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Let us consider a measured wavelet coefficient 
at the scale i.
We assume
that its value, at a given scale and a given position,
results from a noisy process, with a Gaussian distribution with a
mathematical expectation 
, and a standard deviation 
:
Now, we assume that the set of expected coefficients 
for a given
scale also follows a Gaussian distribution, with a null mean and a
standard deviation 
:
The null mean value results from the wavelet property:
We want to get an estimate of 
knowing 
. Bayes' theorem gives:
We get:
where:
the probability 
follows a Gaussian distribution with a mean:
and a variance:
The mathematical expectation of 
is 
.
With a simple multiplication of the coefficients by the constant 
,
we get a linear filter. The algorithm is:
.
of the first plane
from the histogram of 
. As we process oversampled images, the
values of the wavelet image corresponding to the first scale (
)
are due mainly to the noise. The histogram shows a Gaussian peak
around 0. We compute the standard deviation of this Gaussian
function, with a 
clipping, rejecting pixels where the signal
could be significant;
from 
. This
is done from the study of the variation of the noise between two
scales, with an hypothesis of a white gaussian noise;
where 
is the variance of 
.
.
.
.
