Technical astrophotography - Signal and Signal Noise
All astro-photos are composed of multiple frames ("subs") that are combined ("averaged") into a stacked image. Typically, the stacked image will be significantly better than the individual frames. But why this should work is so obvious. For example, suppose one has two equally exposed subs, each of which has a "signal" of 1000DN. Combining these subs together the end result will be (1000DN+1000DN)/2 = 1000DN. Obviously nothing is gained. Why should a combined average work at all? Compared to, say, doubling the exposure time?
The reason turns out to be a subtle property of (almost) all natural light sources. Look at a light bulb. If it is flicked on, electrical current flows through the tungsten filament. It heats up due to ohmic resistance. The tungsten "molecules" emit a photon of visible light every time they collide with each other. But this emission mechanism is essentially a random process. So when a sensor captures this light, the photons arrive in a random order, with some average rate of arrival in some time period. In one time period 1000 photons can arrive, and 1050 the next, and so on, but the average rate can still be constant.
Now comes the crux of the matter. If you want to measure exactly how luminous the light bulb is, you have to expose the sub some finite period of time to receive enough light to show up on the sensor. In the first frame, the bulb measures 500DN. In the next frame, it measures 520DN, then 480DN, and so on. So how bright is the bulb now? The value appears slightly different each time, due to the randomness of the arriving photons. The average of the frames appears to be 500DN, but there is clearly some uncertainty in this estimate.
But how large is this uncertainty? Does this uncertainty depend on the source brightness? The answer is yes.
The French mathematician Poisson showed that this "uncertainty" scales as the square root of the source brightness. In engineering jargon, this is also called "noise". Remarkably, this is a consequence of the statistics of random arrival, and is independent on the process that generates the signal. Essentially, it applies equally to light bulbs as it does to stars and galaxies.
A source \(\) will have an intrinsic noise \(\).
Technical astrophotography - Noise sources
Unfortunately, in the real world, we never measure only the light of a star. There are always additional sources present. Some are inherent to the sensor, like dark current and read-out noise, but there also real sources such as sky background.
The previous discussion also applies to light sources caused by light pollution, even if its a diffuse glow. Because the sky background has a certain signal \(\), it also introduces a definite noise \(\). One can get rid of the fixed average amount \(\) by subtracting it from the image, however, the associated noise cannot be removed this way! Obviously one cannot predict what the exact value of \(\) was per pixel, so by subtracting the average \(\), the "uncertainty of \(\)" remains as noise.
Fortunately, there is a way out. As will be seen next, by combining subs together, this noise can be supressed. In the best case, this noise reduction goes by the square root of the amount of images (\(\)), but in actual practice, the gain is less, sometimes even negative!
Technical astrophotography - SNR
A good indicator of how well a source can be discerned out from all these noises is the Signal to Noise-ratio, or simply SNR. The higher this number is, the better. A point source such as stars get spread out over multiple pixels (but still a small area). If the SNR of those pixels is ~ 2, one cannot be really sure there is actually a star there, or if it just random fluctuation in the (sky-)background or thermal noise.
As a rule of thumb, the following can be said about the significance of a SNR 1)ESO on SNR
To get a feeling of the SNR of a star in an image, do the following. (1) Measure the standard deviation of the background near the star. (2) Perform a PSF fit of the star using (for example) the Moffat function (available in many software packages for astronomical image processing):
here \(\) is the local background, \(\) the amplitude of the source, and \(\) the steepness. The parameter \(\) is constant, typically 2.5~4. The "signal" of the source (in DN) is the integral and can be calculated as \(\) (valid for Moffat with \(\), or Moffat25). This is the value (in DN) the source would have had if all its light were focused in exactly 1 pixel.
The SNR of the point source in the image can now be expressed as \(\), where \(\) is the standard deviation measured in (1).
The following subs were used to combine the final image of this post. Circumstances were less than optimal, with lots of high cirrus clouds. Moisture tends to scatter back a lot of light pollution from below, and the local sky brightness increases by many factors as a result. The resulting SNR of a faint \(\) star (the fainter middle star of the triangle of stars upper-left above M51) is calculated this way in the table below.
The table below shows key measurements to determine the SNR of a faint \(\) star
|Frame||Background||Standard Deviation||Signal (Moffat 2.5)||SNR|
Note that the SNR only applies to this specific (point-)source. It makes no sense to talk about the SNR of the an image. It will be different for every source. In the case of this example, the star is barely seen in Frame 1, with a \(\). As the clouds part, the background becomes darker. Because the background gets darker, the noise in the background reduces likewise. This improves SNR. As can be seen from the table, the effect is rather large on a faint source that is barely detected. On the other hand, a bright source will not be so strongly affected.
References [ + ]
|1.||↑||ESO on SNR|