Capturing faint detail in deep-sky astronomical images is all about
overcoming noise. Noise comes from a variety of sources:
Optimal images will reduce these noise sources as much as possible.
CCDs are designed to generate low dark current, and cameras are cooled to remove
as much thermal noise as possible. Dark frames that capture the remaining
dark noise are subtracted to eliminate it from the final images. Cameras
are designed to have low readout noise. And observers travel to dark sites
to help eliminate light pollution.
Combining Exposures to Reduce Noise
The usual method for reducing noise in astronomical images is to stack
multiple exposures. This works due to the random nature of noise. In
an image, there is signal and there is noise. While the
signal (the light from the object being imaged) stays the same from image to
image, the noise changes. The signal-to-noise ratio (SNR) measures how
much signal there is relative to the noise levels. Decreasing the amount
of noise in an image increases the signal-to-noise ratio and results in a better
Above: Single exposure. Faint star circled has SNR of
Above: 10 stacked exposures. Faint star now has SNR of
15, an increase of 3 times. Noise is noticeably reduced.
Since signal stays constant, averaging two exposures together doubles the signal.
But due to the random nature of noise, averaging two exposures increases the noise by
only √2, about 1.4 times.
Signal to Noise Ratio
An exact equation describing signal to noise ratio is not easily determined
due to the large number of factors and inexact nature of those parameters.
However, for most purposes, it is sufficient to describe SNR as a function of
the following factors (each described more thoroughly below):
Number of Exposures
Sky Background Flux
We can consider each of these factors in turn to see what their effect is and
the quantity or significance of their effects. Also, this will lead us to
a method of determining optimal exposure times and other methods for reducing
noise and increasing SNR.
Note: The above factors determine SNR for random noise sources.
There still exist non-random sources which will be considered later.
This is probably the most important factor. The most obvious way to increase
SNR is simply to increase exposure time. For most deep-sky images,
doubling the exposure time increases the SNR by √2
= 1.4 times. Sky glow, from light pollution sources, prevents us from
taking indefinitely long exposures so SNR must be increased through other means.
Sky glow limitations also imply that there may be an optimal exposure time for a
given imaging system and location, which we will see is true.
Number of Exposures
We saw earlier how stacking multiple exposures
increased SNR. Perhaps stacking multiple exposures taken at the optimal
exposure time would be preferable to a single longer exposure. We will see
that this is true, and for a variety of reasons. There are myriad ways to
combine image files, and they are discussed in more detail below. The
basic method is to average exposures, taking the mean value of each common pixel
to produce a result with less noise. Combining N exposures this way leads
to a SNR increase of √N. As seen in the examples above, averaging 2
exposures yields a √2 = 1.4 increase in SNR, and averaging 10 exposures gives an
increase of √10 = 3.16. It can also be seen that there is a point of
diminishing returns, with an increase in exposures N yielding only a slight
increase in SNR. Going from 2 to 10 exposures gives a 225% increase in SNR,
while increasing to 20 exposures gains only another 140%. As will be seen
below there are other reasons to use a larger number of subframes; for
example, it might be preferable to take ten 5-minute exposures rather than five
The flux is simply the rate at which light from
the target reaches the CCD chip. Flux is often measured in photons per
second, but in the case of a CCD camera, which is converting photons to
electrons, a more practical value is electrons per second, abbreviated e-/sec.
The brighter the object, the greater the flux. With astronomical subjects
the flux is typically low, often not much higher than the sky flux described
Sky Background Flux
This is the flux of the sky glow, determined
primarily by light pollution factors. The sky (whether lit by city lights
or the moon or natural airglow) produces photons that are captured by the CCD
and turned into the background of the image. Note that the background in
an image is not perfectly black but has some value. This value is a
function of the sky background flux and the exposure time. For example,
the sky from a dark site might have a flux of 2e-/sec (this is also a function
of focal ratio since it is measured in terms of what the CCD counts rather than
what the sky itself is producing). In a 5 minute exposure, the
background will reach a value of 300sec x 2e-/sec = 600e-.
Using image processing software, the sky background can be easily measured,
although there is a slight calculation necessary to convert to the background
sky flux in e-/sec.
The above image has an average background sky pixel value of 950. Most
software packages add a pedestal value of 100 to the pixel values in order to
prevent negative numbers. This means the true value is 850. This is
called the Background ADU Count. To convert to a value in electrons, this
number must be multiplied by the camera gain. In this example, an ST-10XME
camera was used, which has a gain of 1.3e-/ADU. Therefore the actual sky
background value is 1105e-. The exposure was 600 seconds, implying a sky
background flux at this observing location of 1.8e-/sec, indicating a dark site.
Again this is a function of the focal ratio, in this case f/7.
This image was taken from a suburban location and has a background ADU count
of 2500. Converting using the equations above gives a value of 3120e-.
Exposure time was again 600 seconds and the flux is 5.2e-/sec, indicating that
the sky is much brighter from this location. (Note that the focal ratio of
the second picture was f/5.4, which means the equalized sky background flux is
3.1e-/sec, indicating the sky was 72% brighter from this location than where the
top picture was taken.)
Measuring Sky Background
In MaxIm DL, use the
Information command to measure the background ADU count. Set the
Aperture Radius to a moderate value such as 6 or 8 to average the values out
over a larger area. Be sure to move the cursor to a few different
background spots and take an average. Estimating to the nearest 50 or
100 should be more than adequate, given all the variables that will affect
an exposure. Be sure to avoid faint areas of nebulosity and keep away
from the corners of the image where there might be vignetting that will
lower the value.
Binning affects the SNR by effectively increasing the sensitivity of the CCD
chip. The effect is similar to making the focal ratio faster.
Binning a CCD 2x2 combines each 2x2 group of pixels into one "super pixel" which
can gather 4 times as much light as a single smaller pixel during a given
exposure. Thus the system becomes 4 times faster when binned 2x2,
equivalent to a 2-stop reduction in focal ratio. However, as described
below, resolution determines SNR as well, and since binning decreases
resolution, it can also decrease SNR.
Resolution is a major factor in determining fine detail SNR such as that of
stars. Increased resolution gives increased SNR. However, sampling
is an important factor as well. (See the section on
Nyquist Theorem for more details on sampling.) Undersampled images
(such as those taken with short focal length scopes and/or binned CCD chips) will
have worse SNR than properly sampled images.
Focal ratio is the primary determinant of imaging speed. Deep-sky
imagers all know the importance of having a fast scope for reducing exposure
times. Focal ratio also affects resolution, assuming a constant aperture
(in other words, using a focal reducer on a given telescope) and thus affects
SNR in the same way. More importantly, focal ratio determines exposure
time necessary to achieve a given sky background flux, the importance of which
will be seen in the next section.
Dark noise is generated within the camera by heat sources. This is the
reason CCD cameras are cooled; colder cameras have less dark current. Most
cameras, even after cooling, still have residual dark noise which is then removed using dark frames.
Assuming a sufficient number of dark frames have been taken and combined to make
a master dark frame, dark noise can essentially be ignored for SNR calculations.
This is an important determinant when it comes to selecting optimal subframe
exposures. Take a look at a simplified equation for SNR:
This equation assumes that dark current is not a significant factor (because
it has been removed) and that the sky flux is large relative to the object flux
(as is the case for faint deep-sky objects). The remaining factors are:
Readout noise is generated by the CCD when the data from the chip is
transferred to the computer. This is a measurable quantity and is quoted
by CCD manufacturers in the spec sheets for their products. The SBIG
ST-10XME used in the above examples has a typical readout noise of 7e-. In terms
of choosing an ideal exposure time, there is little we can do about the object
flux, and the background flux will be more or less constant for a given
location. What we can do, however, is minimize the contribution of the CCD
readout noise to the overall image noise. How to do this is discussed in
the next section, but it can be seen from the equation above that making the sky
background flux, Esky, large compared to the readout noise, Ron,
will minimize the contribution from readout noise.
Sky Limited Exposures
Note: Equations are given in the discussion below but a
calculator is provided at the end of this section for easy calculations.
A very important idea is that of sky limit. An exposure in which the
primary limiting factor is the background sky flux is called sky limited.
In such an exposure the sky background flux is the primary factor in determining
SNR. Astroimager Stan Moore states, "sky limit is the zone where the sky
noise overpowers the readout noise". Just where this point occurs is
somewhat arbitrary but there is a generally accepted guideline:
Stan Moore recommends keeping the readout noise contribution to just 5% of
the total noise, the point where the sky noise is 3 times greater than the
John Smith's excellent article on the subject also adopts this 5% readout
noise rule. The equations which follow are from that article.
A little rearranging of the SNR equation gives us an equation for determining
what John Smith calls the exposure time to overwhelm readout noise, tORN:
Here, the value p is the percent contribution from the readout noise.
Inserting the above recommended value gives the following equation that will
be used for determining optimal subframe exposure times:
The trick now is to determine Esky, the sky background flux. By
measuring the background ADU value from a test image, as shown in the examples
above, the sky background flux can be calculated from the following equation:
Here ADUbkg is the background ADU count, g is the gain, ttest
is the test exposure duration in minutes. Take the first example given
earlier, of the image taken from the dark location. The ADUbkg
value was determined to be 950. 100 is subtracted for the pedestal value
described above. The gain of the ST-10XME camera used is
1.3e- and the exposure was 10 minutes. This gives a value of Esky
= 111e-/min (equal to the 1.8 e-/sec from the example).
This value of Esky is then plugged into the tORN
equations. The readout noise for the ST-10XME is 7e-. For
a 5% contribution from readout noise, the necessary exposure is 4.3 minutes. This implies that
the 10 minute exposure was well beyond that needed to overwhelm the readout
noise from the camera and is a sky limited exposure.
To get the most possible information out of the entire image,
any vignetting in the corners of the image needs to be taken into account.
With a large-format camera and fast optical system, such as a HyperStar system
or Takahashi Epsilon astrograph, there will be some darkening of the image
corners due to light falloff. A flat field is necessary to minimize the
effects of vignetting. The significance of vignetting in determining ideal
exposure times is that the darkest (most vignetted) portion of the image should
be used to calculate the sky background. This means even the most
weakly-illuminated parts of the image will receive enough photons to overwhelm
Other Noise Sources
Once we have determined what is required for a sky limited exposure, is that
really the best exposure time to use for subframes? There is a convincing
argument for using a greater number of shorter subframes, as opposed to a small
number of long subframes. The reasons have to do primarily with other
sources of noise that have not yet been discussed, the non-random noise sources.
In addition to readout noise, dark noise, and sky background noise, there are
several additional significant noise sources:
Hot and Cold Pixels
Hot and Cold Pixels
These are essentially defective pixels in a CCD chip. Hot pixels have a
maximum value, they are filled with charge even if they are not truly gathering
that many photons from the image. They appear as pure white in the final
image. Cold pixels, or dead pixels, register no charge from the photons
that strike them, rendering them pure black in the final image. These are
a non-random noise source because they depend on the position of the defective
pixels in the CCD and will not change from image to image. This also means
that they will not be removed by combining images if the pixels all align
exactly. For this reason, they are best removed by dithering (see below).
Technically, cosmic rays are a random source of noise, because they can occur
anywhere in the image at any time. But they always result in saturated
pixels, which means their value is non-random. High energy particles strike Earth's atmosphere and
release a rain of charged particles and photons which are detected as bright
specks in an image. Because of the high value (100% brightness) of cosmic
rays artifacts, they are not easily removed by certain combination routines such
as averaging images. Better methods exist for combining images that will
provide better cosmic ray removal, but they benefit from a greater number of
Additional problems can arise from pixel defects, including column defects,
and from interlopers into an image such as airplanes, satellites, and meteors.
These effects can all be minimized using certain combining routines.
Image Combining Methods
There are a variety of means for combining subframes into a final exposure,
and each has advantages:
Mean or Average Combine
Sigma Clip Combine
Mean or Average Combine
This method provides the best SNR increase but is worst at removing
non-random noise. If a non-random artifact (esp. hot and cold pixels)
occurs on the same pixel in each image, they will not be removed by this method.
Dithering (see below) is a good method for minimizing hot and cold pixel
For example, for N=5, SNR is proportional to 2.2. Compare this to the
values for the other combine functions.
Median combine rejects the highest and lowest pixel values and thereby
removes extremely bright semi-random artifacts such as cosmic rays.
However, since hot and cold pixels remain the same from image to image, they are
not removed by median combine unless dithering is used. Median combine is
better at artifact removal but at the expense of reduced SNR in terms of random
For example, for N=5, SNR is proportional to 1.78, or only 81% that of the
mean combine method.
MM-clip offers the best non-random noise reduction and can have less SNR
loss than median combine. MM-clip rejects the highest or lowest value
before taking a median value from the remaining pixel values. This
eliminates extreme pixel values from contributing to the median value.
For N=5, median combine actually has less SNR loss. For N=5,
SNR is proportional to 1.73, slightly less than that for median combine.
But for N=6, MM-clip SNR is proportional to 2, whereas median SNR is
proportional to 1.95. So for greater than 6 subframes, it is preferable to
use the Sigma Clip combine routine. This argues for using a greater number
of subframes. In fact, for greater than 11 subframes, MM-clip SNR loss
is less than 10% compared to mean combining, but it has greater non-random noise
Sigma clip is an image combining technique which reduces
extreme pixel values by using data from surrounding pixels. While the end
result is similar to using Min/Max-clip, the methods used to obtain the results
are different. Due to the method used to determine the combined pixel
values the effect on SNR is not predictable. However, the end result is
often very similar to using MM-clip. MM-clip is available in the CCDStack
and Mira software packages, while Sigma Clip is used in MaxIm DL. Either
method is a good choice when combining a large number of subframes.
Dithering is a method of shifting the telescope slightly between exposures to
offset each image slightly. This results in fixed pixel defects like hot
and cold pixels being misaligned in the final composite image and thus removed
by median or Sigma Clip combine methods. Dithering can be done
offsetting the guide star by several pixels when setting up an imaging sequence
using software such as MaxIm DL. The amount of offset can be selected by
the user, and several pixels is usually sufficient. The mount is shifted
between exposures, the guide star reacquired, an the next exposure begins.
The use of dithering is highly recommended, especially for the new generation of
high resolution camera whose large CCD arrays have the attendant high number of
Determining Optimal Exposure Times
So, in the end, what is the best exposure time to use for subframes?
Shorter exposures allow better combination methods for greater noise removal.
Shorter exposures are also an advantage in that, if something goes wrong during
the exposure (wind, tracking errors, etc.), less data is lost. The trade
off between taking more short exposures versus fewer long exposures in terms of
SNR loss is very slight, as John Smith points out on his
recommends the following exposure recommendation:
In other words, if you determine the sky limit exposure time tORN
to be 10 minutes, instead of taking five 10-minute exposures and median
combining, take ten 5-minute exposures and use the Sigma Clip combine method.
And be sure to dither the exposures. This will result in excellent noise
reduction from all possible sources (assuming you take a sufficient number of
dark frames, and taking a flat field or three couldn't hurt).
Below is a link to the Ideal Exposure Calculator that will do all the above
math for you. You will need to have a test exposure from which to measure
the background ADU value using your image processing software.
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