Image Processing Reference
In-Depth Information
g
(
n
,
m
)
+
Restoration
filter
Σ
f
ˆ
(
n
,
m
)
f
(
n
,
m
)
h
(
n
,
m
)
+
N
(
n
,
m
)
Degradation
model
FIGURE 2.49
Image degradation and restoration
filter.
The observed image g
(
n, m
)
is given by
g
(
n, m
) ¼
f
(
n, m
)
*
h
(
n, m
) þ
N
(
n, m
)
(
2
:
135
)
where f
(
n, m
)
is the original image, g
(
n, m
)
is the degraded noisy image, N
(
n, m
)
is
zero mean stationary additive noise process, h
(
n, m
)
is a linear
filter representing the
blur and f
(
n, m
)
filter is designed by minim-
izing a metric that is a measure of closeness of f to f according to some criterion.
Here, we discuss an approach based on minimizing the mean-square error between
the original image and the restored image. This is known as MMSE
is the restored image. The restoration
filter or Wiener
Filter.
2.9.1 W
IENER
F
ILTER
R
ESTORATION
The Wiener
filter is designed by minimizing the following objective function
2
J
¼
E
(j
e
(
n, m
)j
)
(
:
)
2
136
where e
(
n, m
) ¼
f
(
n, m
)
f
(
n, m
)
is the estimation error and E is the expected value
operator. The solution to the above minimization problem can be obtained using
calculus of variation and is given by
(v
x
,
v
y
)
H*
W
(v
x
,
v
y
) ¼
(
2
:
137
)
2
S
N
(v
x
,
v
y
)
S
f
(v
x
,
v
y
)
H
(v
x
,
v
y
)
þ
In Equation 2.137, W
(v
x
,
v
y
)
is the frequency response of the Wiener
filter (restor-
ation
filter), H
(v
x
,
v
y
)
is the blur
(degradation kernel)
frequency response,
S
N
(v
x
,
v
y
)
is the noise power spectrum and Sf
f
(v
x
,
v
y
)
is the power spectrum of the
original image. If there is no additive noise (S
N
(v
x
,
v
y
) ¼
0), then the Wiener
filter.
becomes
H*
(v
x
,
v
y
)
1
H
(v
x
,
W
(v
x
,
v
y
) ¼
2
¼
(
2
:
138
)
v
y
)
H
(v
x
,
v
y
)
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