From left: original image, out of focus image, image sharpened by means of Wiener unfolding
In mathematics , the Wiener deconvolution an application of the Wiener filter for noise problems in the development . They tried the influence of noise in the in the unfolding frequency domain to minimize and is therefore mostly in low signal-to-noise ratios applied.
Wiener deconvolution is widespread in deconvolution applications in the photo field, since the frequency spectrum of images in the visible region is comparatively easy to determine.
The Viennese unfolding is named after Norbert Wiener .
definition Be
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{\ displaystyle y (t) = h (t) * x (t) + n (t)}
where the fold denotes and
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impulse response of a linear time-invariant system
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regardless of is
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The goal is to determine so that :
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{\ displaystyle {\ hat {x}} (t) = g (t) * y (t)}
where is an estimate of with minimized squared error .
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The Wiener filter provides such a solution . The easiest way to describe it is in the frequency domain :
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{\ displaystyle G (f) = {\ frac {H ^ {*} (f) S (f)} {| H (f) | ^ {2} S (f) + N (f)}}}
in which
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Fourier transform of and at frequency .
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spectral power density of the input signal is
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complex conjugate of .
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The filter operation can, as above, be carried out in the time domain or in the frequency domain:
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{\ displaystyle {\ hat {X}} (f) = G (f) Y (f)}
(where is the Fourier transform of ). An inverse Fourier transform of gives .
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It should be noted that in the case of images, the arguments and become two-dimensional; But the result remains the same.
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interpretation The application of the Wiener filter can be seen when the above equation is rewritten:
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{\ displaystyle {\ begin {aligned} 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 {aligned}}}
Here is the inverse of the output system and is the signal-to-noise ratio. With no noise (i.e. infinite signal-to-noise ratio), the term inside the square brackets equals 1, which means that the Wiener filter is simply the inverse of the system, as one might expect. If the noise increases at certain frequencies, i.e. the signal-to-noise ratio falls, the term inside the square brackets also decreases. This means that the Wiener filter attenuates the frequencies depending on their signal-to-noise ratio.
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{\ displaystyle \ \ mathrm {SNR} (f) = S (f) / N (f)}
The above equation assumes that the spectral content of a typical image as well as that of the noise is known. Most of the time, the two sizes are not known, but can be estimated. For example, in photos, the signal (the original image) typically has strong parts of low frequencies and weak parts of high frequencies and the noise parts are evenly distributed over all frequencies.
Derivation As described above, an approximation of the original image is to be generated that minimizes the square error. This can be through
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{\ displaystyle \ \ epsilon (f) = \ mathbb {E} \ left | X (f) - {\ hat {X}} (f) \ right | ^ {2}}
express, where is the expectation operator .
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If replaced, the expression can be rewritten:
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{\ displaystyle {\ begin {aligned} \ 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 {aligned}}}
The square can be expanded and gives:
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{\ displaystyle {\ begin {aligned} \ 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 {aligned}}}
However, it is assumed that the noise is independent of the signal, so:
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{\ displaystyle \ \ mathbb {E} {\ Big \ {} X (f) V ^ {*} (f) {\ Big \}} = \ mathbb {E} {\ Big \ {} V (f) X ^ {*} (f) {\ Big \}} = 0}
The power spectral density is defined as:
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{\ displaystyle \ S (f) = \ mathbb {E} | X (f) | ^ {2}}
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{\ displaystyle \ N (f) = \ mathbb {E} | V (f) | ^ {2}}
This results in:
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{\ displaystyle {\ begin {aligned} \ 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 {aligned}}}
To find the minimum error, one is differentiated and set equal to zero. Since this gives a complex value, it is a constant.
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{\ displaystyle \ {\ frac {d \ epsilon (f)} {dG (f)}} = G ^ {*} (f) N (f) -H (f) {\ Big [} 1-G (f ) H (f) {\ Big]} ^ {*} S (f) = 0}
This equation can be rewritten to obtain the Wiener filter.
credentials Rafael Gonzalez, Richard Woods, and Steven Eddins: Digital Image Processing Using Matlab . Prentice Hall, 2003. Web links
<img src="https://de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">
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![{\ displaystyle {\ begin {aligned} 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 {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/860ae1c471a0f31e946b550665d36f63c822f029)
![{\ displaystyle {\ begin {aligned} \ 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 {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca5382edaacbb0b8f5174ced6218bda106dd1a62)
![{\ displaystyle {\ begin {aligned} \ 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 {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5a5844b72fb422fde5780ecab57903d46a9f747)
![{\ displaystyle {\ begin {aligned} \ 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 {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51f056432b7a258e78aeb64da09402fde5354c9a)
![{\ displaystyle \ {\ frac {d \ epsilon (f)} {dG (f)}} = G ^ {*} (f) N (f) -H (f) {\ Big [} 1-G (f ) H (f) {\ Big]} ^ {*} S (f) = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75c6b4182c7957e6e52dcbf4194cbe7373558265)