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We all know the famous Wiener Filter,

$$G(\omega_1 , \omega_2)=\dfrac{H^*(\omega_1 , \omega_2)}{|H(\omega_1 , \omega_2)|^2 + \dfrac{N(\omega_1 , \omega_2)}{S(\omega_1 , \omega_2)}}$$

How to we use it practically? Meaning, given $y = s * h + n$ where $s$ is the original image, $h$ is a LSI operator, and $n$ is some noise (independent of $s$), how to calulate (or estimate) $N(\omega_1 , \omega_2)$ and $S(\omega_1 , \omega_2)$ in order to reconstruct $s$ from $y$?

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You are speaking about so called Wiener deconvolution, because Wiener filter is set by equation $ w = R^{-1}r $, where:

  • $ R = E[y(n)y(n)^H] $ - autocorrelation matrix of input signal

  • $ r = E[y(n)s(n)^*] $ - crosscorrelation vector of input signal and original signal.

So the practical solution when statistics $R$ and $r$ are unknown is the Method of Least Squares, where only data matrix $F$ and part of original signal $s$ are needed.

In your case you should have some a priori knowledge about formula entities. Try:

https://www.umanitoba.ca/faculties/medicine/radiology/Dick_Gordon_papers/Dhawan%20et%CAal.%20%281985%29.pdf

or:

http://www.eos.ubc.ca/research/cdsst/members/6_96.pdf

It's a pity I only concern this subject in radio applications so there is nothing more to advise.

Hope this helps!

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  • $\begingroup$ I am able to calculate $R$, because I know $y(n)$. How can I calculate (or estimate) $r$? $\endgroup$ – Roi Divon Jul 2 '14 at 17:52
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    $\begingroup$ In order to estimate $r$ you should have original signal $s$ or representative part of this signal (filter learning). If you don't you can try to construct it by yourself making maximum-likelihood decision of $w^H y$ signal. But it only works if your signal of interest could get finite number of values (signal constellation is an example). This is blind or decision-directed method and it isn't work at all situations. When you deal with block of data you can reiterate decision-directed estimation of $w$ N times to achieve better solution and fit your filter closely to Wiener filter. $\endgroup$ – Serj Jul 3 '14 at 2:55

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