Signals and Noise in Images

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You now have enough background to examine the signals and noise that you will encounter in real images from your CCD or digital camera. Before beginning that examination, we need to get certain units of measurement squared away.

We introduced signals and noise in terms of detected photons, electrons, and ADUs; that is, in different units. To understand digital images, it is necessary to know how these different units relate to one another.

• When photons strike a detector, not all of them generate a signal. In a typical CCD, roughly 40% to 80% of the photons are detected—meaning that the photon has liberated an electron. Therefore, the detected photon count, x, translates directly into electrons.

• The statistical uncertainty in the photon count, a, often called shot noise, has units of root-mean-square electrons.

• Dark current electrons, xd, are already in units of electrons.

• The dark current noise, od, has units of root-mean-square electrons.

• Readout noise, Gron, is almost always given in units of root-mean-square electrons to make it easier to compare readout noise to other sources of noise.

• In CCD and CMOS devices, electrons are converted to a voltage that is digitized and sent to your computer as an integer value. For lack of a more mellifluous term, the integer value is called an analog-to-digital unit (ADU) or digital number (DN).

• The gain or conversion factor relates electrons to ADUs, and is denoted by the symbol g. The units of the gain are electrons per ADU. In most CCD cameras, the gain is reasonably close to 1, but the gain may be as large as 500 e~/ADU in webcams, or as small of 0.1 e /ADU in some CCDs.

As a general rule, astronomers use fundamental physical units (electrons) when they talk about properties of the detector and amplifier; and by necessity, they use the arbitrary derived units (ADUs) that the camera generates when they are discussing the images made by cameras.

Let's look at a few examples to see how these conversions work. Suppose 100 photons fall on a pixel of the detector in your CCD, digital camera, webcam, or whatever detector you happen to be using (and remember, the math is the same for all such devices). Typically 60 of those photons will generate an electron, and therefore be detected as a signal x in units of electrons. A device such as a charge detection node (in a CCD) senses the electrons and generates a voltage proportional to the number of electrons, and passes that voltage to an analog-to-digital converter. Suppose that the amplifier and A/D converter together have a gain of 2.5. In our example, the A/D converter adds a bias of 100.

The output signal from the camera will be:

5= - + b= -60 electrons-+ 100 ADU= 124 ADU . (Equ. 2.11)

g 2.5 electrons/ADU

To convert the signal electrons to ADUs, we've divided the signal in electrons by the gain in electrons per ADU, which gives us ADUs. The bias is already in units of ADUs, so we can add the terms.

In the example, we neglected dark current and did not even consider noise, but you can see the need to pay close attention to the units of measurement.

Let's now look at some realistic examples.

2.4.1 Signal and Noise in a Raw Image

You decide to make a raw image. What signals and what sources of noise do you expect to find? The signals are:

Dark Current Noise
Figure 2.4 The raw image is the sum of a photon signal (with Poisson noise), an unwanted dark-current signal (with Poisson noise), and a bias constant (with readout noise). Althought the dark current and the bias can be subtracted, the random noise that they introduce remains.

• The detected photon count, x, in electrons;

The total signal in the raw image, ,S'raw , converted into units of ADUs, is:

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