Capturing faint detail in deep-sky astronomical images is all about overcoming noise.  Noise comes from a variety of sources:

• Heat (dark current)

• CCD readout

• Light pollution/sky background

• Cosmic rays

• Pixel defects

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 picture.

Above:  Single exposure.  Faint star circled has SNR of 5.

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):

• Exposure Time

• Number of Exposures

• Object Flux

• Sky Background Flux

• Binning

• Resolution

• Focal Ratio

• Dark Current

• Readout Noise

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.

Exposure Time

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 10-minute exposures.

Object Flux

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 below.

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.)

Binning

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

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

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 Current

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.

Readout Noise

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:

• N = Number of Exposures

• E_obj = Object Flux

• E_sky = Background Sky Flux

• t = Exposure Time

• R_on = Readout Noise

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, E_sky, large compared to the readout noise, R_on, will minimize the contribution from readout noise.

Sky Limited Exposures

Note:  Equations are given in the discussion below but a JavaScript 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 readout noise.

• 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, t_ORN:

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 E_sky, 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 ADU_bkg is the background ADU count, g is the gain, t_test is the test exposure duration in minutes.  Take the first example given earlier, of the image taken from the dark location.  The ADU_bkg 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 E_sky = 111e-/min (equal to the 1.8 e-/sec from the example).

This value of E_sky is then plugged into the t_ORN 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.

Vignetting

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 readout noise.

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

• Cosmic Rays

• Other Artifacts

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).

Cosmic Rays

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 subframes.

Other Artifacts

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

• Median 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 artifacts.

For example, for N=5, SNR is proportional to 2.2.  Compare this to the values for the other combine functions.

Median Combine

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 noise.

For example, for N=5, SNR is proportional to 1.78, or only 81% that of the mean combine method.

Min/Max-Clip Combine

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, MM-clip 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 reduction.

Sigma Clip

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

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 automatically by 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 pixel defects.

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 website.  He recommends the following exposure recommendation:

• Take 2N-1 exposures of t_ORN/2 duration.

In other words, if you determine the sky limit exposure time t_ORN 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.

Ideal Exposure Calculator

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