# Wiener Filters - Obtaining /Estimating Image and Noise Spectrum

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 $y$?

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:

• I am able to calculate $R$, because I know $y(n)$. How can I calculate (or estimate) $r$? – Roi Divon Jul 2 '14 at 17:52
• 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. – Serj Jul 3 '14 at 2:55