- Wiener deconvolution
In
mathematics , Wiener deconvolution is an application of theWiener filter to thenoise problems inherent indeconvolution . It works in thefrequency domain , attempting to minimize the impact of deconvoluted noise at frequencies which have a poorsignal-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 denotes
convolution . and:* is some input signal (unknown) at time .
* is the knownimpulse response of a linear time-invariant system
* is some unknown additive noise, independent of
* is our observed signalOur goal is to find some so that we can estimate as follows:
:
where is an estimate of that minimises the
mean square error .The Wiener deconvolution filter provides such a . The filter is most easily described in the
frequency domain ::
where:
* and are the
Fourier transform s of and , respectively at frequency .
* is the meanpower spectral density of the input signal
* is the mean power spectral density of the noise
* the superscript denotes complex conjugation.The filtering operation may either be carried out in the time-domain, as above, or in the frequency domain:
:
(where is the Fourier transform of ) 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.
Interpretation
The operation of the Wiener filter becomes apparent when the filter equation above is rewritten:
:
Here, is the inverse of the original system, and is the
signal-to-noise ratio . 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 dependent on their 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, and 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 minimises the mean square error, which may be expressed:
:
where denotes expectation.
If we substitute in the expression for , the following rearrangements can be made:
:
If we expand the quadratic, we get the following:
:
However, we are assuming that the noise is independent of the signal, therefore:
:
Also, we are defining the power spectral densities as follows:
::
Therefore, we have:
:
To find the minimum error value, we differentiate with respect to and set equal to zero. As this is a complex value, acts as a constant.
:
This final equality can be rearranged to give the Wiener filter.
See also
*
Deconvolution
*Wiener filter
*Point spread function
*Blind deconvolution
*Fourier transform References
* Rafael Gonzalez, Richard Woods, and Steven Eddins. "Digital Image Processing Using Matlab". Prentice Hall, 2003.
External links
* [http://www.owlnet.rice.edu/~elec539/Projects99/BACH/proj2/blind/bd.html Comparison of different deconvolution methods.]
* [http://cnx.org/content/m13144/latest/ Deconvolution with a Wiener filter]
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