All my life I have believed that if you're counting photons then the
error of observing N counts is sqrt(N). However, a calculation I just
performed suggests its actually sqrt(N+1).
My purpose here is to understand the weak-image limit of data
processing. Question is: for a given pixel, if one photon is all you
got, what do you "know"?
I simulated millions of 1-second experiments. For each I used a "true"
beam intensity (Itrue) between 0.001 and 20 photons/s. That is, for
Itrue= 0.001 the average over a very long exposure would be 1 photon
every 1000 seconds or so. For a 1-second exposure the observed count (N)
is almost always zero. About 1 in 1000 of them will see one photon, and
roughly 1 in a million will get N=2. I do 10,000 such experiments and
put the results into a pile. I then repeat with Itrue=0.002,
Itrue=0.003, etc. All the way up to Itrue = 20. At Itrue > 20 I never
see N=1, not even in 1e7 experiments. With Itrue=0, I also see no N=1
events.
Now I go through my pile of results and extract those with N=1, and
count up the number of times a given Itrue produced such an event. The
histogram of Itrue values in this subset is itself Poisson, but with
mean = 2 ! If I similarly count up events where 2 and only 2 photons
were seen, the mean Itrue is 3. And if I look at only zero-count events
the mean and standard deviation is unity.
Does that mean the error of observing N counts is really sqrt(N+1) ?
I admit that this little exercise assumes that the distribution of Itrue
is uniform between 0.001 and 20, but given that one photon has been
observed Itrue values outside this range are highly unlikely. The
Itrue=0.001 and N=1 events are only a tiny fraction of the whole. So, I
wold say that even if the prior distribution is not uniform, it is
certainly bracketed. Now, Itrue=0 is possible if the shutter didn't
open, but if the rest of the detector pixels have N=~1, doesn't this
affect the prior distribution of Itrue on our pixel of interest?
Of course, two or more photons are better than one, but these days with
small crystals and big detectors N=1 is no longer a trivial situation.
I look forward to hearing your take on this. And no, this is not a trick.
-James Holton
MAD Scientist
########################################################################
To unsubscribe from the CCP4BB list, click the following link:
https://www.jiscmail.ac.uk/cgi-bin/WA-JISC.exe?SUBED1=CCP4BB&A=1
This message was issued to members of www.jiscmail.ac.uk/CCP4BB, a mailing list
hosted by www.jiscmail.ac.uk, terms & conditions are available at
https://www.jiscmail.ac.uk/policyandsecurity/
