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Commit 07ea5e95 authored by Emmanuel Bertin's avatar Emmanuel Bertin
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Doc: added further paragraphs to the model-fitting section. 

Doc: added a first batch of model-fitting parameters to the list of measurements.
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doc/src/Model.rst
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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.
.. _fig_robustgalfit:
.. figure:: figures/robustgalfit.*
:figwidth: 100%
:align: center
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 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
\tilde{\boldsymbol{m}}(\boldsymbol{q}) = \mathbf{R}(\boldsymbol{x}) (\boldsymbol{m}(\boldsymbol{q})*\boldsymbol{\phi}).
:math:`\mathbf{R}(\boldsymbol{x})` depends on the pixel coordinates :math:`\boldsymbol{x}` of the model centroid:
.. math::
:label: model_resampling
\hat{\boldsymbol{m}}(\boldsymbol{q}) = \mathbf{R}(\boldsymbol{x}) (\boldsymbol{m}(\boldsymbol{q})*\boldsymbol{\phi}).
\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:
Minimization
~~~~~~~~~~~~
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.
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.
.. _models_def:
The models that can be fit are:
Models
------
- Exponential disk
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`.
.. math::
The present version of |SExtractor| supports the following models
\Sigma_{\tt ExpDisk}(R) = \Sigma(0) \exp \left (- {R\over h}\right )
\label{expdisk}
- :param:`BACKOFFSET`: flat background offset
- Sérsic (:math:`R^{1/n}`) spheroid (bulg)
Relevant measurement parameters: :param:`FLUX_BACKOFFSET`, :param:`FLUXERR_BACKOFFSET`
.. math::
.. math::
:label: backoffset_model
m_{\tt BACKOFFSET}(r) = m_0
\Sigma_{\tt Sersic}(R) = \Sigma(0) \exp \left [- b(n)\,\left({R\over
R_e}\right)^{1/n}\right ] \ ,
\label{sersic}
where, for the :raw-latex:`\cite{sersic:1968}` model, :math:`b(n)` is the solution of
- :param:`POINT_SOURCE`: point source
.. math::
Relevant measurement parameters: :param:`FLUX_POINTSOURCE`, :param:`FLUXERR_POINTSOURCE`, :param:`MAG_POINTSOURCE`, :param:`MAGERR_POINTSOURCE`, :param:`FLUXRATIO_POINTSOURCE`, :param:`FLUXRATIOERR_POINTSOURCE`
2 \gamma[2\,n,b(n)] = \Gamma(2\,n)
\label{bofn}
.. 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`
An accurate approximation for the solution for :math:`b(n)` of equation (bofn) is :raw-latex:`\citep{ciotti:bertin:1999}`
.. math::
:label: spheroid_model
.. math::
m_{\tt SPHEROID}(r) = m_0 \exp \left(- b(n)\,\left({R\over R_e}\right)^{1/n}\right),
b(n) = 2\,n - {1\over3} + {4\over 405\,n} + {46\over 25515\,n^2} + {131\over
1148175\,n^3}
where, for the :cite:`1968adga_book_S` model, :math:`b(n)` is the solution to
- :raw-latex:`\cite{devaucouleurs48}` spheroid (bulge, eq. [[sersic]], with :math:`n=4`)
.. math::
:label: bofn
- Exponential disk + Sérsic (:math:`R^{1/n}`) spheroid (bulge)
2 \gamma[2\,n,b(n)] = \Gamma(2\,n)
- Point source
An accurate approximation for the solution for :math:`b(n)` of :eq:`bofn` is :cite:`1999AA_352_447C`:
- Background (constant)
.. 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
......@@ -100,9 +196,8 @@ 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.
Models are measured according to the following table.
.. math::
..
Models are measured according to the following table.
\begin{aligned}
\hbox{{\tt FLUX\_BACKOFFSET} or {\tt FLUXERR\_BACKOFFSET}} &\to& \hbox{background}
......@@ -113,54 +208,47 @@ Models are measured according to the following table.
\hbox{{\tt SPHEROID\_xxx} without {\tt SPEHEROID\_SERSICN[ERR]}} &\to&
\hbox{de Vaucouleurs (}n=4 \hbox{ S\'ersic)} \nonumber \\
\hbox{{\tt MODEL\_xxx} only} &\to& \hbox{S\'ersic [???]} \nonumber \\
\hbox{{\tt SPHEROID\_xxx} and {\tt DISK\_xxx}}&\to& \hbox{S\'ersic spheroid + exponential disk [???]} \nonumber \end{aligned}
Table [modeltriggers] should be interpreted as meaning that if one of
the parameters given in the parameter file (e.g. default.param) includes
the string on the left of the arrow, the model to the right of the arrow
is triggered. For example, when including parameters that contain the
string ‘MODEL’, both galaxies and stars are fit with convolutions of
Sérsic models with the PSF. If no SPHEROID\_xxx or DISK\_xxx parameter
is present, but the model-fitting process is nevertheless triggered by
the presence of other measurement parameters or relevant
CHECKIMAGE\_TYPEs , a single component with Sérsic profile and
adjustable Sérsic index :math:`n` is fitted.
The number of parameters that are fit are 2 for the global center, 4 per
model for the scale, normalization, aspect ratio and position angle,
plus the index for the Sérsic model. For example, fitting a Sérsic +
exponential disk involves a fitting 11 parameters.
\hbox{{\tt SPHEROID\_xxx} and {\tt DISK\_xxx}}&\to& \hbox{S\'ersic spheroid + exponential disk [???]} \nonumber \end{aligned}
Table [modeltriggers] should be interpreted as meaning that if one of
the parameters given in the parameter file (e.g. default.param) includes
the string on the left of the arrow, the model to the right of the arrow
is triggered. For example, when including parameters that contain the
string ‘MODEL’, both galaxies and stars are fit with convolutions of
Sérsic models with the PSF. If no SPHEROID\_xxx or DISK\_xxx parameter
is present, but the model-fitting process is nevertheless triggered by
the presence of other measurement parameters or relevant
CHECKIMAGE\_TYPEs , a single component with Sérsic profile and
adjustable Sérsic index :math:`n` is fitted.
The number of parameters that are fit are 2 for the global center, 4 per
model for the scale, normalization, aspect ratio and position angle,
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:
The measurement parameters related to model-fitting follow the usual
SExtractor rules:
Flux measurements are available in ADUs (FLUX\_xxx parameters) or
magnitudes (MAG\_xxx parameters), Coordinates and radii are available in
pixels or celestial units (provided that the FITS image header contains
the appropriate WCS information).
Flux measurements are available in ADUs (FLUX\_xxx parameters) or
magnitudes (MAG\_xxx parameters), Coordinates and radii are available in
pixels or celestial units (provided that the FITS image header contains
the appropriate WCS information).
xxxMODEL\_yyy measurement parameters deal with the global fitted model,
i.e. the sum of all components (e.g. chi-square per d.o.f. CHI2\_MODEL,
PSF-corrected ellipticities E1/2MODEL\_IMAGE, EPS1/2 MODEL\_IMAGE).
xxxMODEL\_yyy measurement parameters deal with the global fitted model,
i.e. the sum of all components (e.g. chi-square per d.o.f. CHI2\_MODEL,
PSF-corrected ellipticities E1/2MODEL\_IMAGE, EPS1/2 MODEL\_IMAGE).
:math:`1\,\sigma` error estimates xxxERR\_yyy are provided for most
measurement parameters; they are obtained by marginalizing the full
covariance matrix of the fit.
:math:`1\,\sigma` error estimates xxxERR\_yyy are provided for most
measurement parameters; they are obtained by marginalizing the full
covariance matrix of the fit.
Since the model fitting involves convolution with the PSF, it is
imperative to launch PSFEx before launching SExtractor. In practice, the
sequence of operations is:
Since the model fitting involves convolution with the PSF, it is
imperative to launch PSFEx before launching SExtractor. In practice, the
sequence of operations is:
#. Run SExtractor to prepare PSFEx;
#. Run SExtractor to prepare PSFEx;
#. Run PSFEx to prepare model fits in SExtractor;
#. Run PSFEx to prepare model fits in SExtractor;
#. Run SExtractor with model fit parameters.
#. Run SExtractor with model fit parameters.
doc/src/Param.rst
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......@@ -86,9 +86,10 @@ of their meaning.
: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>`
......@@ -153,4 +154,95 @@ of their meaning.
_`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
doc/src/Position.rst
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......@@ -224,7 +224,7 @@ parameters can be derived from the 2nd order moments:
:figwidth: 100%
:align: center
Meaning of shape parameters.
Meaning of basic shape parameters.
.. _poserr_iso_def:
......
doc/src/keys.rst
View file @ 07ea5e95
......@@ -19,6 +19,9 @@
.. |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
......
doc/src/references.bib
View file @ 07ea5e95
......@@ -27,6 +27,30 @@
adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}
@ARTICLE{1999AA_352_447C,
author = {{Ciotti}, L. and {Bertin}, G.},
title = "{Analytical properties of the R^(1/m) law}",
journal = {A&A},
eprint = {astro-ph/9911078},
keywords = {GALAXIES: ELLIPTICAL AND LENTICULAR, CD, GALAXIES: FUNDAMENTAL PARAMETERS, GALAXIES: KINEMATICS AND DYNAMICS, GALAXIES: PHOTOMETRY},
year = 1999,
month = dec,
volume = 352,
pages = {447-451},
adsurl = {http://adsabs.harvard.edu/abs/1999A&A...352..447C},
adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}
@ARTICLE{duchon1979,
author = {{Duchon}, C.~E.},
title = "{Lanczos Filtering in One and Two Dimensions}",
journal = {Journal of Applied Meteorology},
year = 1979,
month = aug,
volume = 18,
pages = {1016-1022},
doi = {10.1175/1520-0450(1979)018<1016:LFIOAT>2.0.CO;2},
}
@ARTICLE{2002AA_395_1061G,
author = {{Greisen}, E.~W. and {Calabretta}, M.~R.},
title = "{Representations of world coordinates in FITS}",
......@@ -153,6 +177,16 @@ booktitle = {Conference on Applications of Digital Image Processing to Astronomy
adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}
@BOOK{1968adga_book_S,
author = {{Sérsic}, J.~L.},
title = "{Atlas de Galaxias Australes}",
keywords = {GALAXIES, GROUPS OF GALAXIES, ATLASES},
publisher = {Cordoba, Argentina: Observatorio Astronomico, 1968},
year = 1968,
adsurl = {http://adsabs.harvard.edu/abs/1968adga.book.....S},
adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}
@PHDTHESIS{1996PhDT_68B,
author = {{Bertin}, E.},
title = "{Photom\'etrie automatique de galaxies et contraintes sur leur \'evolution r\'ecente}",
......@@ -163,4 +197,10 @@ booktitle = {Conference on Applications of Digital Image Processing to Astronomy
adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}
@misc{lourakis04LM,
author = {M.I.A. Lourakis},
title = {levmar: Levenberg-Marquardt nonlinear least squares algorithms in {C}/{C}++},
url = {http://www.ics.forth.gr/~lourakis/levmar/},
year = {2004}
}
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