next up previous contents
Next: Adaptive filtering from the Up: Noise reduction from the Previous: The Wiener-like filtering in

   
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:
$\displaystyle P(W_i/w_h) = \frac{1}{\sqrt{2\pi}T_i}e^{-\frac{(W_i- w_h)^2}{2T_i^2}}$     (14.84)

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

$\displaystyle P(W_i/w_i \mbox{ and } w_h) = \frac{1}{\sqrt{2\pi}\beta_i}
e^{-\f...
...i w_i)^2}{2\beta_i^2}} \frac{1}{\sqrt{2\pi}T_i}
e^{-\frac{(W_i-w_h)^2}{2T_i^2}}$     (14.85)

with:
$\displaystyle \beta_i^2 = \frac{S_i^2B_i^2}{S^2 + B_i^2}$     (14.86)

and:
$\displaystyle \alpha_i = \frac{S_i^2}{S_i^2+B_i^2}$     (14.87)

This follows a Gaussian distribution with a mathematical expectation:

$\displaystyle W_i = \frac{T_i^2}{B_i^2+T_i^2+Q_i^2} w_i +
\frac{B_i^2}{B_i^2+T_i^2+Q_i^2} w_h$     (14.88)

with:
$\displaystyle Q_i^2 = \frac{T_i^2B_i^2}{S_i^2}$     (14.89)

Wi is the barycentre of the three values wi, wh, 0 with the weights Ti2, Bi2, Qi2. The particular cases are:

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.
$W_i = \frac{T_i^2}{B_i^2+T_i^2+Q_i^2} w_i + \frac{B_i^2}{B_i^2+T_i^2+Q_i^2} w_h$
8.
i = i - 1. If i > 0 go to 4
9.
Reconstruct the picture


next up previous contents
Next: Adaptive filtering from the Up: Noise reduction from the Previous: The Wiener-like filtering in
http://www.eso.org/midas/midas-support.html
1999-06-15