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<H2><A NAME="SECTION00682000000000000000"> </A>
<A NAME="ccd:combining-methods"> </A>
<BR>
Combining Methods
</H2>
Except for summing the frames together, combining frames may require
correcting for variations between the frames due to different exposure
times, sky backgrounds, extinctions, and positions. Currently, scaling
and shifting corrections are included. The scaling corrections may be
done by exposure times or by using statistics in each frame over a
selected part of the image. The statistics can reveal (depending
on the keyword <TT>`exp'_STA</TT>) setting, (where <TT>`exp'</TT> is the
exposure type) for each image the mean, median, or the mode. In the
following we refer to the value by <I>MMM</I>. Additive shifting is also
done by computing the statistics in the frames.
<P>
The region of the frames in which the statistics is computed can
be specified by the keyword <TT>`exp'_SEC</TT>. By default the whole
frame is used. A scaling correction is used when the flux level or
sensitivity is varying. The offset correction is used when the sky
brightness is varying independently of the object brightness. If the
frames are not scaled then special routines combine the frames more
efficiently.
<P>
Below follows a simple overview how the weighting, scaling
and offset parameters are determined. All obviously depend on the
settings of the keywords <TT>`exp'_SCA</TT> <TT>`exp'_OFF</TT>,
<TT>`exp'_WEI</TT>, and <TT>`exp'_EXP</TT>. The overview makes clear that
offset corrections will only be applied if the scaling correction
is switched off. The same is true for applying an exposure time
correction.
<P>
<PRE>
==========================================================================
o_i = 0.0
w_i = 1.0
s_i = 1.0
exp_SCA=yes
s_i = M_i
exp_WEI=yes
w_i = sqrt(N*s_i)
exp_SCA=no
exp_EXP=yes
s_i = e_i
exp_WEI=yes
w_i = sqrt(N*s_i)
exp_OFF=yes
o_i = M_i/s_i
exp_WEI=yes
w_i = sqrt(N*s_i/o_i)
s_i = s_i/s_mean
o_i = (o_i - o_mean) * s_mean
w_i = w_i/w_sum
--------------------------------------------------------------------------
key: o_i: offset for frame i
o_mean: mean offset over all input frames
s_i: scale factor for frame i
s_mean: mean scale factor over all input frames
w_i: weight factor for frame i
w_sum: sum over all weight factors of all input frames
e_i: exposure time of frame i
M_i: MMM of frame i
N: number of of frames previously combined
==========================================================================
</PRE>
<P>
In the combining no checks are done on the reduction status of the input
frames and no attempts are made for any calibration correction like for
bias or dark. Hence, in more complicated reduction sequences the user should
be sure not to combine <I>e.g.</I> flat fields that have been
corrected for bias and dark with flats fields that are not corrected.
<P>
Except for medianing and summing, the frames are combined by
averaging. The average may be weighted by <BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{equation}
weight =
(N * scale) ** 1/2
\end{equation} -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP>
<I>weight</I> =
(<I>N</I> * <I>scale</I>) ** 1/2
</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(3.10)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
where <I>N</I> is the number of frames
previously combined (the command records the number of frames
combined in the frame descriptor), <I>scale</I> is the scale factor
depending on the keyword settings listed above (<TT>s_i</TT> or <TT> s_i/o_i</TT>). In most of the applications <I>N</I> = 1, <I>i.e.</I> the
input calibration frames are the original ones and not the result of
previous combinings.
<P>
There are a number of algorithms which may be used as well as applying
statistical weights. The algorithms are used to detect and reject
deviant pixels, such as cosmic rays. The choice of algorithm depends
on the data, the number of frames, and the importance of rejecting
cosmic rays. The more complex the algorithm the more time consuming
the operation. For every method pixels above and below specified
thresholds can be rejected. These thresholds are stored in the keyword
<TT>`exp'_MET</TT>. If used the input frames are combined with pixels above
and below the specified threshold values (before scaling) excluded.
The sigma frame, if requested, will also have the rejected pixels excluded.
<P>
The following list summarizes the algorithms. Further algorithms are
available elsewhere in MIDAS (see <TT>COMPUTE/...</TT>, <TT>AVERAGE/...</TT>),
or may be added in time.
<UL>
<LI>Sum - sum the input frames.
<BR>
The input frames are combined by summing. Summing is the only
algorithm in which scaling and weighting are not used. Also no
sigma frame is produced.
<P>
<LI>Average - average the input frames.
<BR>
The input frames are combined by averaging. The frames may be scaled
and weighted. There is no pixel rejection. A sigma frame is produced
if more than one frame is combined.
<P>
<LI>Median, MMedian - (mean) median the input frames.
<BR>
The input frames are combined by medianing each pixel. Unless the frames
are at the same exposure level they should be scaled. The sigma frame
is based on all input frames and is only a first approximation of the
standard deviations in the median estimates.
The second method does an averaging around the found median in a
certain interval in order to take into account the distribution of
the values near the median. This is in effect the same what <TT> AVERAGE/IMAGE</TT> also does using the parameter setting 'options = median,low,high'. The required data interval has to be defined by the
<TT>exp_CLP</TT> keyword and is assumed to specify relative limits to the
determined median - same as in <TT>AVERAGE/IMAGE</TT> (both limits positive).
<P>
<LI>Minreject, maxreject, minmaxreject - reject extreme pixels.
<BR>
At each pixel after scaling the minimum, maximum, or both are
excluded from the average. The frames should be scaled and
the average may be weighted. The sigma frame requires at least two
pixels after rejection of the extreme values. These are relatively fast
algorithms and are a good choice if there are many frames (>15).
<P>
<LI>Sigclip - apply a sigma clipping algorithm to each pixel.
<BR>
The input frames are combined by applying a sigma clipping algorithm
at each pixel. The frames should be scaled. This only rejects highly
deviant points and so includes more of the data than the median or
minimum and maximum algorithms. It requires many frames (>10-15) to
work effectively. Otherwise the bad pixels bias the sigma significantly.
The mean used to determine the sigmas is based on the "minmaxrej"
algorithm to eliminate the effects of bad pixels on the mean. Only one
iteration is performed and at most one pixel is rejected at each
point in the output image. After the deviant pixels are rejected the
final mean is computed from all the data. The sigma frame excludes the
rejected pixels.
<P>
<LI>Avsigclip - apply a sigma clipping algorithm to each pixel.
<BR>
The input frames are combined with a variant of the sigma clipping
algorithm which works well with only a few frames. The images should
be scaled. For each line the mean is first estimated using the
"minmaxrej" algorithm. The sigmas at each point in the line are scaled
by the square root of the mean, that is a Poisson scaling of the noise
is assumed. These sigmas are averaged to get a line estimate of the
sigma. Then the sigma at each point in the line is estimated by
multiplying the line sigma by the square root of the mean at that point.
As with the sigma clipping algorithm only one iteration is performed and
at most one pixel is rejected at each point. After the deviant pixels
are rejected the file mean is computed from all the data. The sigma
frame excludes the rejected pixels.
</UL>
<P>
The "avsigclip" algorithm is the best algorithm for rejecting cosmic
rays, especially with a small number of frames, but it is also the
most time consuming. With many frames (>10-15) it might be advisable
to use one of the other algorithms ("maxreject", "median", "minmaxrej")
because of their greater speed.
<P>
The choice of the most optimal combining algorithm will clearly depend
on the nature of the data and on the exposure type. Therefore, for
every supported exposure type the CCD context contains a default
combining setup. Currently, there are five combining setups stored
in the CCD keywords, all starting with a specific two letter prefix: for
bias <TT>BS_</TT>, dark <TT>DK_</TT>, dome flats <TT>FF_</TT>, sky flats
<TT>SK_</TT>, and for all other exposure types <TT>OT_</TT>. At initialization
these keywords are filled with sensible defaults. Below we will shortly
comment on combining the various calibration frames and list the default
keywords settings.
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<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
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