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.
Note that in the case of images, the arguments and 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, is the inverse of the original system, is the signal-to-noise ratio, and 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:
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