Wei-Hao,
Thank you for an
excellent explanation on some key topics. I disagree with you just in one point:
your English is excellent. If you ever decide to write up those stories on
statistics and imaging, please let me know; I'll read them avidly
;-)
Regards,
Juan
______________________________________________________________________
Juan
Conejero, Pleiades Astrophoto Team
PixInsight Home Page: http://pleiades-astrophoto.com/pixinsight/
----- Original Message -----
Sent: Tuesday, September 20, 2005 1:43
PM
Subject: Re: [APML] What's the Best Way
to "Combine" Two Images ofDifferingTimes?
[...]
You are not really terribly
wrong. Indeed, it is exactly the "different signal
intensities" you
just mention taking care of the weighting. The longer
exposure
images look brighter. And therefore in a straight add, they
automatically have higher weights. The raw images are already
weighted
by the exposure times so there is no need to give additional
weights.
This can be viewed in another way. In the best possible
weighting method,
stacking an A min exposure and an B min exposure should
give a result
exactly identical to a single (A+B) min exposure. How
to achieve this?
Just directly add the raw images together without
any weights. The final
result doesn't care if the "integration" is
done on the CCD chip or in
softwares. Just collect and add the
photons together, and we will
have the best possible S/N.
Even
another different way to look at this. Above you mentioned "exposure
time improves S/N." This is true. And in a good linear device,
the S/N
improves as the square root of exposure time. (This is the
nature of both
photons and dark electrons.) Therefore, if we
calibrate the images first
(i.e., bring them to similar brightness) and
weight the images with 1/rms^2
as I mentioned earlier, what we really do
is weighting the images with
the exposure times. This is identical to
directly adding the raw images.
This also shows why error weighted mean
will give the best results, although
the idea of error weighted mean isn't
really restricted to photons. In general
statistics, a good mean is
an error weighted mean. In astronomy, an error
weighted mean is an
exposure time weighted mean.
Hope this clears your questions.
There are lots of stories to tell about
statistics and imaging. I
once tried to write up series of essays
on this for amateur astronomers and
I gave up. This is not because the
topic itself is difficult to
write, but because my English isn't good enough.
Writing academic papers is
already challenging enough. It's too much
for me to write something
that is easy to understand for the
amateur.
Cheers,
Wei-Hao
--
________________________________________________________________
Wei-Hao
Wang :)
Institute for Astronomy at University of
Hawaii
Address:
2680 Woodlawn Drive
Personal Website:
Honolulu, HI
96822
http://www.ifa.hawaii.edu/~wang
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