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:

The relation which gives the coefficient W_i knowing w_i and w_h is:

with:

and:

This follows a Gaussian distribution with a mathematical expectation:

with:

W_i is the barycentre of the three values w_i, w_h, with the weights T_i², B_i², Q_i². 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 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 B_i from B₀.

5.

S_i² = s_i² - B_i² where s_i² 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