boldsymbol{x}` of the model centroid:

.. math::

  :label: model_resampling

  \mathbf{R}_{ij}(\boldsymbol{x}) =  h\left(\boldsymbol{x}_j - \eta.(\boldsymbol{x}_i - \boldsymbol{x})\right),

where :math:`h` is a 2-dimensional interpolant (interpolating function), :math:`\boldsymbol{x}_i` is the coordinate vector of image pixel :math:`i`, :math:`\boldsymbol{x}_j` the coordinate vector of model sample :math:`j`, and :math:`\eta` is the image-to-model sampling step ratio (sampling factor) which is by default defined by the PSF model sampling.

We adopt a Lánczos-4 function :cite:`duchon1979` as interpolant.

Regularization

~~~~~~~~~~~~~~

.. _model_minimization_def:

      \Sigma_{\tt Sersic}(R) = \Sigma(0) \exp \left [- b(n)\,\left({R\over

          R_e}\right)^{1/n}\right ] \ ,

      \label{sersic}

Minimization

~~~~~~~~~~~~

where, for the :raw-latex:`\cite{sersic:1968}` model, :math:`b(n)` is the solution of

Minimization of the loss function :math:`\lambda(\boldsymbol{q})` is carried out using the `Levenberg-Marquardt algorithm <http://en.wikipedia.org/wiki/Levenberg%E2%80%93Marquardt_algorithm>`_, and more specifically the |LevMar|_ implementation :cite:`lourakis04LM`.

The fit is done inside a disk which diameter is scaled to include the isophotal footprint of the object, plus the FWHM of the PSF, plus a 20 % margin.

The number of iterations is returned in the :param:`NITER_MODEL` measurement parameter.

It is generally a few tens.

The final value of the modified chi square term in :eq:`loss_func`, divided by the number of degrees of freedom, is returned in :param:`CHI2_MODEL`.

The :param:`FLAGS_MODEL` parameter flags various issues which may happen during the fitting process (see the flags section for details on how flags are managed in |SExtractor|):

.. csv-table:: :param:`FLAGS_MODEL` flag description

  :header: "Value", "Meaning"

  :widths: 3 60

  1, "the unconvolved, supersampled model raster exceeds 512×512 pixels and had to be resized"

  2, "the convolved, resampled model raster exceeds 512×512 pixels and had to be resized"

  4, "not enough pixels are available for model fitting on the measurement image (less pixels than fit parameters)"

  8, "at least one of the fitted parameters hits the lower bound"

  16, "at least one of the fitted parameters hits the upper bound"

:math:`1\,\sigma` error estimates are provided for most measurement parameters; they are obtained by marginalizing the full covariance matrix of the fit.

.. _models_def:

Models

------

Models contain one or more components, which share their central coordinates. For instance, a galaxy model may be composed of a spheroid (bulge) and a disk components. Both components are concentric but they may have different scales, aspect ratios and position angles. Adding a component is done simply by invoking one of its measurement parameters in the parameter file, e.g., :param:`DISK_SCALE_IMAGE`.

The present version of |SExtractor| supports the following models

- :param:`BACKOFFSET`: flat background offset

  Relevant measurement parameters: :param:`FLUX_BACKOFFSET`, :param:`FLUXERR_BACKOFFSET`

.. math::

  :label: backoffset_model

      2 \gamma[2\,n,b(n)] = \Gamma(2\,n)

      \label{bofn}

  m_{\tt BACKOFFSET}(r) = m_0

  An accurate approximation for the solution for :math:`b(n)` of equation (bofn) is :raw-latex:`\citep{ciotti:bertin:1999}`

- :param:`POINT_SOURCE`: point source

  Relevant measurement parameters: :param:`FLUX_POINTSOURCE`, :param:`FLUXERR_POINTSOURCE`, :param:`MAG_POINTSOURCE`, :param:`MAGERR_POINTSOURCE`, :param:`FLUXRATIO_POINTSOURCE`, :param:`FLUXRATIOERR_POINTSOURCE`

.. math::

  :label: pointsource_model

  m_{\tt POINTSOURCE}(r) = m_0 \delta(r)

- :param:`DISK`: exponential disk

  Relevant measurement  parameters:

  :param:`FLUX_DISK`, :param:`FLUXERR_DISK`, :param:`MAG_DISK`, :param:`MAGERR_DISK`,

  :param:`FLUXRATIO_DISK`, :param:`FLUXRATIOERR_DISK`,

  :param:`FLUX_MAX_DISK`, :param:`MU_MAX_DISK`,

  :param:`FLUX_EFF_DISK`, :param:`MU_EFF_DISK`,

  :param:`FLUX_MEAN_DISK`, :param:`MU_MEAN_DISK`,

  :param:`DISK_SCALE_IMAGE`, :param:`DISK_SCALEERR_IMAGE`,

  :param:`DISK_SCALE_WORLD`, :param:`DISK_SCALEERR_WORLD`,

  :param:`DISK_ASPECT_IMAGE`, :param:`DISK_ASPECTERR_IMAGE`,

  :param:`DISK_ASPECT_WORLD`, :param:`DISK_ASPECTERR_WORLD`,

  :param:`DISK_INCLINATION`, :param:`DISK_INCLINATIONERR`,

  :param:`DISK_THETA_IMAGE`, :param:`DISK_THETAERR_IMAGE`,

  :param:`DISK_THETA_WORLD`, :param:`DISK_THETAERR_WORLD`,

  :param:`DISK_THETA_SKY`, :param:`DISK_THETA_J2000`, :param:`DISK_THETA_B1950`

.. math::

  :label: disk_model

  m_{\tt DISK}(r) = m_0 \exp \left( - {r\over h}\right) 

- :param:`SPHEROID`: Sérsic (:math:`R^{1/n}`) spheroid

  :param:`FLUX_SPHEROID`, :param:`FLUXERR_SPHEROID`, :param:`MAG_SPHEROID`, :param:`MAGERR_SPHEROID`,

  :param:`FLUXRATIO_SPHEROID`, :param:`FLUXRATIOERR_SPHEROID`,

  :param:`FLUX_MAX_SPHEROID`, :param:`MU_MAX_SPHEROID`,

  :param:`FLUX_EFF_SPHEROID`, :param:`MU_EFF_SPHEROID`,

  :param:`FLUX_MEAN_SPHEROID`, :param:`MU_MEAN_SPHEROID`,

  :param:`SPHEROID_SCALE_IMAGE`, :param:`SPHEROID_SCALEERR_IMAGE`,

  :param:`SPHEROID_SCALE_WORLD`, :param:`SPHEROID_SCALEERR_WORLD`,

  :param:`SPHEROID_ASPECT_IMAGE`, :param:`SPHEROID_ASPECTERR_IMAGE`,

  :param:`SPHEROID_ASPECT_WORLD`, :param:`SPHEROID_ASPECTERR_WORLD`,

  :param:`SPHEROID_INCLINATION`, :param:`SPHEROID_INCLINATIONERR`,

  :param:`SPHEROID_THETA_IMAGE`, :param:`SPHEROID_THETAERR_IMAGE`,

  :param:`SPHEROID_THETA_WORLD`, :param:`SPHEROID_THETAERR_WORLD`,

  :param:`SPHEROID_THETA_SKY`, :param:`SPHEROID_THETA_J2000`, :param:`SPHEROID_THETA_B1950`

  :param:`SPHEROID_SERSICN`, :param:`SPHEROID_SERSICNERR`

.. math::

  :label: spheroid_model

      b(n) = 2\,n - {1\over3} + {4\over 405\,n} + {46\over 25515\,n^2} + {131\over

        1148175\,n^3}

  m_{\tt SPHEROID}(r) = m_0 \exp \left(- b(n)\,\left({R\over R_e}\right)^{1/n}\right),

-  :raw-latex:`\cite{devaucouleurs48}` spheroid (bulge, eq. [[sersic]], with :math:`n=4`)

where, for the :cite:`1968adga_book_S` model, :math:`b(n)` is the solution to

-  Exponential disk + Sérsic (:math:`R^{1/n}`) spheroid (bulge)

.. math::

  :label: bofn

-  Point source

  2 \gamma[2\,n,b(n)] = \Gamma(2\,n)

-  Background (constant)

An accurate approximation for the solution for :math:`b(n)` of :eq:`bofn` is :cite:`1999AA_352_447C`:

.. math::

For these models, SExtractor can compute fluxes and magnitudes, as well

as sizes (disk scale length for the disks and effective — projected

half-light — radii for the spheroids), characteristic surface

magnitudes, and Sérsic index, as well as their uncertainties.

  b(n) = 2\,n - {1\over3} + {4\over 405\,n} + {46\over 25515\,n^2} + {131\over 1148175\,n^3}

The models are concentric (they assume the same center) and are all

convolved with the PSF, given by the .psf file, which must be determined

by first running PSFEx (see below).

Experience shows that the de Vaucouleurs spheroid + exponential disk

combination provides fairly accurate and robust fits for moderately

resolved faint galaxies. An adjustable Sérsic index may offer lower

residuals on spheroids and/or well-resolved galaxies, but makes the fit

less robust and more sensitive to PSF model errors.

Unfortunately, the Sérsic profile is very cuspy in the center for

The Sérsic profile is very cuspy in the center for

:math:`n>2`. To avoid huge wings in the FFTs when convolving the profile

with the PSF, the profile is split between a 3rd order polynomial,

analytically fit to match, in intensity and its 1st and 2nd spatial

position angle when the ellipticity is very low. The Sérsic index is

allowed values between 1 and 10.

..

   Models are measured according to the following table.

.. math::

   \begin{aligned}

   \hbox{{\tt FLUX\_BACKOFFSET} or {\tt FLUXERR\_BACKOFFSET}} &\to& \hbox{background}

   \nonumber \\

  plus the index for the Sérsic model. For example, fitting a Sérsic +

  exponential disk involves a fitting 11 parameters.

Experience shows that the de Vaucouleurs spheroid + exponential disk

combination provides fairly accurate and robust fits for moderately

resolved faint galaxies. An adjustable Sérsic index may offer lower

residuals on spheroids and/or well-resolved galaxies, but makes the fit

less robust and more sensitive to PSF model errors. One might think of

adding some mechanism to lock or unlock the Sérsic index automatically

in future versions of SExtractor.

  The measurement parameters related to model-fitting follow the usual

  SExtractor rules:

  :header: "Name", "Unit", "Description"

  :widths: 15 10 30

  NUMBER,, Running object number

  ID_PARENT,..., Parent ID (before deblending)

  EXT_NUMBER,..., FITS extension number

  _`NUMBER`,, Running object number

  _`ID_PARENT`,..., Parent ID (before deblending)

  _`EXT_NUMBER`,..., FITS extension number

  _`FLAGS`,..., Extraction flags

  _`FLUX_ISO`, count, :ref:`Isophotal flux <flux_iso_def>`

  _`FLUXERR_ISO`, count, :ref:`RMS error estimate for the isophotal flux <flux_iso_def>`

  _`MAG_ISO`, magnitude, :ref:`Isophotal magnitude <flux_iso_def>`

  _`AWIN_IMAGE`, pixel, :ref:`Windowed image major axis <shape_win_def>`

  _`BWIN_IMAGE`, pixel, :ref:`Windowed image minor axis <shape_win_def>`

  _`THETAWIN_IMAGE`, degree, :ref:`Windowed image position angle <shape_win_def>`

  _`VECTOR_MODEL`, ..., :ref:`Model-fitting coefficients <models_def>`

  _`VECTOR_MODELERR`, ..., :ref:`Model-fitting coefficient uncertainties <models_def>`

  _`MATRIX_MODELERR`, ..., :ref:`Model-fitting covariance matrix <model_minimization_def>`

  _`CHI2_MODEL`, ..., :ref:`Reduced modified Chi2 of the fit <model_minimization_def>`

  _`FLAGS_MODEL`, ..., :ref:`Model-fitting flags <model_minimization_def>`

  _`NITER_MODEL`, ..., :ref:`Number of model-fitting iterations <model_minimization_def>`  

  _`FLUX_MODEL`, count, :ref:`Flux from model-fitting <models_def>`

  _`FLUXERR_MODEL`, count, :ref:`RMS error estimate for the model-fitting flux <models_def>`

  _`MAG_MODEL`, magnitude, :ref:`Magnitude from model-fitting <models_def>`

  _`MAGERR_MODEL`, count, :ref:`RMS error estimate for the model-fitting magnitude <models_def>`

  _`FLUX_MAX_MODEL`, count, :ref:`Peak model flux above the background <models_def>`

  _`FLUX_EFF_MODEL`, count, :ref:`Effective model flux above the background <models_def>`

  _`FLUX_EFF_MODEL`, count, :ref:`Mean effective model flux above the background <models_def>`

  _`MU_MAX_MODEL`, mag.arcsec\ :sup:`-2`, :ref:`Peak model surface brightness above the background <models_def>`

  _`MU_EFF_MODEL`, mag.arcsec\ :sup:`-2`, :ref:`Effective model surface brightness above the background <models_def>`

  _`MU_MEAN_MODEL`, mag.arcsec\ :sup:`-2`, :ref:`Mean effective model surface brightness above the background <models_def>`

  _`XMODEL_IMAGE`, pixel, :ref:`x coordinate from model-fitting <models_def>`

  _`YMODEL_IMAGE`, pixel, :ref:`y coordinate from model-fitting <models_def>`

..

  #XMODEL_WORLD             Fitted position along world x axis                        [deg]

  #YMODEL_WORLD             Fitted position along world y axis                        [deg]

  #ALPHAMODEL_SKY           Fitted position along right ascension  (native)           [deg]

  #DELTAMODEL_SKY           Fitted position along declination (native)                [deg]

  #ALPHAMODEL_J2000         Fitted position along right ascension (J2000)             [deg]

  #DELTAMODEL_J2000         Fitted position along declination (J2000)                 [deg]

  #ALPHAMODEL_B1950         Fitted position along right ascension (B1950)             [deg]

  #DELTAMODEL_B1950         Fitted position along declination (B1950)                 [deg]

  #ERRX2MODEL_IMAGE         Variance of fitted position along x                       [pixel**2]

  #ERRY2MODEL_IMAGE         Variance of fitted position along y                       [pixel**2]

  #ERRXYMODEL_IMAGE         Covariance of fitted position between x and y             [pixel**2]

  #ERRX2MODEL_WORLD         Variance of fitted position along X-WORLD (alpha)         [deg**2]

  #ERRY2MODEL_WORLD         Variance of fitted position along Y-WORLD (delta)         [deg**2]

  #ERRXYMODEL_WORLD         Covariance of fitted position X-WORLD/Y-WORLD             [deg**2]

  #ERRCXXMODEL_IMAGE        Cxx error ellipse parameter of fitted position            [pixel**(-2)]

  #ERRCYYMODEL_IMAGE        Cyy error ellipse parameter of fitted position            [pixel**(-2)]

  #ERRCXYMODEL_IMAGE        Cxy error ellipse parameter of fitted position            [pixel**(-2)]

  #ERRCXXMODEL_WORLD        Cxx fitted error ellipse parameter (WORLD units)          [deg**(-2)]

  #ERRCYYMODEL_WORLD        Cyy fitted error ellipse parameter (WORLD units)          [deg**(-2)]

  #ERRCXYMODEL_WORLD        Cxy fitted error ellipse parameter (WORLD units)          [deg**(-2)]

  #ERRAMODEL_IMAGE          RMS error of fitted position along major axis             [pixel]

  #ERRBMODEL_IMAGE          RMS error of fitted position along minor axis             [pixel]

  #ERRTHETAMODEL_IMAGE      Error ellipse pos.angle of fitted position (CCW/x)        [deg]

  #ERRAMODEL_WORLD          World RMS error of fitted position along major axis       [deg]

  #ERRBMODEL_WORLD          World RMS error of fitted position along minor axis       [deg]

  #ERRTHETAMODEL_WORLD      Error ellipse pos.angle of fitted position (CCW/world-x)  [deg]

  #ERRTHETAMODEL_SKY        Native fitted error ellipse pos. angle (east of north)    [deg]

  #ERRTHETAMODEL_J2000      J2000 fitted error ellipse pos. angle (east of north)     [deg]

  #ERRTHETAMODEL_B1950      B1950 fitted error ellipse pos. angle (east of north)     [deg]

  #X2MODEL_IMAGE            Variance along x from model-fitting                       [pixel**2]

  #Y2MODEL_IMAGE            Variance along y from model-fitting                       [pixel**2]

  #XYMODEL_IMAGE            Covariance between x and y from model-fitting             [pixel**2]

  #ELLIP1MODEL_IMAGE        Ellipticity component from model-fitting                 

  #ELLIP2MODEL_IMAGE        Ellipticity component from model-fitting                 

  #POLAR1MODEL_IMAGE        Ellipticity component (quadratic) from model-fitting     

  #POLAR2MODEL_IMAGE        Ellipticity component (quadratic) from model-fitting     

  #ELLIP1ERRMODEL_IMAGE     Ellipticity component std.error from model-fitting       

  #ELLIP2ERRMODEL_IMAGE     Ellipticity component std.error from model-fitting       

  #ELLIPCORRMODEL_IMAGE     Corr.coeff between ellip.components from model-fitting   

  #POLAR1ERRMODEL_IMAGE     Polarisation component std.error from model-fitting      

  #POLAR2ERRMODEL_IMAGE     Polarisation component std.error from model-fitting      

  #POLARCORRMODEL_IMAGE     Corr.coeff between polar. components from fitting        

  #X2MODEL_WORLD            Variance along X-WORLD (alpha) from model-fitting         [deg**2]

  #Y2MODEL_WORLD            Variance along Y_WORLD (delta) from model-fitting         [deg**2]

  #XYMODEL_WORLD            Covariance between X-WORLD and Y-WORLD from model-fitting [deg**2]

  #ELLIP1MODEL_WORLD        Ellipticity component from model-fitting                 

  #ELLIP2MODEL_WORLD        Ellipticity component from model-fitting                 

  #POLAR1MODEL_WORLD        Polarisation component from model-fitting                

  #POLAR2MODEL_WORLD        Polarisation component from model-fitting                

  #ELLIP1ERRMODEL_WORLD     Ellipticity component std.error from model-fitting       

  #ELLIP2ERRMODEL_WORLD     Ellipticity component std.error from model-fitting       

  #ELLIPCORRMODEL_WORLD     Corr.coeff between ellip.components from model-fitting   

  #POLAR1ERRMODEL_WORLD     Polarisation component std.error from model-fitting      

  #POLAR2ERRMODEL_WORLD     Polarisation component std.error from model-fitting      

  #POLARCORRMODEL_WORLD     Corr.coeff between polar. components from fitting        

  #CXXMODEL_IMAGE           Cxx ellipse parameter from model-fitting                  [pixel**(-2)]

  #CYYMODEL_IMAGE           Cyy ellipse parameter from model-fittinh                  [pixel**(-2)]

  #CXYMODEL_IMAGE           Cxy ellipse parameter from model-fitting                  [pixel**(-2)]

  #CXXMODEL_WORLD           Cxx ellipse parameter (WORLD) from model-fitting          [deg**(-2)]

  #CYYMODEL_WORLD           Cyy ellipse parameter (WORLD) from model-fitting          [deg**(-2)]

  #CXYMODEL_WORLD           Cxy ellipse parameter (WORLD) from model-fitting          [deg**(-2)]

  #AMODEL_IMAGE             Model RMS along major axis                                [pixel]

  #BMODEL_IMAGE             Model RMS along minor axis                                [pixel]

  #THETAMODEL_IMAGE         Model position angle (CCW/x)                              [deg]

  #AMODEL_WORLD             Model RMS along major axis (WORLD units)                  [deg]

  #BMODEL_WORLD             Model RMS along minor axis (WORLD units)                  [deg]

  #THETAMODEL_WORLD         Model position angle (CCW/WORLD-x)                        [deg]

  #THETAMODEL_SKY           Model position angle (east of north) (native)             [deg]

  #THETAMODEL_J2000         Model position angle (east of north) (J2000)              [deg]

  #THETAMODEL_B1950         Model position angle (east of north) (B1950)              [deg]

  #SPREAD_MODEL             Spread parameter from model-fitting                      

   :figwidth: 100%

   :align: center

   Meaning of shape parameters.

   Meaning of basic shape parameters.

.. _poserr_iso_def:

.. |Intel| replace:: Intel\ :sup:`®`\ 

.. _Intel: http://intel.com

.. |LevMar| replace:: :program:`LevMar`

.. _LevMar: http://users.ics.forth.gr/~lourakis/levmar

.. |MEF| replace:: :abbr:`MEF (Multi-Extension FITS)`

.. _MEF: http://www.stsci.edu/hst/HST_overview/documents/datahandbook/intro_ch23.html