## What Is Noise

Astronomers and engineers use the word "noise" in a special technical sense. To the layperson, noise is people yelling, bad music, the roar of traffic. In the vocabulary of science, noise is the random variation in a signal. Noise is present in all signals; you hear it in the hiss of a vinyl record, see it in the "snow" in the television signal from a weak or distant station, or more to the point, you experience it as graininess in CCD images.

Digital images are made by measuring the number of photons falling on a light sensor. The photons generate photoelectrons in the individual photosites

(pixels) of the sensor, and the output from each photosite is proportional to the number of photoelectrons detected during the time you expose the sensor to light. At first, you might expect the number of photons to be exactly proportional to the length of the exposure. After all, when you put gasoline in the tank of your car, if you pump twice as long, you expect twice as much gas. Why are photons any different? The answer lies in the way that photons arrive—each reaching the sensor independent of any other photon. Instead of arriving in a steady flow like the ticking of a clock ("click-click-click-click") photons arrive at random, like the irregular rattling sound of a Geiger counter ("click-a-click, clickitty, clack") or the rattle of rain on a tin roof.

In normal daytime activities, so many photons bombard us that their arrival appears to be a steady flow—even though it is not. Only when the number of photons becomes small does their random arrival make a significant difference. Under a dark sky, the sky background signal usually consists of a few hundred photons per minute. Since it usually takes integration time of a minute or more to accumulate that many photons, you can count the number that arrive each second on the fingers of one hand.

Let's see how this affects astronomical images. Suppose that you shoot ten images of a galaxy, one right after another. You bring the images up on your computer and measure the pixel value of the same spot in each image.

What do you see?

Instead of the seeing the same value in each image, you measure a different number in each one. This is because each image is not a precise measure of the rate at which photons arrive, but instead is a sample of the photon stream—a snapshot recording the number of photons that arrived during that particular exposure. The number of photons varies from one sample to the next.

An astonishingly simple rule describes the expected variation:

• If the average signal consists of x photons, an individual sample will contain x± Jk photons.

But what does the notation "x ± Jk " mean?

The "x " term, read as "x-bar," is the mean value. It is the average rate over a long time interval, the mean value of a large number of samples. The " " term is the standard deviation, a statistical measure of the departure of a typical sample from the mean value. Finally, the "plus or minus" symbol, "±", means that an individual sample may be either larger or smaller than the mean value.

The notation "x ± Jk " implies that the distribution of sample values about the mean follows a bell-shaped curve called the normal distribution. In the normal distribution:

• 68.3% of samples will lie between x - Jk and x + Jk,

• 95.4% of samples will lie between x-ljk and x + ijk, and

• 99.7% of samples will lie between x - 3 Jk and x + 3 Jk.

In other words, roughly two-thirds of the time, the measured sample value Pixel value of pixel (100, 100) in ten sequential images.

Figure 2.1 In a seemingly identical sequence of images, the number of photons that arrive during a constant exposure time varies randomly. The more photons, the smaller the percentage variation. To make an image, it is necessary to expose long enough to gather enough photons for a good image.

Pixel value of pixel (100, 100) in ten sequential images.

 Image #1 