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>
>@Zdenek: A SNR of 3.5 or a little bit more might be enough for some 
>applications ... I was planning to measure the photon count per 
>pixel (using this method :
>http://labrigger.com/blog/2010/07/30/measuring-the-gain-of-your-imaging-system/ 
>), but I always was busy with other things, so I cannot give you 
>numbers for my imaging system.

Hi all,

Thanks to Labrigger for working on this important topic.

However, I have read his analysis and think that the assumption that 
one can use this procedure to measure the number of photoelectrons 
(PE: i.e., detected photons) created at the photocathode (PC) of the 
PMT may be an over-simplification.

The analysis depends on the assumption that the only source of noise 
in the data recorded in the "image" of a flat white field is Poisson 
Noise associated with the small number of PEs produced at the 
photocathode.  This might be true if PMTs were free from 
multiplicative noise but in fact Poisson Noise also affects every 
stage in the multiplication of a single PE after it leaves the PC. In 
the very unusual case that the voltage between the PC and the first 
dynode is 500-600 volts (and that this dynode has both the optimal 
shape and the best GaAs surface), the gain of this stage may be 25 
+/-5 or 20% additional noise. More commonly, this gain will be closer 
to  4 +/-2 or 50% additional noise. More noise is added at each stage 
and even though these noise terms are not additive (they are combined 
as the sqrt of the sum of the squares), it is not at all uncommon for 
this process to double or even triple the variation present in the 
resulting signal beyond what one would expect from Poisson Noise 
applied only to the number of PE. Furthermore, this added noise will 
be somewhat larger if the system is working at a relatively high 
signal level because then the PMT will be turned down, the gain/stage 
correspondingly lower and the Poisson Noise proportionally higher.

Offsetting this error to some extent is the finite bandwidth of the 
entire amplifier system (PMT plus the electronics between the final 
dynode and the ADC). This bandwidth is in general unknown but may be 
adjusted by the computer to more-or-less match what the computer 
estimates is needed to pass the finest optical details that the 
system can transmit on the basis of settings for wavelength, 
objective NA, zoom/pixel size, and even PMT setting (high PMT voltage 
implies a noisy signal that may benefit from the artificial, 
1-dimensional smoothing that attends lower bandwidth).

Clearly, because bandwidth limits the maximum excursion that can be 
transmitted between one pixel and its neighbour, it will tend to 
reduce the apparent noise present in the digitized signal. The 
magnitude of this clipping is unknown but may vary with the 
parameters mentioned above.

This is relevant because, unlike the optical signal, the Poisson 
Noise signal that we are searching for shows no correlation between 
adjacent pixels. In particular, following the blog's suggestion of 
using a high zoom (to reduce fixed pattern noise) may cause the 
computer to limit the bandwidth more than using a lower zoom.

Although, as noted above, because these two factors bias the results 
in opposite directions, their effects may cancel each other out to 
some extent. However, we need to know a lot more about how the 
components are actually operating before we can decide whether and to 
what extent this is true.

The analysis also assumes that there is no fixed patterns noise in 
the image of a "flat white field" as might be caused, for instance, 
by field curvature, spherical aberration, vignetting, dust or other 
optical parameters that may change detected signal across the field 
of view.  I note that many of these sources of non-Poisson Noise can 
be substantially reduced by recording two sequential frames and 
obtaining a measure of the noise by subtracting one from the other.

For the analysis to work, it is also important to set the brightness 
control (DC - offset) so that zero signal corresponds to closely to 
zero intensity in the image memory.

I should note that multiplicative noise ceases to be a factor in 
systems employing either hybrid PMT (where the first stage gain is 
about 10,000) or effective photon-counting (i.e. a photon counting 
where the recorded peak pixel signal is at least 10x smaller than the 
saturation count rate of the system as set by pulse-pileup.).

One can avoid multiplcative noise by recording the data using  a CCD 
(but NOT on an EM-CCD used with the electronic gain turned on) and 
the record-two-then-subtract approach can again be used to reduce 
inevitable fixed pattern noise.  However, this sensor will probably 
work best when recording a fairly large signal (at least 10% of 
peak?) so that read noise will be relatively insignificant. And as 
above, the results will again be limited by the finite bandwidth of 
the FET amplifier between the read-node and the ADC. Finally, when 
using a CCD for quantitative measurements, it is particularly 
important to remember that they are usually set up so that zero light 
corresponds to 20-50 computer intensity units.

The noise performance of sCMOS detectors is both non-Gaussian and 
depends strongly on the extent to which the internal pixel-by-pixel 
variations in gain and offset are detected and corrected. This will 
make their use for this type of measurement somewhat more difficult 
unless the signal levels are well away from the noise floor.

Bottom line: Although the procedure may indeed give a useful 
benchmark that we might call the "effective gain" of the signal path, 
the measurement is subject to influence by a number of imaging 
parameters and will not really allow one to measure how many 
recorded-signal-intensity-units correspond to one PE.

Jim Pawley
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