i. The inverse filtering approach makes no explicit provision for handling noise. Weiner filtering is an approach that incorporates both the degradation function and statistical characteristics of noise into the restoration process.

ii. The method is founded on considering images and noise as random variables, and the objective is to find an estimate $f ̌ $ of the uncorrupted image f such that the mean square error between them is minimized

iii. This error measure is given by

$e^2 = \{(f-\hat f)\} $ (Equation 1)

Where E{ .} is the expected value of the argument.

iv. It is assumed that the noise and the image are uncorrelated; that one or the other has zero mean; and that the intensity levels in the estimate are a linear function of the levels in the degraded image.

v. Based on these conditions, the minimum of the error function in Eq. (1) is given in the frequency domain by the expression

$\hat F (u,v)=[\dfrac {H^*(u,v)S_f(u,v)}{S_f(u,v)|H(u,v)|^2 + S_n(u,v)}]G(u,v)\\ =[\dfrac {H^*(u,v)}{|H(u,v)|^2+S_n(u,v)/S_f(u,v)}]G(u,v)\\ =[\dfrac 1{H(u,v)}\dfrac {|H(u,v)|^2}{|H(u,v)|^2+S_n(u,v)/S_f(u,v)}]G(u,v)$

Equation 2

vi. Here we used the fact that the product of a complex quantity with its conjugate is equal to the magnitude of the complex quantity squared. This result is known as the Wiener filter, after N.

vii. Wiener [1942], who first proposed the concept in the year shown. The filter, which consists of the terms inside the brackets, also is commonly referred to as the minimum mean square error filter or the least square error filter.

viii. Now from the first line in Eq.(2) we can see that the Wiener filter does not have the same problem as the inverse filter with zeros in the degradation function, unless the entire denominator is zero for the same value(s) of u and v. The terms in Eq. (2) are as follows:

ix. H(u, v) = degradation function

$H*(u, v) =$ complex conjugate of H(u, v)

$|H(u, v)|^2 = H*(u, v)H(u, v)$

$S_n(u, v) = |N(u, v)|^2 = $ power spectrum of the noise

$S_f(u,v) = |F(u, v)|^2 =$ power spectrum of the undegraded image

$G(u, v)$ is the transform of the degraded image.

x. The restored image in the spatial domain is given by the inverse Fourier transform of the frequency-domain estimate F(u, v). Note that if the noise is zero, then the noise power spectrum vanishes and the Wiener filter reduces to the inverse filter.

xi. A number of useful measures are based on the power spectra of noise and of the undegraded image.

xii. One of the most important in Wiener filtering is the signal-to-noise ratio, approximated using frequency domain quantities such as

$SNR=\dfrac {∑_{u=0}^{M-1} ∑_{v=0}^{N-1}|F(u,v) |^2}{∑_{u=0}^{M-1}∑_{v=0}^{N-1}|N(u,v) |^2 } ….. $ (Equation 3)

xiii. This ratio gives a measure of the level of information bearing signal power (i.e., of the original, undegraded image) to the level of noise power.

xiv. Images with low noise tend to have a high SNR and, conversely, the same image with a higher level of noise has a lower SNR. This ratio by itself is of limited value, but it is an important metric used in characterizing the performance of restoration algorithms.

xv. The mean square error given in statistical form in Eq. (5.8–1) can be approximated also in terms a summation involving the original and restored images:

$MSE =\dfrac 1{MN}\sum\limits_{x=0}^{M-1}\sum\limits_{y=0}^{N-1}[f(x,y)-\hat f (x,y)]^2$ ----- (Equation 4)

xvi. When we are dealing with spectrally white noise, the spectrum $|N(u, v)|^2$ is a constant, which simplifies things considerably. However, the power spectrum of the undegraded image seldom is known. An approach used frequently when these quantities are not known or cannot be estimated is to approximate Eq. (2) by the expression

$\hat F(u,v)=[ \dfrac 1{H(u,v)} \dfrac { |H(u,v) |^2}{|H(u,v) |^2+K}] G(u,v) $ Equation 5

xvii. In this manner Wiener filtering is used, in order to handle noise present in the image.