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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:
- Compute the wavelet transform of the data. We get
.
- Estimate the standard deviation of the noise
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;
- Set i to 0.
- Estimate the standard deviation of the noise
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
.
-
.
-
.
- i = i + 1 and go to 4.
- Reconstruct the picture from
.
Next: Hierarchical Wiener filtering
Up: Noise reduction from
Previous: The convolution from
Pascal Ballester
Tue Mar 28 16:52:29 MET DST 1995