 What is Binning good for?

In unbinned imaging, the signal from each pixel is associated with a given amount of read noise.  For instance, 4 pixels would be associated with 4 separate read noise events, since each pixel is read individually. Binning is a procedure in which several pixels are grouped into a function unit, which has the effect of reducing the impact of read noise on the signal to noise ratio (SNR). In 2x2 binning, for example, an array of 4 pixels forms a functional "superpixel," accumulating the same signal as 4 individual pixels, but associated with only 1 read noise event (since the entire superpixel is read out as a unit).  However, the larger functional pixel size with binning results in lower resolution of the final image.  Below I express these concepts mathematically and derive some illustrative graphs of the effects of binning at a dark site and at a light polluted site.  If you are not interested in the math, just skip to the summary discussed in point #2 below.  Another excellent site that explains these concepts may be found here.

`SNR binned and unbinned `
1.  From equation 6 on my subexposure duration page, note that the signal to noise ratio per pixel in the total (cumulative) exposure is:
SNR (Tot) =
sqrt[K*tsub]*(Obj) / sqrt[(Sky+Obj)*tsub + R2],

where K is the total (cumulative) exposure time, tsub is the subexposure duration, Obj is the object flux in e/pixel/min, Sky is the sky flux in e/pixel/minute, and R is the read noise in e RMS.

2.  Let's derive the equation for  the SNR (binned 2x2) contributed by the signal from a binned 4 pixel unit (the "superpixel"), over a total (cumulative) exposure K.  It would simply be:
SNR (binned 2x2, i.e., 4 pixels) = sqrt[K*tsub]*4*(Obj) / sqrt[4*[(Sky+Obj)*tsub]+ R2]

Compared to just one pixel, the signal is obviously 4 fold greater since we are now collecting light over an area of 4 pixels.  But notice something important about the noise.  The noise increases, but this is due only to the increased contribution of photon noise, not read noise (note the yellow parentheses).  This is the power of binning, since the read noise contribution does not increase, even though 4 pixels are being read out.  They are viewed as a unit.  More on this topic can be found near the end of the page on this informative website.

3.  Now let's derive the comparable equation for the SNR of 4 pixels read out individually (i.e., unbinned):
SNR (4 pixels, but unbinned) =
sqrt[K*tsub]*4*(Obj) / sqrt[4*[(Sky+Obj)*tsub+ R2]]

Notice that it's the same signal (i.e., 4 fold greater than one pixel), but look at what's happened to the noise.  Since each pixel is being read out, the entire noise component (photon plus read noise) is increased 4 fold (note the yellow parentheses).

4.  Using these two equations and applying some arbitrary (but reasonable) initial values for the variables at a dark site versus light polluted site yields the following graphs:

Graph of binning at a dark site.  Conditions are stated in the figure legend and include a total (cumulative) exposure time of 6 hours for both curves.  Dark site in this example is defined as sky flux = 5 e/pixel/minute.  The exact number is not as important as the principle illustrated by the shape of the curves.

Graph of binning at a light polluted site
As above, conditions are stated in the figure legend and include a total (cumulative) exposure time of 6 hours for both curves.  Light polluted site in this example is defined as sky flux = 100 e/pixel/minute.  The exact number is not as important as the principle illustrated by the shape of the curves.

`Summary`