In the previous blog the focus was on binning (charge, voltage and digital domain) in the case the readout noise was dominating over the photon-shot noise. In other words, for the case of small signals or low light levels. This time, the situation for a shot-noise limited condition is considered. And actually the story can be very short : it does not matter when or where the binning is done, in all cases the result is exactly the same.
For the charge domain : if n x n charge packets are added together, each with m electrons, then after binning the final charge packet holds :
n x n x m electrons.
The photon-shot noise in each individual charge packet was
sqrt(m) electrons,
so the SNR for every individual charge packet was
SNR = m/sqrt(m) = sqrt(m).
After binning the total photon-shot noise is equal to
sqrt(n x n x m) electrons
and the SNR will be equal to :
SNR = n x n x m/sqrt(n x n x m) = sqrt (n x n x m).
After binning in the charge domain, the increase in SNR will be
sqrt(n x n x m)/sqrt(m) = n !
For the voltage domain or digital domain : if n x n signals are added together, each with m electrons, then before binning the output signal would have been k x m V or DN, with k being the conversion gain from input (charge) to output (Volts or Digital Numbers). Then after binning the final signal will be
n x n x k x m V or DN.
The photon-shot noise of each individual signal before binning was
k x sqrt(m) V or DN,
so the SNR for the individual signal before binning was
SNR = (k x m)/(k x sqrt(m)) = sqrt(m).
After binning the total photon-shot noise is equal to
k x sqrt(n x n x m)
and the SNR will be equal to :
SNR = (n x n x k x m)/(k x sqrt(n x n x m)) = sqrt (n x n x m).
After binning in the voltage of digital domain, the increase in SNR will be
sqrt(n x n x m)/sqrt(m) = n !
Conclusion : if there is enough light so that the performance of the sensor or the camera is shot-noise limited, it does not matter how the binning is realized, charge domain, voltage domain or digital domain. The increase of the SNR after binning is always equal to a factor n, being the kernel size in the case of n x n binned pixels.
Albert, 31-05-2016.
Hi, great explanation, very useful! Does the theory still apply with rectangular binning (e.g. 2×4)?
Yes, the theory is independent of the number of pixels that are binned.
Albert.