Image Processing Reference
In-Depth Information
3.2 Inverse filtering
Inverse filtering
,or
deconvolution
is another
fundamental
issue in signal processing.
This problem arises
for
example
in direct-filter design in spline
interpolation
(Nagahara & Yamamoto, 2011).
Suppose a filter
P
)
−
1
. However, real
(
z
)
is given. Symbolically, the inverse filter of
P
(
z
)
is
P
(
z
design is not that easy.
Example 2.
Suppose P
(
z
)
is given by
z
+
0.5
P
(
z
)=
0.5
.
−
z
)
−
1
becomes
(
)
=
(
Then, the inverse Q
z
:
P
z
z
−
0.5
)
−
1
(
)=
(
=
Q
z
P
z
0.5
,
+
z
which is stable and causal. Then suppose
z
−
2
P
(
z
)=
0.5
,
z
−
then the inverse is
z
−
0.5
)
−
1
Q
(
z
)=
P
(
z
=
.
z
−
2
This has the pole at
|
z
| >
1
, and hence the inverse filter is unstable. On the other hand, suppose
1
(
)=
P
z
0.5
,
−
z
then the inverse is
)
−
1
(
)=
(
=
−
Q
z
P
z
z
0.5,
which is noncausal.
By these examples, the inverse filter
P
)
−
1
may unstable or noncausal. Unstable or noncausal
filters are difficult to implement in real digital device, and hence we adopt approximation
technique; we design an FIR digital filter
Q
(
z
1. Since FIR filters are
always stable and causal, this is a realistic way to design an inverse filter. Our problem is now
formulated as follows:
(
z
)
such that
Q
(
z
)
P
(
z
)
≈
Problem 2
(Inverse filtering)
.
Given a filter P
(
z
)
which is necessarily not bi-stable or bi-causal (i.e.,
)
−
1
can be unstable or noncausal), find an FIR filter Q
P
(
z
(
z
)
which minimizes
Q
1
W
e
j
ω
)
e
j
ω
)
−
e
j
ω
)
(
QP
−
1
)
W
∞
=
max
ω∈
[
(
P
(
(
,
0,
π
]
where W is a given stable weighting function.
The procedure to solve this problem is shown in Section 4.
