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 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:
y
(
t
)
=
h
(
t
)
∗
x
(
t
)
+
n
(
t
)
{\displaystyle \ y(t)=h(t)*x(t)+n(t)}
where
∗
{\displaystyle *}
convolution and:
x
(
t
)
{\displaystyle \ x(t)}
t
{\displaystyle \ t}
h
(
t
)
{\displaystyle \ h(t)}
impulse response of a linear time-invariant system
n
(
t
)
{\displaystyle \ n(t)}
independent of
x
(
t
)
{\displaystyle \ x(t)}
y
(
t
)
{\displaystyle \ y(t)}
Our goal is to find some
g
(
t
)
{\displaystyle \ g(t)}
x
(
t
)
{\displaystyle \ x(t)}
x
^
(
t
)
=
g
(
t
)
∗
y
(
t
)
{\displaystyle \ {\hat {x}}(t)=g(t)*y(t)}
where
x
^
(
t
)
{\displaystyle \ {\hat {x}}(t)}
x
(
t
)
{\displaystyle \ x(t)}
mean square error .
The Wiener deconvolution filter provides such a
g
(
t
)
{\displaystyle \ g(t)}
frequency domain :
G
(
f
)
=
H
∗
(
f
)
S
(
f
)
|
H
(
f
)
|
2
S
(
f
)
+
N
(
f
)
{\displaystyle \ G(f)={\frac {H^{*}(f)S(f)}{|H(f)|^{2}S(f)+N(f)}}}
where:
G
(
f
)
{\displaystyle \ G(f)}
H
(
f
)
{\displaystyle \ H(f)}
Fourier transforms of
g
{\displaystyle \ g}
h
{\displaystyle \ h}
f
{\displaystyle \ f}
S
(
f
)
{\displaystyle \ S(f)}
power spectral density of the input signal
x
(
t
)
{\displaystyle \ x(t)}
N
(
f
)
{\displaystyle \ N(f)}
n
(
t
)
{\displaystyle \ n(t)}
the superscript
∗
{\displaystyle {}^{*}}
complex conjugation . The filtering operation may either be carried out in the time-domain, as above, or in the frequency domain:
X
^
(
f
)
=
G
(
f
)
Y
(
f
)
{\displaystyle \ {\hat {X}}(f)=G(f)Y(f)}
(where
X
^
(
f
)
{\displaystyle \ {\hat {X}}(f)}
x
^
(
t
)
{\displaystyle {\hat {x}}(t)}
inverse Fourier transform on
X
^
(
f
)
{\displaystyle \ {\hat {X}}(f)}
x
^
(
t
)
{\displaystyle \ {\hat {x}}(t)}
Note that in the case of images, the arguments
t
{\displaystyle \ t}
f
{\displaystyle \ f}
Interpretation
The operation of the Wiener filter becomes apparent when the filter equation above is rewritten:
G
(
f
)
=
1
H
(
f
)
[
|
H
(
f
)
|
2
|
H
(
f
)
|
2
+
N
(
f
)
S
(
f
)
]
=
1
H
(
f
)
[
|
H
(
f
)
|
2
|
H
(
f
)
|
2
+
1
S
N
R
(
f
)
]
{\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,
1
/
H
(
f
)
{\displaystyle \ 1/H(f)}
S
N
R
(
f
)
=
S
(
f
)
/
N
(
f
)
{\displaystyle \ \mathrm {SNR} (f)=S(f)/N(f)}
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:
ϵ
(
f
)
=
E
|
X
(
f
)
−
X
^
(
f
)
|
2
{\displaystyle \ \epsilon (f)=\mathbb {E} \left|X(f)-{\hat {X}}(f)\right|^{2}}
where
E
{\displaystyle \ \mathbb {E} }
expectation .
If we substitute in the expression for
X
^
(
f
)
{\displaystyle \ {\hat {X}}(f)}
ϵ
(
f
)
=
E
|
X
(
f
)
−
G
(
f
)
Y
(
f
)
|
2
=
E
|
X
(
f
)
−
G
(
f
)
[
H
(
f
)
X
(
f
)
+
V
(
f
)
]
|
2
=
E
|
[
1
−
G
(
f
)
H
(
f
)
]
X
(
f
)
−
G
(
f
)
V
(
f
)
|
2
{\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}}}
If we expand the quadratic, we get the following:
ϵ
(
f
)
=
[
1
−
G
(
f
)
H
(
f
)
]
[
1
−
G
(
f
)
H
(
f
)
]
∗
E
|
X
(
f
)
|
2
−
[
1
−
G
(
f
)
H
(
f
)
]
G
∗
(
f
)
E
{
X
(
f
)
V
∗
(
f
)
}
−
G
(
f
)
[
1
−
G
(
f
)
H
(
f
)
]
∗
E
{
V
(
f
)
X
∗
(
f
)
}
+
G
(
f
)
G
∗
(
f
)
E
|
V
(
f
)
|
2
{\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, we are assuming that the noise is independent of the signal, therefore:
E
{
X
(
f
)
V
∗
(
f
)
}
=
E
{
V
(
f
)
X
∗
(
f
)
}
=
0
{\displaystyle \ \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
)
=
E
|
X
(
f
)
|
2
{\displaystyle \ S(f)=\mathbb {E} |X(f)|^{2}}
N
(
f
)
=
E
|
V
(
f
)
|
2
{\displaystyle \ N(f)=\mathbb {E} |V(f)|^{2}}
Therefore, we have:
ϵ
(
f
)
=
[
1
−
G
(
f
)
H
(
f
)
]
[
1
−
G
(
f
)
H
(
f
)
]
∗
S
(
f
)
+
G
(
f
)
G
∗
(
f
)
N
(
f
)
{\displaystyle \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
)
{\displaystyle \ G(f)}
d
ϵ
(
f
)
d
G
(
f
)
=
G
∗
(
f
)
N
(
f
)
−
H
(
f
)
[
1
−
G
(
f
)
H
(
f
)
]
∗
S
(
f
)
=
0
{\displaystyle \ {\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
References
Rafael Gonzalez, Richard Woods, and Steven Eddins. Digital Image Processing Using Matlab . Prentice Hall, 2003.
External links
.png)
![{\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/enwiki/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/enwiki/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/enwiki/api/rest_v1/media/math/render/svg/a5a5844b72fb422fde5780ecab57903d46a9f747)
![{\displaystyle \epsilon (f)={\Big [}1-G(f)H(f){\Big ]}{\Big [}1-G(f)H(f){\Big ]}*S(f)+G(f)G^{*}(f)N(f)}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/b18ba324bec4c8443ef8b14bb8dc69a59ba1fc42)
![{\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/enwiki/api/rest_v1/media/math/render/svg/75c6b4182c7957e6e52dcbf4194cbe7373558265)
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