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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
Rein Warmels
Mon Jan 22 15:08:15 MET 1996