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*t_{sub}]*(Obj)_{ }/
sqrt[(Sky+Obj)*t_{sub }+
R^{2}],

where K is the total (cumulative) exposure time, t_{sub} 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*t_{sub}]*4*(Obj)_{ }/
sqrt[4*[(Sky+Obj)*t_{sub}]+
R^{2}]

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*t_{sub}]*4*(Obj)_{ }/
sqrt[4*[(Sky+Obj)*t_{sub}+ R^{2}]]

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.

SNR (Tot) = sqrt[K*t

where K is the total (cumulative) exposure time, t

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*t

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*t

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`

1.
Binning is
mainly useful
for reducing the subexposure
duration, but this is
relevant only if you
feel that the calculated subexposure duration (unbinned) is
unacceptably long.
This
might occur, for
instance, when imaging at a dark site, using a high read noise camera,
a narrowband filter with a tight
bandpass, and/or a
high f
ratio.
However, if the noise contribution
from sky glow is already dominant in your subs, reducing the effective
read noise contribution with 2x2 binning will have very
little effect. For instance, if you have a low read noise camera
and are taking a luminance image at a light polluted site, your
calculated subexposures will typically be quite short, and binning
would provide little advantage but will compromise resolution (image
scale at 2x2 binning is twice that of unbinned, etc.). Also, note
that binning will not appreciably improve the final signal to noise ratio of the
cumulative exposure, assuming that your subs are photon limited.
For photon limited subs, an upper limit on the S/N ratio is imposed by
the equation shown in line 8 of my subexposure duration page.
The downside of binning is a reduction in resolution if your unbinned
images are already undersampled (or borderline undersampled). So
only use binning if you are
very well sampled to start with, and if you feel the need to shorten
your subexposure duration due to the reasons mentioned above. On
the other hand, if your calculated subexposure durations (unbinned) are
within a perfectly acceptable range (however you wish to define that,
based upon your mount, accuracy of polar alignment, and patience), then
you do not need to bin.

2. The other use of binning is to achieve an image scale that is more realistic for the seeing conditions. For instance, if your image scale unbinned is 0.3 arcsec/pixel, but if your seeing is at best 2.0 arcseconds, then you are wasting effort by imaging unbinned (i.e., you are oversampled, resulting in decreased sensitivity without a gain in resolution, and at the same time placing more demands on your mount and guider). This is because an unbinned image scale of 0.3 arcsec/pixel represents a sampling rate appropriate for seeing conditions of roughly 0.3 x 3 (Nyquist), or about 0.9 arcseconds. You can't resolve 0.9 arcseconds if your seeing is 2.0 arcseconds. Binning can be very helpful in this situation. If you bin such a system 2x2, the image scale becomes 0.3 x 2, or 0.6 arcsec/pixel. This is appropriate for a seeing of roughly 0.6 x 3 (Nyquist), or about 1.8 arcseconds, which is now well matched for your seeing of 2.0 arcseconds. Your images will not suffer any loss of resolution compared to the 0.3 arcsec/pixel image scale (because seeing is the limiting factor), but your system will be more manageable from the standpoint of autoguiding and file size.

2. The other use of binning is to achieve an image scale that is more realistic for the seeing conditions. For instance, if your image scale unbinned is 0.3 arcsec/pixel, but if your seeing is at best 2.0 arcseconds, then you are wasting effort by imaging unbinned (i.e., you are oversampled, resulting in decreased sensitivity without a gain in resolution, and at the same time placing more demands on your mount and guider). This is because an unbinned image scale of 0.3 arcsec/pixel represents a sampling rate appropriate for seeing conditions of roughly 0.3 x 3 (Nyquist), or about 0.9 arcseconds. You can't resolve 0.9 arcseconds if your seeing is 2.0 arcseconds. Binning can be very helpful in this situation. If you bin such a system 2x2, the image scale becomes 0.3 x 2, or 0.6 arcsec/pixel. This is appropriate for a seeing of roughly 0.6 x 3 (Nyquist), or about 1.8 arcseconds, which is now well matched for your seeing of 2.0 arcseconds. Your images will not suffer any loss of resolution compared to the 0.3 arcsec/pixel image scale (because seeing is the limiting factor), but your system will be more manageable from the standpoint of autoguiding and file size.

Steve