Info

The noise in the output signal is:

The noise in the dark-subtracted image is:

Figure 2.5 Although subtracting a dark frame adds noise, it also removes the distracting pattern of dark current and hot pixels. The "ideal" image (upper left) has zero dark current; the image (upper right) has a mean of 60 electrons dark current. Subtracting a dark frame adds noise but improves the result.

Figure 2.5 Although subtracting a dark frame adds noise, it also removes the distracting pattern of dark current and hot pixels. The "ideal" image (upper left) has zero dark current; the image (upper right) has a mean of 60 electrons dark current. Subtracting a dark frame adds noise but improves the result.

We can now finally determine the signal-to-noise ratio, SNRimage, at the sky background level of 240 ADUs to be:

Although an image with this signal-to-noise ratio would look somewhat grainy, faint deep-sky objects would be readily visible in it. Note that this is a sky-limited image because the overwhelmingly dominant source of noise comes from the photon statistics of the sky background, whose 600 electrons contribute 9.8 ADUs worth of noise. For comparison, the readout noise of 8 electrons root-mean-square adds only 3.2 ADUs of noise, and the 60 electrons of dark current add only another 3.1 ADUs of noise.

Under dark skies, however, dark current noise or readout noise can easily become the dominant noise source. At rural dark-sky sites, you might find a sky contribution of 40 electrons in a 60-second exposure. Photon statistics would then account for a mere 2.5 ADUs of noise, but the readout noise and dark current noise would stiil be 3.2 and 3.1 ADUs, respectively. Under these conditions, your images would be detector limited or photon limited.

How good is this image? Against a noise level of 11 ADUs, a single pixel 11 ADUs brighter than the mean level of the sky background—240 ADUs—would not stand out enough to be visible; but a few dozen pixels or a few hundred pixels representing a faint galaxy image would appear as clearly brighter than the surrounding sky. A large, bright nebula would be clear and unmistakable.

In this example, subtracting the dark frame raised the noise level from 10.76 to 11.64 ADUs. As a practical matter, the dark-subtracted image with 11.4 ADUs of random noise would certainly look a lot nicer than a raw image with 24 ADUs worth of dark current patterning and 10.74 ADUs of random noise.

However, it is not necessary to accept even this modest loss. By making several dark frames and averaging them together, you can reduce the adark to a negligible level. Applying Equation 2.7 to averaging dark frames, you can see that by averaging ten dark frames, you could reduce Gdark to 1.4 ADUs. Subtracting the averaged dark frame from the raw image yields a Gimage of 10.66 ADUs, demonstrating that if it's done properly, dark-frame subtraction causes negligible loss in image quality and removes the hot-pixel noise pattern. If you do your dark frames right, you'll preserve every photon your camera captures!

2.4.4 The Effect of the Sky Background on Signal and Noise

In the preceding section, we have considered various noise sources and their impact on the signal-to-noise ratio. We hinted that the sky background has a significant impact on a camera's ability to detect extended astronomical objects such as galaxies and nebulae. In this section, we examine the role of the sky background on imaging performance.

In Section 2.4.1 we defined the count of photons detected, x . In astronomical observation, however, photons come from two sources: an object of interest, and parasitic illumination from the night sky:

We now define a ratio between object photons and sky photons as:

■^sky in which OSR stands for 6>bject-to-£ky 7?atio. Sky brightness varies greatly from rural to suburban to urban sites, and also varies with the phase of the Moon. Furthermore, the range of object brightness is enormous. Suffice it to say that for very bright objects (the Moon and planets), the OSR is 1000 or more, and for exceedingly faint objects, it may be as low as 0.001.

Because the OSR varies over such a wide range, you may wish to determine the OSR of objects in your own images specific to your skies and local conditions. From a calibrated image, measure the following:

• ^object > the pixel value of the object in ADUs; and

Remember that when you measure the pixel value of the object, it is the sum of the object and the sky. Because of this, the OSR must be calculated as follows:

^sky

To illustrate the impact of sky brightness, we'll now consider the signal-to-noise ratio for three cases: a dark rural sky, a fairly good suburban sky, and a bright urban sky. Our celestial object is a galaxy that yields a photon flux of 250 electrons per pixel in 60 seconds. For this exercise, we'll use a camera with the gain, dark current, and readout noise from the preceding sections.

Dark Rural Sky. In this setting, we assume a signal of 40 electrons per pixel in 60 seconds, so for our celestial object, the OSR is 6.3; that is, the object is considerably brighter than the sky background. In a dark-subtracted image, a sky pixel measures 16 ± 5.4 ADUs and an object pixel measures 116 ± 8.3 ADUs. The object is 100 ADUs brighter than the surrounding sky, and it will stand out clearly against the noise in the dark rural sky background.

Fairly Good Suburban Sky. For our suburban sky, we take a signal of 600 electrons per pixel in 60 seconds. Our celestial object is now less bright than the sky background; its OSR for this sky is 0.42. In a dark-subtracted image, a sky pixel measures 240 ± 10.9 ADUs and an object pixel measures 340 ± 12.6 ADUs. The object is still 100 ADUs brighter than the surrounding sky, but both the sky and the object noise are greater than they were under the rural sky.

Bright Urban Sky. Urban skies are detrimental to good imaging because they are so bright. For the urban sky, we assume a signal of 4000 electrons per pixel in 60 seconds. The OSR of the object has fallen to 0.06. In a dark-subtracted image, a sky pixel measures 1600 ± 25.7 ADUs and an object pixel measures 1700 ± 26.5 ADUs. Our object is still 100 ADUs brighter than the surrounding sky, but the noise levels have become a significant fraction of the object's pixel value, and the image looks quite noisy. Nevertheless, even though the urban sky is 100 times brighter than the rural sky, the celestial object remains visible.

In the section that follows, we investigate the improvement obtained when you average multiple images. One seemingly paradoxical result is that it is neces sary to accumulate more total exposure time under a bright sky than that required to achieve the same result under a dark sky.

2.4.5 Signal and Noise in Multiple Averaged Images

In preceding sections, you observed that by averaging several dark frames you could avoid adding noise. In this section, you will see that by shooting an ample number of dark frames and by averaging multiple raw images, you can cut noise and build image quality.

Begin by reviewing the signal and noise information you have on hand:

• you know Graw, the raw-image sky noise in ADUs,

• you know Sdark, the dark-frame signal in ADUs, and

• you know odark , the dark-frame noise in ADUs.

Suppose that you decide to shoot a number, Nrsw , of raw images and some other number, Ndark, of dark frames. You average the raw images, you average the dark frames, and then you subtract them.

Here is the general equation for the signal:

When you average your raw images and dark frames, the resulting signal level does not depend on the number of raw images and dark frames. Here is the general equation for the noise:

The fact that the 7Vraw and 7Vdark are inside the square-root operation means that more raw images and more dark frames improve the image quality only as the square root—so getting a factor of two improvement requires four times more images. This difficulty notwithstanding, you're happy to shoot lots of images if it means that you'll have outstanding results.

Here is the signal and noise information for the suburban-sky example:

The signal will always be:

combined combined

The noise depends on the number of raw images and dark frames:

Figure 2.6 Under bright urban and suburban skies, the dominant source of image noise is photon statistics. Under dark rural skies, however, dark current and readout noise may dominate. Under dark skies, therefore, making multiple dark frames decreases noise and produces better images.

Suppose that you make 10 raw images and 10 dark frames—what signal-to-noise ratio can you expect? Evaluating the equation gives oC0mbined= 3.68 ADUs, producing sky-level signal-to-noise ratio of 65—suggesting that can expect a combined image of excellent quality!

But—are 10 raw images and 10 dark frames the best use of precious telescope time? Intuitively, the answer is "no" because the noise contribution from the dark frame is so much smaller than that of the raw images. If you're willing to accept equal noise contributions from dark frames and raw images, then if you shoot some number of raw images, jVraw, you should shoot Niark dark frames:

Section 2.5: Signal and Noise Effects from Flat-Fielding

^dark

varawy

In the suburban skies example, you only need to shoot one dark frame for every 6 raw images. With 17 raw images and 3 dark frames, Equation 2.27 gives a total noise of cjcombined= 3.66 ADUs, for a signal-to-noise ratio of 65—another excellent-quality image.

For the dark-sky example, for which Scombined = 16 ADUs and (Traw= 2.5 ADUs, your images are noise limited. You must make enough dark frames to subtract the dark current as accurately as possible; for dark skies, Equation 2.28 calls for 3.1 dark frames for each raw image. If you shoot 10 raw images and 10 dark frames, the result is tfcombined= 1.98 ADUs—considerably less noise than you could attain under suburban skies. Paradoxically, the signal-to-noise ratio has fallen to 8.1 at the sky-brightness level, but that's only because the sky is so very dark. To make a direct comparison, compute the signal-to-noise ratio for a nebula with a pixel value of 240 ADUs, and you will see that the signal-to-noise ratio rises to an outstanding 121.

If you take Equation 2.25 seriously for the dark rural sky and decide to shoot 10 raw images and 32 dark frames, you get acombined= 1.11 ADUs and a signal-to-noise ratio for the sky background of 14.4; but for a 240-ADU nebula, the signal-to-noise ratio is 216. Dark skies make a huge difference in image quality, especially if you shoot enough dark frames to make photon statistics the dominant source of noise.

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