|
|
| Line 1: |
Line 1: |
| [[File:Image restoration (motion blur, Wiener filtering).png|thumb|350px|From left: Original image, blurred image, image deblurred using Wiener deconvolution.]]
| | Hi there, I am Alyson Boon even though it is not the title on my beginning certificate. He functions as a bookkeeper. Her family life in Ohio but her spouse desires them to move. The preferred hobby for him and his kids is fashion and he'll be beginning some thing else alongside with it.<br><br>Feel free to surf to my weblog: authentic psychic readings ([http://www.chk.woobi.co.kr/xe/?document_srl=346069 www.chk.woobi.co.kr]) |
| 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 deconvoluted 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 ==
| |
| | |
| Given a system:
| |
| | |
| :<math>\ y(t) = h(t)*x(t) + n(t)</math>
| |
| | |
| where <math>*</math> denotes [[convolution]] and:
| |
| | |
| *<math>\ x(t)</math> is some input signal (unknown) at time <math>\ t </math>.
| |
| *<math>\ h(t)</math> is the known [[impulse response]] of a [[LTI system theory|linear time-invariant]] system
| |
| *<math>\ n(t)</math> is some unknown additive noise, [[statistical independence|independent]] of <math>\ x(t)</math>
| |
| *<math>\ y(t)</math> is our observed signal
| |
| | |
| Our goal is to find some <math>\ g(t)</math> so that we can estimate <math>\ x(t)</math> as follows:
| |
| | |
| :<math>\ \hat{x}(t) = g(t)*y(t)</math>
| |
| | |
| where <math>\ \hat{x}(t)</math> is an estimate of <math>\ x(t)</math> that minimizes the [[mean square error]].
| |
| | |
| The Wiener deconvolution filter provides such a <math>\ g(t)</math>. The filter is most easily described in the [[frequency domain]]:
| |
| | |
| :<math>\ G(f) = \frac{H^*(f)S(f)}{ |H(f)|^2 S(f) + N(f) }</math>
| |
| | |
| where:
| |
| | |
| * <math>\ G(f)</math> and <math>\ H(f)</math> are the [[Fourier transform]]s of <math>\ g</math> and <math>\ h</math>, respectively at frequency <math>\ f </math>.
| |
| * <math>\ S(f)</math> is the mean [[power spectral density]] of the input signal <math>\ x(t)</math>
| |
| * <math>\ N(f)</math> is the mean power spectral density of the noise <math>\ n(t)</math>
| |
| * the superscript <math>{}^*</math> denotes [[complex conjugate|complex conjugation]].
| |
| | |
| The filtering operation may either be carried out in the time-domain, as above, or in the frequency domain:
| |
| | |
| :<math>\ \hat{X}(f) = G(f)Y(f)</math>
| |
| | |
| (where <math>\ \hat{X}(f)</math> is the Fourier transform of <math>\hat{x}(t)</math>) and then performing an [[inverse Fourier transform]] on <math>\ \hat{X}(f)</math> to obtain <math>\ \hat{x}(t)</math>.
| |
| | |
| Note that in the case of images, the arguments <math>\ t </math> and <math>\ f </math> 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: | |
| | |
| :<math>
| |
| \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}
| |
| </math>
| |
| | |
| Here, <math>\ 1/H(f)</math> is the inverse of the original system, and <math>\ \mathrm{SNR}(f) = S(f)/N(f)</math> 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 minimizes the mean square error, which may be expressed:
| |
| | |
| :<math>\ \epsilon(f) = \mathbb{E} \left| X(f) - \hat{X}(f) \right|^2</math>
| |
| | |
| where <math>\ \mathbb{E}</math> denotes [[Expected value|expectation]].
| |
| | |
| If we substitute in the expression for <math>\ \hat{X}(f)</math>, the above can be rearranged to
| |
| | |
| :<math> | |
| \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}
| |
| </math>
| |
| | |
| If we expand the quadratic, we get the following:
| |
| | |
| :<math>
| |
| \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}
| |
| </math>
| |
| | |
| However, we are assuming that the noise is independent of the signal, therefore:
| |
| | |
| :<math>\ \mathbb{E}\Big\{X(f)V^*(f)\Big\} = \mathbb{E}\Big\{V(f)X^*(f)\Big\} = 0</math>
| |
| | |
| Also, we are defining the power spectral densities as follows:
| |
| | |
| :<math>\ S(f) = \mathbb{E}|X(f)|^2</math>
| |
| :<math>\ N(f) = \mathbb{E}|V(f)|^2</math>
| |
| | |
| Therefore, we have:
| |
| | |
| :<math>
| |
| \begin{align}
| |
| \epsilon(f) & = \Big[ 1-G(f)H(f) \Big]\Big[ 1-G(f)H(f) \Big]^* S(f) \\
| |
| & {} + G(f)G^*(f)N(f)
| |
| \end{align}
| |
| </math>
| |
| | |
| To find the minimum error value, we calculate the [[Wirtinger derivatives|Wirtinger derivative]] with respect to <math>\ G(f)</math> and set it equal to zero.
| |
| | |
| :<math>\
| |
| \frac{d\epsilon(f)}{dG(f)} = G^*(f)N(f) - H(f)\Big[1 - G(f)H(f)\Big]^* S(f) = 0
| |
| </math>
| |
| | |
| This final equality can be rearranged to give the Wiener filter.
| |
| | |
| == See also ==
| |
| | |
| * [[Deconvolution]]
| |
| * [[Wiener filter]]
| |
| * [[Point spread function]]
| |
| * [[Blind deconvolution]]
| |
| * [[Fourier transform]]
| |
| {{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 ==
| |
| * [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]
| |
| | |
| [[Category:Signal processing]]
| |
| [[Category:Image noise reduction techniques]]
| |
| [[Category:Estimation theory]]
| |
Hi there, I am Alyson Boon even though it is not the title on my beginning certificate. He functions as a bookkeeper. Her family life in Ohio but her spouse desires them to move. The preferred hobby for him and his kids is fashion and he'll be beginning some thing else alongside with it.
Feel free to surf to my weblog: authentic psychic readings (www.chk.woobi.co.kr)