# Explain about Wiener filter used for image restoration.

1 month ago

The inverse filtering approach makes no explicit provision for handling noise. This approach incorporates both the degradation function and statistical characteristics of noise into the restoration process. The method is founded on considering images and noise as random processes, and the objective is to find an estimate *f *of the uncorrupted image f such that the mean square error between them is minimized. This error measure is given by

## e^{2} = E {(f- *f *)^{2}}

where E{•} is the expected value of the argument. It is assumed that the noise and the image are uncorrelated; that one or the other has zero mean; and that the gray levels in the estimate are a linear function of the levels in the degraded image. Based on these conditions, the minimum of the error function is given in the frequency domain by the expression

where 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. 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. The Wiener filter does not have the same problem as the inverse filter with zeros in the degradation function, unless both H(u, v) and S_{η}(u, v) are zero for the same value(s) of u and v.

The terms in above equation are as follows: 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_{η} (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.

As before, H (u, v) is the transform of the degradation function and G (u, v) is the transform of the degraded image. 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.

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 the equation as

where K is a specified constant.

###### Raju Singhaniya

Oct 15, 2021