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Concept
Wiener deconvolution
Summary
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 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.
Given a system:
where denotes convolution and:
is some original signal (unknown) at time .
is the known impulse response of a linear time-invariant system
is some unknown additive noise, independent of
is our observed signal
Our goal is to find some so that we can estimate as follows:
where is an estimate of that minimizes the mean square error
with denoting the expectation.
The Wiener deconvolution filter provides such a . The filter is most easily described in the frequency domain:
where:
and are the Fourier transforms of and ,
is the mean power spectral density of the original signal ,
is the mean power spectral density of the noise ,
, and are the Fourier transforms of , and , and , respectively,
the superscript denotes complex conjugation.
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 to obtain .
Note that in the case of images, the arguments and above become two-dimensional; however the result is the same.
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.
Official source
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