In
mathematics, Wiener deconvolution is an application of the
Wiener filter
In signal processing, the Wiener filter is a filter used to produce an estimate of a desired or target random process by linear time-invariant ( LTI) filtering of an observed noisy process, assuming known stationary signal and noise spectra, and ...
to the
noise problems inherent in
deconvolution
In mathematics, deconvolution is the operation inverse to convolution. Both operations are used in signal processing and image processing. For example, it may be possible to recover the original signal after a filter (convolution) by using a deco ...
. It works in the
frequency domain, attempting to minimize the impact of deconvolved noise at frequencies which have a poor
signal-to-noise ratio
Signal-to-noise ratio (SNR or S/N) is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. SNR is defined as the ratio of signal power to the noise power, often expressed in de ...
.
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
Norbert Wiener (November 26, 1894 – March 18, 1964) was an American mathematician and philosopher. He was a professor of mathematics at the Massachusetts Institute of Technology (MIT). A child prodigy, Wiener later became an early researcher i ...
.
Definition
Given a system:
:
where
denotes
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
and:
*
is some original signal (unknown) at time
.
*
is the known
impulse response
In signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an impulse (). More generally, an impulse response is the react ...
of a
linear time-invariant
In system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance; these terms are briefly define ...
system
*
is some unknown additive noise,
independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independ ...
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
In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between ...
:
,
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 transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
s 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 In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively it may be viewed as the statement that if we know all frequency and phase information a ...
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,
is the
signal-to-noise ratio
Signal-to-noise ratio (SNR or S/N) is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. SNR is defined as the ratio of signal power to the noise power, often expressed in de ...
, 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:
:
.
The equivalence to the previous definition of
, can be derived using
Plancherel theorem
In mathematics, the Plancherel theorem (sometimes called the Parseval–Plancherel identity) is a result in harmonic analysis, proven by Michel Plancherel in 1910. It states that the integral of a function's squared modulus is equal to the integ ...
or
Parseval's theorem
In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. It originate ...
for the
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
.
If we substitute in the expression for
, 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
and
, we have:
:
To find the minimum error value, we calculate the
Wirtinger derivative
In complex analysis of one and several complex variables, Wirtinger derivatives (sometimes also called Wirtinger operators), named after Wilhelm Wirtinger who introduced them in 1927 in the course of his studies on the theory of functions of se ...
with respect to
and set it equal to zero.
:
This final equality can be rearranged to give the Wiener filter.
See also
*
Information field theory Information field theory (IFT) is a Bayesian statistical field theory relating to signal reconstruction, cosmography, and other related areas. IFT summarizes the information available on a physical field using Bayesian probabilities. It uses comp ...
*
Deconvolution
In mathematics, deconvolution is the operation inverse to convolution. Both operations are used in signal processing and image processing. For example, it may be possible to recover the original signal after a filter (convolution) by using a deco ...
*
Wiener filter
In signal processing, the Wiener filter is a filter used to produce an estimate of a desired or target random process by linear time-invariant ( LTI) filtering of an observed noisy process, assuming known stationary signal and noise spectra, and ...
*
Point spread function
*
Blind deconvolution
*
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
{{Commons, File:Image restoration (motion blur, Wiener filtering).png, An example of Wiener deconvolution on motion blured image (and source codes in MATLAB/GNU Octave).
References
* Rafael Gonzalez, Richard Woods, and Steven Eddins. ''Digital Image Processing Using Matlab''. Prentice Hall, 2003.
External links
Comparison of different deconvolution methods.Deconvolution with a Wiener filter
Signal estimation
Image noise reduction techniques