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How you do this will depend a bit on how much light you've got. I generally tend to model pixels as the sum of a constant, Gaussian distributed, noise source which models read noise etc, and a Poisson noise source which depends on the number of photo-electrons detected. For most modern detectors, the read noise component is almost negligible. If you've got reasonable signal levels (ie signal + background > ~ 10 photons) the Poisson noise can be very well approximated as a Gaussian noise source with a std. deviation equal to the sqrt of the number of photo-electrons in a given pixel. In this case you can use standard Gaussian error propagation and say that:
Err(A - B) = sqrt(Err(A)^2 + Err(B)^2)
or approximately sqrt(N_A + N_B), where N_A and N_B are the number of photoelectrons in each of the pixels.
For electron multiplying devices (ie EMCCDs and photomultipliers) you'll also need to take the excess noise factor into account.
For very low photon numbers, you're going to have to be a bit more rigorous I don't know the details off the top of my head, and how rigorous you need to be will depend on the exact application (The main problem with approximating a Poisson distribution with a Gaussian based on the sqrt of the number of observed photo-electrons is that it will give an observed count of 0 a variance of 0, which is usually incorrect. A semi-empirical way of fixing this which works for most practical applications is to take the std. deviation as sqrt(1 + N) ).
hope this helps,
David
________________________________
From: Xiang Hao <
[hidden email]>
To:
[hidden email]
Sent: Thursday, 12 July 2012 4:30 PM
Subject: How to analyze SNR in subtraction images?
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Dear all,
our lab now are building up a VAAS (Virtual Adaptable Aperture System)
confocal microscope. As the final image is generated by subtracting two images
obtained with different-sized pin hole, we want to analyze the signal-to-noise
ratio (SNR) of the system. We check the reference and have already known the
aspects that may influence SNR from the conspicuous work by C.J.R Sheppard et
al, such as the dark current of detector, the background noise, and the shot
noise. Among all these aspects, some are constant, while the others depend on
the quantum efficiency which obeys a Poisson distribution. Therefore, this problem
becomes a little complex. I just want to know if anyone has ever considered
about this problem and can give me some inspiration, or if there are some
corresponding papers that have been published. Thank you for your help!
Hao,Xiang
State Key Laboratory of Modern Optical Instrumentation
Zhejiang University
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