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Although the signal bias can be subtracted, it introduces another complication: you need to measure the signal bias before you can subtract it. In the grocery store, you know that if you tap the scale lightly, the needle will move and settle at a slightly different bias value. To determine the bias accurately, you might decide to tap the scale and read the bias value half-a-dozen times; and then you would need to calculate the mean of the bias readings.

If you were fanatical about buying that pound of cashews, you would also compute the standard deviation in the bias readings, thereby recognizing and acknowledging the fact that your measurement of the bias also has an associated uncertainty that contributes to the uncertainty in the measured weight of the nuts.

The same thing happens with images from digital cameras and in CCD images: for accurate measurements, you need to determine the zero point by taking bias frames and computing their mean value. You can then subtract the bias to obtain a measurement of the true signal level. With most digital cameras and CCDs, the bias value is small and steady, and adds only a small dose of uncertainty to the signal in your images.

Equations 2.4 and 2.5 assume that the signal consists entirely of photon counts. If you have studied statistics, you are already familiar with the Gaussian distribution. The Gaussian distribution gets its name from the legendary German mathematician (and astronomer) Carl Friedrich Gauss (1777-1855). The general equation for the Gaussian distribution is:

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where f(x) is the function describing the probability that x events will occur, x is the mean number of events, e is the base of natural logarithms (2.718...), and a is the standard deviation of x. Whereas Equation 2.5 is specific to event probabilities such as photon counting, the Gaussian function makes no assumptions about the signal—given measured values for x and a , it will generate the expected distribution of signal values.

2.3.5 Unwanted Signals and More Sources of Noise

In addition to detecting photons, CCDs and CMOS devices generate an unwanted signal called dark current. Dark current is a true signal in that it conveys useful information about the detector and—unlike noise—dark current is not random. When we make images, the raw image combines the desired photon signal with the unwanted dark current signal as well as the signal bias.

Since most astronomers are interested only in the photon signal, we resent the need to make dark frames (images in which no light is allowed to reach the detector). Dark frames are, however, needed to measure as accurately as possible the dark current from every pixel, but only so that we can subtract the dark current from our raw images.

Dark current arises from lattice defects that generate extra electrons in the crystal lattice of the silicon detector. Each electron liberated is a statistically independent event, so dark current obeys the same Poisson statistical model that detected photons do. This means that dark current has noise. If we sample a dark current containing a mean of xd electrons, the electron count will have a statistical uncertainty of Gd = J¥d electrons. (To highlight the similarity of the statistics, we'll use the variables x and o to denote both detected photons and dark-current electrons, but we'll distinguish those associated with the dark current by the "d" subscript.)

In the preceding sections, we have mentioned signal bias several times. Bias is another unwanted signal, but it's not due to a random process. Bias is a constant output offset, b ADUs, originating in the electronics that detect and amplify the signal from the detector. However, the bias is accompanied by readout noise, a random variation in the amplifier circuits that is imposed on everything output from the detector. Unlike the noise associated with counting photons or counting dark current electrons, readout noise adds a constant uncertainty regardless of the amplitude of the bias and signal levels. We'll denote readout noise as oron . Readout noise is usually expressed in units of root-mean-square electrons.

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