Wiener deconvolution

In mathematics, Wiener deconvolution is an application of the Wiener filter to the noise problems inherent in deconvolution. It works in the frequency domain, attempting to minimize the impact of deconvolved noise at frequencies which have a poor signal-to-noise ratio.
The Wiener deconvolution method has widespread use in image deconvolution applications, as the frequency spectrum of most visual images is fairly well behaved and may be estimated easily.
Wiener deconvolution is named after Norbert Wiener.
Definition.
Given a system:
where formula_2 denotes convolution and:
Our goal is to find some formula_9 so that we can estimate formula_3 as follows:
where formula_12 is an estimate of formula_3 that minimizes the mean square error
with formula_15 denoting the expectation.
The Wiener deconvolution filter provides such a formula_9. The filter is most easily described in the frequency domain:
where:
The filtering operation may either be carried out in the time-domain, as above, or in the frequency domain:
and then performing an inverse Fourier transform on formula_34 to obtain formula_12.
Note that in the case of images, the arguments formula_4 and formula_37 above become two-dimensional; however the result is the same.
Interpretation.
The operation of the Wiener filter becomes apparent when the filter equation above is rewritten:
Here, formula_39 is the inverse of the original system, formula_40 is the signal-to-noise ratio, and formula_41 is the ratio of the pure filtered signal to noise spectral density. When there is zero noise (i.e. infinite signal-to-noise), the term inside the square brackets equals 1, which means that the Wiener filter is simply the inverse of the system, as we might expect. However, as the noise at certain frequencies increases, the signal-to-noise ratio drops, so the term inside the square brackets also drops. This means that the Wiener filter attenuates frequencies according to their filtered signal-to-noise ratio.
The Wiener filter equation above requires us to know the spectral content of a typical image, and also that of the noise. Often, we do not have access to these exact quantities, but we may be in a situation where good estimates can be made. For instance, in the case of photographic images, the signal (the original image) typically has strong low frequencies and weak high frequencies, while in many cases the noise content will be relatively flat with frequency.
Derivation.
As mentioned above, we want to produce an estimate of the original signal that minimizes the mean square error, which may be expressed:
The equivalence to the previous definition of formula_43, can be derived using Plancherel theorem or Parseval's theorem for the Fourier transform.
If we substitute in the expression for formula_34, the above can be rearranged to
If we expand the quadratic, we get the following:
However, we are assuming that the noise is independent of the signal, therefore:
Substituting the power spectral densities formula_48 and formula_49, we have:
To find the minimum error value, we calculate the Wirtinger derivative with respect to formula_18 and set it equal to zero.
This final equality can be rearranged to give the Wiener filter.