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Hierarchical Wiener filtering
In the above process, we do not use the information between the wavelet
coefficients at different scales. We modify the previous
algorithm by introducing a prediction w_{h} of the wavelet coefficient from
the upper scale. This prediction could be determined from the regression
[2] between the two scales but better results are obtained
when we only set w_{h} to W_{i+1}. Between the expectation
coefficient W_{i} and the prediction, a dispersion exists where we
assume that it is a Gaussian distribution:



(14.84) 
The relation which gives the coefficient W_{i} knowing w_{i} and w_{h} is:



(14.85) 
with:



(14.86) 
and:



(14.87) 
This follows a Gaussian distribution with a mathematical expectation:



(14.88) 
with:



(14.89) 
W_{i} is the barycentre of the three values w_{i}, w_{h}, 0 with the
weights T_{i}^{2}, B_{i}^{2}, Q_{i}^{2}. The particular cases are:
 If the noise is large (
)
and even if the correlation
between the two scales is good (T_{i} is low), we get
.
 if
then
.
 if
then
.
 if
then
.
At each scale, by changing all the wavelet coefficients w_{i} of the
plane by the estimate value W_{i}, we get a Hierarchical Wiener
Filter. The algorithm is:
 1.
 Compute the wavelet transform of the data. We get w_{i}.
 2.
 Estimate the standard deviation of the noise B_{0} of the first plane
from the histogram of w_{0}.
 3.
 Set i to the index associated with the last plane: i = n
 4.
 Estimate the standard deviation of the noise B_{i} from B_{0}.
 5.

S_{i}^{2} = s_{i}^{2}  B_{i}^{2} where s_{i}^{2} is the variance of w_{i}
 6.
 Set w_{h} to W_{i+1} and compute the standard deviation T_{i}of w_{i}  w_{h}.
 7.

 8.
 i = i  1. If i > 0 go to 4
 9.
 Reconstruct the picture
Next: Adaptive filtering from the
Up: Noise reduction from the
Previous: The Wienerlike filtering in
Petra Nass
19990615