World Library  
Flag as Inappropriate
Email this Article

Wiener deconvolution

Article Id: WHEBN0008103917
Reproduction Date:

Title: Wiener deconvolution  
Author: World Heritage Encyclopedia
Language: English
Subject: Wiener filter, Norbert Wiener, Signal processing, List of statistics articles
Collection: Estimation Theory, Image Noise Reduction Techniques, Signal Processing
Publisher: World Heritage Encyclopedia

Wiener deconvolution

From left: Original image, blurred image, image deblurred using 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 1
  • Interpretation 2
  • Derivation 3
  • See also 4
  • References 5
  • External links 6


Given a system:

\ y(t) = h(t)*x(t) + n(t)

where * denotes convolution and:

Our goal is to find some \ g(t) so that we can estimate \ x(t) as follows:

\ \hat{x}(t) = g(t)*y(t)

where \ \hat{x}(t) is an estimate of \ x(t) that minimizes the mean square error.

The Wiener deconvolution filter provides such a \ g(t). The filter is most easily described in the frequency domain:

\ G(f) = \frac{H^*(f)S(f)}{ |H(f)|^2 S(f) + N(f) }


The filtering operation may either be carried out in the time-domain, as above, or in the frequency domain:

\ \hat{X}(f) = G(f)Y(f)

(where \ \hat{X}(f) is the Fourier transform of \hat{x}(t)) and then performing an inverse Fourier transform on \ \hat{X}(f) to obtain \ \hat{x}(t).

Note that in the case of images, the arguments \ t and \ f 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:

\begin{align} G(f) & = \frac{1}{H(f)} \left[ \frac{ |H(f)|^2 }{ |H(f)|^2 + \frac{N(f)}{S(f)} } \right] \\ & = \frac{1}{H(f)} \left[ \frac{ |H(f)|^2 }{ |H(f)|^2 + \frac{1}{\mathrm{SNR}(f)}} \right] \end{align}

Here, \ 1/H(f) is the inverse of the original system, and \ \mathrm{SNR}(f) = S(f)/N(f) 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.


As mentioned above, we want to produce an estimate of the original signal that minimizes the mean square error, which may be expressed:

\ \epsilon(f) = \mathbb{E} \left| X(f) - \hat{X}(f) \right|^2

where \ \mathbb{E} denotes expectation.

If we substitute in the expression for \ \hat{X}(f), the above can be rearranged to

\begin{align} \epsilon(f) & = \mathbb{E} \left| X(f) - G(f)Y(f) \right|^2 \\ & = \mathbb{E} \left| X(f) - G(f) \left[ H(f)X(f) + V(f) \right] \right|^2 \\ & = \mathbb{E} \big| \left[ 1 - G(f)H(f) \right] X(f) - G(f)V(f) \big|^2 \end{align}

If we expand the quadratic, we get the following:

\begin{align} \epsilon(f) & = \Big[ 1-G(f)H(f) \Big] \Big[ 1-G(f)H(f) \Big]^*\, \mathbb{E}|X(f)|^2 \\ & {} - \Big[ 1-G(f)H(f) \Big] G^*(f)\, \mathbb{E}\Big\{X(f)V^*(f)\Big\} \\ & {} - G(f) \Big[ 1-G(f)H(f) \Big]^*\, \mathbb{E}\Big\{V(f)X^*(f)\Big\} \\ & {} + G(f) G^*(f)\, \mathbb{E}|V(f)|^2 \end{align}

However, we are assuming that the noise is independent of the signal, therefore:

\ \mathbb{E}\Big\{X(f)V^*(f)\Big\} = \mathbb{E}\Big\{V(f)X^*(f)\Big\} = 0

Also, we are defining the power spectral densities as follows:

\ S(f) = \mathbb{E}|X(f)|^2
\ N(f) = \mathbb{E}|V(f)|^2

Therefore, we have:

\epsilon(f) = \Big[ 1-G(f)H(f) \Big]\Big[ 1-G(f)H(f) \Big]^ * S(f) + G(f)G^*(f)N(f)

To find the minimum error value, we calculate the Wirtinger derivative with respect to \ G(f) and set it equal to zero.

\ \frac{d\epsilon(f)}{dG(f)} = G^*(f)N(f) - H(f)\Big[1 - G(f)H(f)\Big]^* S(f) = 0

This final equality can be rearranged to give the Wiener filter.

See also


  • 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
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.