Difference between binning and averaging (2)

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.

2 Responses to “Difference between binning and averaging (2)”

  1. Jordan White says:

    Hi, great explanation, very useful! Does the theory still apply with rectangular binning (e.g. 2×4)?

  2. albert says:

    Yes, the theory is independent of the number of pixels that are binned.
    Albert.

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