No imaging system is perfect. Noise is introduced into the imaging
process via the use of real lenses and cameras that differ in
operation from the pinhole camera model. Moreover, lighting and
atmospheric conditions can also effect the resulting image. In
addition, digital images suffer deviations in image values introduced
by sampling. Thus, measurements are affected by fluctuations in the
signal being measured, and these fluctuations are described according
to some probability distribution, *p*(*x*).

Since *p*(*x*) is a probability distribution, it always satisfies

and

The *mean* or first moment of the distribution is given by

but implying

The spread of the distribution is given by the second
moment or *variance*:

The *cumulative probability distribution*

tells us the probability that the measurement will be less than or
equal to *x*. Thus, the probability density distribution is just
the derivative of the cumulative probability distribution, that is,
*p*(*x*) = *P*'(*x*).

One way to improve accuracy is to average several measurements, assuming that the `noise' in them will be independent and tend to cancel out. To see how this is indeed the case, we consider the following analysis.

Let *x* = *x _{1}*+

We first look at the probability of getting a value between *x* and . If the sum *x _{1}* +

But *x _{2}* can also take on a range of values. The probability that

To get the probability that the sum lies between *x* and we have to integrate the product over all *x _{2}*. Thus

or

where *t* is a dummy variable of integration.

By a similar argument we can show symmetrically that

This is called the *convolution* of *p _{1}* and

We can use this result to prove that the mean of the sum of several random variables is equal to the sum of the means, and the variance of the sum is the sum of the variances.

If we calculate the average of *N* independent measurements each having
mean and standard deviation , then

So the average will have mean and standard deviation .

This latter result is because the variance of the sum of the
distributions is just , and so the standard deviation of
the sum is . Thus the distribution has
standard deviation and hence we obtain a
more accurate measurement by taking the average of *N* independent
measurements.

The usual probability distribution we use to model noise in an image is the normal or Gaussian distribution with mean and standard deviation given by

Of course, digital images are *sampled* versions of continuous
or analog images, and the sampling is done both spatially and in the
luminance values (where it is known as quantization).