<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
<!--Converted with LaTeX2HTML 98.1p1 release (March 2nd, 1998)
originally by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds
* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
* with significant contributions from:
Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
<HTML>
<HEAD>
<TITLE>Theory</TITLE>
<META NAME="description" CONTENT="Theory">
<META NAME="keywords" CONTENT="vol2">
<META NAME="resource-type" CONTENT="document">
<META NAME="distribution" CONTENT="global">
<META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1">
<LINK REL="STYLESHEET" HREF="vol2.css">
<LINK REL="next" HREF="node85.html">
<LINK REL="previous" HREF="node83.html">
<LINK REL="up" HREF="node82.html">
<LINK REL="next" HREF="node85.html">
</HEAD>
<BODY >
<!--Navigation Panel-->
<A NAME="tex2html2498"
HREF="node85.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
SRC="icons.gif/next_motif.gif"></A>
<A NAME="tex2html2495"
HREF="node82.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
SRC="icons.gif/up_motif.gif"></A>
<A NAME="tex2html2489"
HREF="node83.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
SRC="icons.gif/previous_motif.gif"></A>
<A NAME="tex2html2497"
HREF="node1.html">
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
SRC="icons.gif/contents_motif.gif"></A>
<BR>
<B> Next:</B> <A NAME="tex2html2499"
HREF="node85.html">Overview of ROMAFOT</A>
<B> Up:</B> <A NAME="tex2html2496"
HREF="node82.html">Crowded Field Photometry</A>
<B> Previous:</B> <A NAME="tex2html2490"
HREF="node83.html">Introduction</A>
<BR>
<BR>
<!--End of Navigation Panel-->
<H1><A NAME="SECTION00820000000000000000"> </A>
<A NAME="theory"> </A>
<BR>
Theory
</H1>
ROMAFOT uses a linearised least squares analytical reconstruction method
on two-dimensional data frames. Introductions of modern tests of
significance (e.g. <IMG
WIDTH="54" HEIGHT="78" ALIGN="MIDDLE" BORDER="0"
SRC="img179.gif"
ALT="$\chi^{2}$">)
to measure the discrepancy between
observations and hypothesis can be found in many excellent text books
(see e. g. Bevington, 1969). Hence, it is not really useful to give
any theoretical summary here.
<P>
Although minimisation of the absolute deviations or, more precisely,
the square of deviations between data and a parametric model is not
the only technique to derive optimum values for parameters, it is
generally used because it is straightforward and theoretically justified.
There are no basic difficulties in extrapolating this method for the
case of a model function with non-linear parameters if <IMG
WIDTH="54" HEIGHT="78" ALIGN="MIDDLE" BORDER="0"
SRC="img180.gif"
ALT="$\chi^{2}$">
is continuous in parameter space. Problems may arise if local minima
occur for ``reasonable'' values of parameters and methods need to be
employed to search for absolute minima.
<P>
Nevertheless, if the PSF parameters are reasonably estimated one can
assume that the localisation of the region of absolute minimum is
generally well determined. This is an important condition since the method
ROMAFOT uses requires that high order terms in the expansion of the fitting
function are negligible, a condition satisfied if the starting point
is close to the minimum.
<P>
This method is to expand the fitting function to first order in a Taylor
expansion, where the derivatives are taken with respect to the parameters
and evaluated for the initial guess in this space. The calculation of the
model parameters to fit the data has led to the solution of the usual
system:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{displaymath}
\frac{\delta\,\Sigma}{\delta\,P_{j}} = 0,
\end{displaymath} -->
<IMG
WIDTH="85" HEIGHT="52"
SRC="img181.gif"
ALT="\begin{displaymath}\frac{\delta\,\Sigma}{\delta\,P_{j}} = 0,
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where <IMG
WIDTH="46" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
SRC="img182.gif"
ALT="$\Sigma$">
is the summation over all the <I>n</I> points of the square of
the difference between the model function and the data, and <I>P</I><SUB><I>j</I></SUB> is
the current parameter (
<!-- MATH: $j = 1,....,m$ -->
<I>j</I> = 1,....,<I>m</I>). As the method requires
an approximation of the function instead of the function itself,
the parameter adjustments the system reinstitutes are not exactly those
which correct the trial values, and the operations must be iterated.
<P>
Using the technique just outlined, ROMAFOT carries out stellar photometry
by fitting a sum of elementary Point Spread Functions, analytically
determined, to a two dimensional array. The sky contribution is taken
into account with an analytical plane, possibly tilted. This plane is
fitted (simultaneously) with the stars.
<P>
The analytical formulation of the PSF has obvious advantages in case
of undersampled images allowing the comparison of the intensity of each
pixel to the <I>integral</I> of the function over the pixel area.
ROMAFOT does not attempt to discriminate between stars and galaxies in
the phase of searching the objects. It prefers to take into account their
photometric contribution and delete them from the final catalogue by
checking the fit parameters with a suitable program. This program also
eliminates cosmic rays and defects, if present.
<P>
In several parts in this documentation reference is made to the sigma
(<IMG
WIDTH="44" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
SRC="img183.gif"
ALT="$\sigma$">)
of either the Gaussian or the Moffat fitting function. In
both cases this sigma is <B>NOT</B> the sigma in the statistical sense
(i.e. the standard deviation) but the Full Width Half Maximum (FWHM)
of the distribution. In case of the Moffet distribution, sigma is a
function of the parameter <IMG
WIDTH="44" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
SRC="img184.gif"
ALT="$\beta$">,
i.e. not equal to the FWHM except
for <IMG
WIDTH="32" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img185.gif"
ALT="$\beta=3.1$">.
This definition of sigma will be used throughout this
chapter.
<P>
<HR>
<!--Navigation Panel-->
<A NAME="tex2html2498"
HREF="node85.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
SRC="icons.gif/next_motif.gif"></A>
<A NAME="tex2html2495"
HREF="node82.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
SRC="icons.gif/up_motif.gif"></A>
<A NAME="tex2html2489"
HREF="node83.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
SRC="icons.gif/previous_motif.gif"></A>
<A NAME="tex2html2497"
HREF="node1.html">
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
SRC="icons.gif/contents_motif.gif"></A>
<BR>
<B> Next:</B> <A NAME="tex2html2499"
HREF="node85.html">Overview of ROMAFOT</A>
<B> Up:</B> <A NAME="tex2html2496"
HREF="node82.html">Crowded Field Photometry</A>
<B> Previous:</B> <A NAME="tex2html2490"
HREF="node83.html">Introduction</A>
<!--End of Navigation Panel-->
<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
</ADDRESS>
</BODY>
</HTML>
