The Wiener-like filtering in the wavelet space

  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:

1. Compute the wavelet transform of the data. We get .
2. 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;
3. Set i to 0.
4. 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;
5. where is the variance of .
6. .
7. .
8. i = i + 1 and go to 4.
9. Reconstruct the picture from .