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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 wh 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 wh to Wi+1. Between the expectation coefficient Wi and the prediction, a dispersion exists where we assume that it is a Gaussian distribution:
 (14.84)

The relation which gives the coefficient Wi knowing wi and wh is:

 (14.85)

with:
 (14.86)

and:
 (14.87)

This follows a Gaussian distribution with a mathematical expectation:

 (14.88)

with:
 (14.89)

Wi is the barycentre of the three values wi, wh, 0 with the weights Ti2, Bi2, Qi2. The particular cases are:
• If the noise is large ( ) and even if the correlation between the two scales is good (Ti is low), we get .
• if then .
• if then .
• if then .

At each scale, by changing all the wavelet coefficients wi of the plane by the estimate value Wi, we get a Hierarchical Wiener Filter. The algorithm is:

1.
Compute the wavelet transform of the data. We get wi.
2.
Estimate the standard deviation of the noise B0 of the first plane from the histogram of w0.
3.
Set i to the index associated with the last plane: i = n
4.
Estimate the standard deviation of the noise Bi from B0.
5.
Si2 = si2 - Bi2 where si2 is the variance of wi
6.
Set wh to Wi+1 and compute the standard deviation Tiof wi - wh.
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 Wiener-like filtering in
Petra Nass
1999-06-15