In many instances it is reasonable to
assume that the image noise can be modelled by a zero mean Normal
distribution 
.
In this case, analysing the difference in intensity images is straightforward.
Differencing followed by taking the absolute value will produce
the Normal distribution 
for positive values only.
Sometimes it is preferred to difference edge maps rather than intensity images as they can be more robust for change detection under varying illumination [1, 21, 25]. Unfortunately this makes determining the distribution of the noise more troublesome. First we note that the noise in the edge maps can be modelled by a Rayleigh distribution [24]
if the edge response is of the form
.
Denoting the noise in two edge maps as independent random variables
a and b we wish to calculate the density of
the noise in the difference image 
.
Initially we consider the symmetric function
.
Its density 
equals the convolution of the densities
of 
and 
[14]
For the Rayleigh functions 
we obtain,
using Mathematica,
where 
is the generalised Laguerre polynomial.
Since 
, then
(this assumes that 
is even, although it is actually undefined
for negative values in the form given above).
Figure 1: Probability density function of and N(0,1)
Looking at the plot of drawn bold alongside a Normal distribution
(figure 1) we see that they are similar, especially near the tails.
We are mainly interested in the distribution function towards the tails since
we wish to threshold out most of the noise.
Therefore we can reasonably approximate 
by a Normal distribution.
When thresholding at 
the probability of incorrectly
classifying a pixel as motion is
This enables us to choose a suitable threshold 
for a given acceptable
proportion of false motion pixels.
In practise the variance of the noise is often unknown so we need to estimate it from the image. Since the difference image will contain not just noise but also appreciable amounts of signal due to the motion a robust estimation technique is required. Similar to our previous work in estimating noise levels in edge maps [16] we use the Least Median of Squares (LMedS) method applied to the difference image histogram. Its advantages are that it is efficient (at least for 1D data), and has a high breakdown point. This latter property enables it to return the correct result even when large amounts of outliers (i.e. true motion) are present. It is straightforward [17] to derive the following relation between the LMedS and the expected standard deviation of the noise:
