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
The mean or first moment of the distribution is given by
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 = x1+x2, the sum of two independent variables with probability distributions p1(x1) and p2(x2). Then what can we say about p(x), the probability distribution of x?
We first look at the probability of getting a value between x and . If the sum x1 + x2 is between x and ,and we are given a value of x2, then x1 must be between x-x2 and . The probability of this occurring is .
But x2 can also take on a range of values. The probability that x2 lies between a particular x2 and is
To get the probability that the sum lies between x and we have to integrate the product over all x2. Thus
where t is a dummy variable of integration.
By a similar argument we can show symmetrically that
This is called the convolution of p1 and p2 and is denoted by .
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).